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IS A partition of unity IFF, TO BOO T, FOR EACH '\: a)
° and
. , f>. t f · net A 3 {
L
1.2.44 THEOREM. IF K IS A COMPAC T S UBSE T OF A LOCALLY COMPAC T HAUSDORFF SPACE X AND U clef {U),} ),EA IS AN OPEN COVER OF K, FOR K REGARDED AS A TOPOLOGICAL SPACE, THERE IS A PAR TI TION OF UNI TY
For each '\, K), cle =f
g)' g)'
U
N(x,JcU),
N (Xi) is a compact subset of U), and 1 .2.41
g),
implies there is a such that K), < < U),. Since there are only finitely many pairwise different K)" there are only finitely many At each x . K, S x cle=f ", } cle cle f f {
=
{g),}
=
1.2.46 THEOREM. (Dini) HYPO THESIS: X IS COMPAC T; IS A MAP n : A x X ('\, x) H n('\, x) E ffi. SUCH THA T:
3
g)'.
A
IS A DISE T; n
Section 1.2. Topology and Continuity
17
a) FOR EACH '\, n('\, X) E C(X, JR.) ; b) FOR EACH x, n('\ , x) IS A NET, {A < IL}:::;" {n('\, x) n('\, x) CONVERGES TO ZERO. CONCLUSION: IF E > 0, FOR SOME '\( E) , IF n('\, x) < E.
,\ >
�
n(lL, X) } , AND
'\( E) AND x E X, THEN
PROOF. If the conclusion is false, for some positive E and each
,\
in
A,
K>. �f { x : n('\, x) � E } is closed and nonempty. The hypothesis implies that { K>.hEA enjoys the fip, whence �f n K>. j. 0. For x in and each >'EA ,\
S
in A, n('\, x)
�
S
E, a contradiction.
[ 1.2.47 Note. Paraphrased, Dini's Theorem says that a diset of nonnegative continuous functions converging monotonely to zero on a compact space converges uniformly to zero.]
0
1.2.48 Example. If X is compact, {fn } nEN C C(X, JR.) , and fn + 0, fn �O.
A S A, properties such as openness, closedness, connectedness, etc., may differ for S according as S is viewed as a subset of A or as a subset
[ 1.2.49 Note. For a topological space (X, T), a subset of X in its induced topology T A, and a subset of topological
of X.
S
S
Thus, if X = JR., A = (0, 00) and = (0, 1], then is a relatively closed subset of and is not a closed subset of X. Similarly if A = [0, (0 ) and = [0, 1), is a relatively open subset of A and is not an open subset of X .]
A S
1.2.50 Exercise.
S
Compactness is an absolute topological property: if (X, T) is a topological space, C X, and S is T Acompact iff S is Tcompact. [ 1.2.51 Remark. Owing to 1.2.49, the following locution entrenched in the language of mathematics. The subset of A (contained in X) is relatively compact iff S n A is compact.]
A
S e A,
IS
S
1.2.52 Exercise. There exist relatively compact sets that are not com pact.
18
Chapter 1. Fundamentals
[Hint: The set S �f (0.2, 0.3) is a relatively compact subset of the subset A � (0.1, 0.4) of ffi..] 1.3. Baire C ategory Arguments
1.3.1 DEFINI TION. A SUBSE T S OF A TOPOLOGICAL SPACE X IS: a) nowhere dense IFF EACH NEIGHBORHOOD CON TAINS A NEIGHBORHOOD THAT DOES NOT MEET S; b) OF THE first (Baire) category IFF S IS THE UNION OF COUNTABLY MANY NOWHERE DENSE SETS; c) OF THE second (Baire) category IF S IS NOT OF THE FIRST CATEGORY. 1.3.2 Exercise. A union of finitely many nowhere dense sets is nowhere dense. 1.3.3 Exercise. A subset S of a topological space X is nowhere dense iff X \ S = X. 1.3.4 THEOREM. A COMPLETE METRIC SPACE (X, d) IS OF THE SECOND CATEGORY. PROOF. If X = U An and each A n is nowhere dense, for any N (x) , by nEN induction there are sequences { xn }nEN and {N (xn ) }nEN such that: a) { N (Xn ) } nEN C N(x); b) for each n,
sup { d(y, z) :
lim Xn ex{Xn } nEN is a Cauchy sequence. Since X is complete, �f n>(X) ists. On the one hand, for some no , E An o and on the other hand, w E N (xn o+ d e N (xn o) , a contradiction since N (xn o) n An o = 0. 0 W
W
1.3.5 Example. a) In ffi., Q is a set of the first category and IT �f ffi. \ Q is of the second category. b) For an enumeration {qn }nEN of Q, if 2 � m E N, is the complement Nm in ffi. of the set Sm �f U (qn  m  n , qn + m
nowhere dense. Thus A �f
nEN
n)
U Nm is of the first category and 1.3.4 implies
mEN
Section 1.3. Baire Category Arguments
B �f ffi. \
19
A n Sm is of the second category. If E > 0, B is contained in =
m EN
a countable union of open intervals and the sum of their lengths is less than
E.
1.3.6 THEOREM. ( Baire's Category Theorem ) IF (X, d) IS A COMPLETE METRIC SPACE AND { Un } nEN IS A SEQUENCE OF DENSE OPEN SUBSETS OF X, THEN U �f n Un IS DENSE IN X.
nEN
PROOF. If each Un is replaced by Vn �f U1 n . . . n Un, each Vn is open and dense, n Vn = n Un �f U, and, to boot, Vn ::) Vn +1 • Thus it may be
nEN
nEN
assumed at the outset that Un ::) Un + 1 • Since each Un is dense, if V is a nonempty open subset of X, induction leads to a sequence {xn }nEN in X and a sequence { En } nEN such that: a ) 0 < E l < B (X l , E I ) C UI n v; b ) if n > l , 0 < En < 2  n , and B (Xn , En ) C B (xn  l , En  I ) ° n Un . Then { xn } nEN is a Cauchy sequence and, since X is complete, there is an x such that n+ lim d (x, xn ) = o . Furthermore,
1,
=
nEN
nEN
whence x E V n U: U meets every nonempty open set V.
o
1.3.7 THEOREM. IF (X, d) AND (Y, J) ARE METRIC SPACES, T E y X , AND T(X) = Y, T IS OPEN IFF FOR EACH Y IN Y, WHENEVER A SEQUENCE {Yn }nEN IN Y CONVERGES TO Y AND Y = T(x) , FOR SOME SEQUENCE {Xn}nEN CONVERGING TO x, T (Xn ) = Yn , n EN. PROOF. If T is open, Yn + Y as n + 00 , T(x) = Y, and
{
1}
Nk(X) cle=f � : d(�, x) < k ' then T [Nk (X)] �f Uk is open and Y E Uk. For some least nk , if n 2': nk , then Yn E rh. By definition, Nk (X) contains a �n such that T (�n ) Yn . Since Uk ::) Uk +l , nk :::; nk +l . If nk < n < nk+ l , then Yn E Uk. Hence T 1 (Yn ) n Nk (x) j. 0. Thus the formula =
Xn
=
{
an element of T 1 (Yn )
1
if :::; n < n l if n = nk an element of T 1 (Yn ) n Nk (X ) if nk < n < nk +l 
�n

defines a sequence { xn }nEN such that T (xn ) = Yn, n EN. Furthermore if k EN and n 2': nk, then Xn E Nk (x), whence Xn + x as n + 00.
Chapter 1. Fundamentals
20
If T is not open and Y E Y, for some x in X, Y �f T(x), while for some N(x) and each n in N, T[N(x)]1; Nn (y ) �f 1] : 15(1], y) < i.e., in Nn (y) there is a Yn not in T[N(x)]. Hence Yn + Y as n + 00, but if T (xn ) = Yn , then Xn tJ N(x), i.e., some sequence {Yn } nEN converges to Y and no sequence {xn } nEN such that T (xn ) = Yn , n EN, converges to x. o
� },
{
1.4. Homotopy, Simplices, Fixed Points
The results in the current Section are widely applicable in many parts of mathematics. The main result is Brouwer's Fixed Point Theorem. The machinery that leads to a proof is important in and of itself. • •
•
Homotopy plays a central role in Section 5.5 where the relation of holomorphy and homotopy is explored. Simplices or rectangular versions of them occur in Section 4.6 in the treatment of differentiation, in Section 5.3 where the fundamental theorems and formul& of Cauchy are discussed, and in Section 7.1 to provide a direct approach to Runge's Theorem. In Section 4.7, Brouwer's Fixed Point Theorem appears to be essen tial in the derivation of the formula for change of variables in multi dimensional integration. More generally, Brouwer's Fixed Point The orem is central for many of the developments in general and algebraic topology, e.g., dimension theory, antipodal point theory, etc. [HW,
Kel, Sp, Thc].
1.4.1 DEFINITION. FOR TOPOLOGICAL SPACES X AND Y, A SUBSET A OF X, AND SOME F IN C(X X [0, 1] ' Y) THE MAPS h AND r5} IN C(X, Y) ARE F homotopic in A IFF FOR EACH (x, ) IN X X [0, 1] ' F(x, ) E AND s
{ x E X} '* {{F(x, O) = 'Y(x)} 1\ { F(x, 1) = r5(x) } } .
s
A
(1.4.2)
A
THE MAP F IS A homotopy in OF 'Y INTO 15; THE CIRCUMSTANCE DE SCRIBED IS SYMBOLIZED 'Y "' F, A 15, 'Y "' F 15, OR 'Y '" 15. 1.4.3 Exercise. If 'Y '" F, A 15 and 15 "'G, A 1], 'Y "'Go F, A 1], i.e., 'Y '" 15 sym bolizing that for some F, 'Y '" F, A 15, '" is an equivalence relation among the elements of C(X, Y) . The homotopy equivalence class of a curve 'Y is h}. 1.4.4 Example. If 'Y (t) then 'Y '" F,e 15.
=
8
8
3e2 7rit , 15 (t) = e27rit and F (t, ) �f (3 2 ) e2 7rit , 
Section 1.4. Homotopy, Simplices, Fixed Points
For x �f (
XO,"
21
xn ) in ffi.n + l , d as in 1.2. 10, and a positive r, the sets
"
Bn+ l (x,r) �f { y : d(x, y):Sr } (a closed (n + I)ball), Bn +l (x,rt �f { y : d(x, y)
In particular,Bn + 1 �f Bn +1 ( O, 1) is the n + Iball and is homeomorphic to every n + Iball. Similarly, sn �f sn (o, 1) is the nsphere and is home omorphic to every nsphere.
1.4.5 Exercise. a) The n + Icell, [0, It+ 1 is homeomorphic to Bn + l . b) The boundary 8[0, I]n + l is homeomorphic to sn . The object of the subsequent discussion is to prove that the identity map id : sn sn and any consta.nt map c : sn sn for which c (sn ) is a single vector in sn are not homotopic in sn . The technique involves the decomposition of sn into small pieces, the spherical nsimplices described below.
r+
r+
{
{xo , x l , . . . , xd OF VECTORS IN � AJ  1 , DE. k, � AJ > 0, ° :S J:S J=o IFF {Xo , . . . , Xk } IS linearly indepen
1.4.6 DEFINITION. FOR A k + ITUPLE
IDln+ 1 , m.
THE SET •
� AJXJ a
\
.
.
•
\ .
\
}
NOTED (Xo, ... , Xk) IS A ksimplex dent. OTHERWISE, a IS A degenerate ksimplex. THE closed ksimplex resp. closed degenerate ksimplex IS
an cle=f
_
{L k
J=O
AjXj
DENOTED [xo , . . . , X k ]. FOR A SUBSET { io , . . . , ip } OF { O , I, . . . , k }, THE CORRESPONDING p face OF a IS
A
OFACE IS A vertex; A IFACE IS AN edge.
22
Chapter 1. Fundamentals
[ 1.4.7 Note. Every face of an resp. an is a relatively open subset of an resp. an . Thus an edge is homeomorphic to (0, 1); a 2face is homeomorphic to B2 (0, l r , etc. ] 1.4.8 Exercise. If a in ffi.Tn is a ksimplex: a) m 2: k; b) a is an open subset of the hyperplane Ha �f Xo + span {X l  Xo , . . . , Xk  xo } in the topology induced on Ha by ffi.Tn ; c) each vector in a belongs to one and only one face of a; d) a is a closed subset of ffi.Tn and is homeomorphic to Bk . A (possibly degenerate) simplex is a convex set. When x j. 0, p(x) � Esn is the mdial projection of x onto sn .
1I�1 2 1.4.9 Exercise. If {xo , . . . , xn } Csn , d ( xi , xj) < 1 , ° � i, j � n, and X E (Xo , . . . , Xn ) cle=f a, then x j. 0. (The set p ( a) is the spherical simplex (uniquely) determined by an . Its faces are the radial projections of the faces of an .) n [Hint: If L t i = 1 and t i 2: 0, ° � i � n, then i=O n n L ti Xi = Lti (Xi  Xo ) + Xo· i=O i=O The inequality
II X YI1 2 2: I II x I1 2  IIY I 1 2 1
applies (v. 3. 1.2 and (3.2.12)). ] When sn is the union of nondegenerate spherical nsimplices and any n  Iface is in the intersection of precisely two spherical nsimplices, the spherical nsimplices constitute a triangulation of sn . 1.4.10 Exercise. When 1 � i � n + 1, the vectors °
ei �f
1
�
ith component
° determine 2n + 1 nondegenerate nsimplices (±e l , . . . , ±en + l ) . Their radial projections constitute a triangulation of sn . 1.4. 11 Exercise. The diameter of the nsimplex a �f (xo , . . . , xn ) is
23
Section 1.4. Homotopy, Simplices, Fixed Points
Thus diam (a) is the maximum of the lengths of the (finitely many) edges of a, whence diam (a) is the length of some edge of a . 1.4. 12 DEFINITION. THE barycenter OF THE nSIMPLEX a cle =f (Xo , ..., Xn)
f 1 � Xk. FOR THE PARTIAL ORDER < DEIS THE VECTOR b (a) cle = n+ 1 � k=o FINED AMONG THE FACES OF a BY
THE SIMPLICES OF THE barycentric subdivision OF a ARE THOSE AND ONLY THOSE DETERMINED BY BARYCENTERS {b (ap,) , ... , b (apr)} SUCH THAT ap1 < ap2 .. . < apr. THE UNION OF THE SIMPLICES OF THE BARYCENTRIC SUBDIVISION OF a IS DENOTED a'. M ORE GENERALLY, aU) DENOTES THE UNION OF nSIMPLICES OF THE BARYCENTRIC SUBDIVISION OF ALL THE SIMPLICES IN aUI), 2 :::; j.

1.4.13 LEMMA. IF a IS AN nSIMPLEX AND 7 IS AN nSIMPLEX IN a', n THEN diam (7) :::; diam (a). n+ l PROOF. If b (ap) and b (aq) are vertices of 7 and ap < aq, via appropri ate labelling, ap = (xo, ..., xp) and aq = (xo,..., xp,Xp+I, . .., x q). Direct calculation leads to the equation
(
p
q
1 1 qXk   L Xk b (ap)  b (aq) = p L 1 1 + q  P k=p+ q+ p k=O I
{IL p
Since
1
q
X , L X P + 1 k=O k q  P k=p+l k
)
}
1.4.14 COROLLARY. IF a IS AN nSIMPLEX AND 7 IS AN nSIMPLEX IN diam (a). aU), diam (7):::; n 1 PROOF. Mathematical induction, based on the conclusion of 1.4.13 applies. o
(:r
24
Chapter 1. Fundamentals
1.4.15 Exercise. If E > 0, sn admits a triangulation T such that the diameter of each simplex of T does not exceed E. (The triangulation T is of mesh E.) If F is a homotopy in sn of id to c and T is a triangulation of sn , to each s and each vertex x of T there corresponds a vector F (x, s) �f Xs of sn . Because sn x [0, 1 ] is compact, F is uniformly continuous (v. Section 1 .6). Hence, if the mesh of T is small, and a �f (xo, . . . , xn ) is a spherical n + Isimplex in T, for all s in [0, 1 ] ' the diameter of the (possibly degenerate) simplex aF(.,s) �f (F (xo, s) , ... , F ( xn , s)) is also small. of [ 1 .4.16 Note. For fixed s , the set TF(.,s) �f {aF("S)} aET spherical simplices does not necessarily constitute a triangulation of sn . The union of the constituent spherical simplices need not cover sn ; the intersection of two or more of them can have a nonempty interior relative to S nthey can overlap.]
When x E Sn and x is not on the boundary of any aFt,s), the cardinal ity of the set of spherical nsimplices aF(.,s) to which x belongs is denoted N (x, F (·, s),
T).
1.4.17 Exercise. a) If Sf is near s, then N (x, F (·, s ) ,
T)
=
N (x, F ( . , S f ) ,
T).
b) When x remains fixed and traverses [0, 1 ] ' the number N (x, F (·, s ) , T) remains constant. c) Where N (·, F (·, 0), T) is defined, N (·, F (·, 0), T) = 1. [Hint: The function N (·, F (·, ) T) is continuous and N+ valued. ] s
s
,
1.4.18 Exercise. If T is a triangulation of sn and F is any map of the vertices of T into TF in the manner described above (F is a vertex map), there is a vertex map Ff such that: a) for TF', each spherical nsimplex is determined by linearly independent vectors ; b) if x is on no boundary found in TF, x is on no boundary found in TF' and N (x, F, T) = N (x, Ff, T). [Hint: A hyperplane in ffi.n + l is nowhere dense. ]
1.4.19 Exercise. If neither x nor y is on any boundary found in TF and the vectors determining each spherical nsimplex of TF are linearly independent, there is a path connecting x and y on sn , meeting each n  Iface of TF at most finitely often. If k > 1, meets no n  kface. [Hint: The union of the spherical nsimplices determined by TF is connected. The hypothesis of linear independence implies that an escape from one spherical nsimplex to another can always be managed through an n  Iface. There are only finitely many spherical nsimplices in TF.] 1f
1f
25
Section 1.4. Homotopy, Simplices, Fixed Points 1f
1.4.20 Exercise. At each crossing of from one spherical nsimplex to another, the number N( · , F, T) changes by ±2 or o . [Hint: If the adjoining nsimplices overlap, crossing through their common n  Iface involves either entering or leaving both. If the adjoining nsimplices do not overlap, the number N( · , F, T) does not change. ] 1.4.21 Exercise. For a given triangulation T and a vertex map F such that the spherical nsimplices of T F are determined by linearly independent vectors, the number N(x, F, T) is, modulo 2, independent of x. In sum, the vertex map F and the triangulation T determine a number N (F, T) such that for all x in sn ,
N(x, F, T)
==
N(F, T)
(mod 2).
Since, modulo 2, N( · , F( . , 0), T) = 1 and N( · , F( · , 1 ) , T) = 0, the homotopy F does not exist: id and c are not homotopic. [ 1.4.22 Remark. When J E C (sn , sn ) and T is a triangulation of sn , J confined to the vertices of T is a vertex map Ff. The number N (Ff, T) (mod 2) is the degree modulo 2 of the pair {J, T}. In fact, N (Ff , T) does not depend on T and is an intrinsic property of the pair {J, sn }. If orientation of simplices is taken into account, a more general Brouwer degree (taking values in Z) of the map J can be defined. The Brouwer degree, which may be viewed as counting the number (positive or negative) of times J wraps sn about itself, is a fun damental and important topological invariant of the pair {J, sn }. For example, if n E Z, the map n : S 1 3 (cos B, sin B)
r+ (cos nB, sin nB) E S 1
wraps S1 about itself n times. Among other applications of the Brouwer degree is a proof of the Fundamental Theorem of Algebra (FTA) [D u , D uG, Sp] . ]
r+ r+
l
1.4.23 Exercise. There is no map g : Bn + 1 S n such that g s n = id . [Hint: The map F : sn x [0, 1 ] 3 (x, s) g[(1  s)x ] is a forbid den homotopy. ] Brouwer's Fixed Point Theorem says that a continuous map J of Bn + 1 into itself leaves some point x fixed: J(x) = x.
Chapter 1. Fundamentals
26
1.4.24 Exercise. If E C ( [I, 1], [1, 1]), for some x in
g
g(x) =
x,
[1, 1]'
i.e., Brouwer's Fixed Point Theorem is valid when n o . On the other hand, slight modifications of the hypothesis invalidate the conclusion. 1.4.25 Exercise. The map
=
maps B 2 (O, 1)° onto itself and leaves no in B2 ( O, It fixed. 1.4.26 Exercise. The map z
f : (D( O, 1 ) \ {O}) :1 Z
r+ �lZ
z
maps D( O, 1) \ {O} onto itself and leaves no fixed. Here is an intuitive argument for Brouwer's Fixed Point Theorem. If the conclusion is false, for some continuous f and each x in Bn + l , f (x ) j. x. The halfline starting at f (x ) and through x meets sn in a point x) . The map is continuous and I id . Intuition suggests does some tearing, i.e., is not continuous. As the following material reveals, via the technology developed above, the intuitive argument just offered can be made rigorous.
g g
g(
g Sn =
g
1.4.27 THEOREM. (Brouwer's Fixed Point Theorem) IF f : IS CONTINUOUS, FOR SOME x IN Bn+ l , f (x ) x.
=
Bn+ 1 r+ Bn+ 1
PROOF. Otherwise, for each x, f (x ) j. x and the line
{ f (x) + t [ x  f (x)] : t � O } meets sn in a vector g ( x) . The map g : Bn+ 1 r+ Bn+ 1 is continuous and if x E sn , then g(x) = x: g l s n = id , a contradiction of 1.4.23. 0 1.4.28 Exercise. If X and Y are homeomorphic and each f in C ( X, X) leaves some in X fixed, each g in C ( Y, Y) leaves some in Y fixed. x
y
The fixed point property is a topological invariant An important aspect of the development above is that for the purpose in hand, discussions of continuous maps may be reduced to discussions of maps of finite sets of points, e.g., the vertices of the triangulations T. A significant block of topology is devoted to a finitistic approachsimplicial approximation. In this direction, the work, e.g., of Alexander, tech, Eilen berg, Hurewicz, Lefschetz, Massey, Mayer, Spanier, Steenrod, and Vietoris,
Section 1.4. Homotopy, Simplices, Fixed Points
27
gave rise to modern algebraic topology [Bro , Du, DuG, Sp] ; v. also [F, The].
In the discussion below, a similar finitistic analysis provides an alter native derivation of Brouwer's Fixed Point Theorem. 1.4.29 LEMMA. ( Sperner ) IF an cle=f (xo, ..., Xn) IS AN nSIMPLEX AND
IS SUCH THAT FOR EACH pFACE ap OF an, W [b(ap)] IS A VERTEX OF ap, i.e., IF W IS A Sperner map , THEN FOR SOME Tn �f (Yo,..., Yn) OF a�,
( FOR SOME PERMUTATION
1f
OF {O, ...,n}, w ( Yd = X7r (k) , O <::;k <::;n )
[ 1.4.30 Note. Sperner's Lemma is purely combinatorial. No topological or algebraic properties, e.g., continuity, linearity, are assumed for w.]
PROOF. When n = 0 the conclusion is automatic since w = id . If the re sult obtains when n = m 1, a relabeling of the vertices permits the as sumption that w [b (am)] = Xm. Furthermore, for w confined to the sim plex am I �f (xo, ..., xm  d , Sperner's Lemma obtains. Hence there is an (m  I ) simplex Tm I �f (Yo,...,Ym d contained in a:n I and such that , I. However Tm is the face of some {w (Yk)} :'=OI {Xd:� msimplex Tm I in a:". The vertices of Tm are the vertices of Tm  I together with b (am). Since w [b (am)] = Xm , W maps the vertices of Tm onto the vertices of am. 
=
o
1.4.31 Exercise. If n = 1 , barycentric subdivision is simply bisection and Sperner's Lemma is a triviality. Since each vertex of a�k) is a barycenter b a�,kI) of some a�kI), a Sperner map wn .· anrk) an(kI) carries b an(kI) onto a vertex of an(k I), which is a barycenter b a�k 2) of some a�k 2 ). Hence a Sperner map
r+
r+
(
)
(
(
)
(
)
)
Wn I : a�kI) a�k 2) carries b a�kI) onto a vertex of a�k 2 ). The composition Wn I 0 Wn : a�k) a�k 2 ) is a Sperner map. By induction, there is a Sperner map w (k) �f WI 0 ..· 0 Wn : a�k) an
r+
r+
28
Chapter 1. Fundamentals 1
1
3
2
2
Figure 1.4.1. 1.4.32 Example. The marking of the vertices in Figure 1.4.1 is quite arbitrary. 1.4.33 Exercise. The counterpart of Sperner's Lemma obtains for w (k).
[Hint: For a Isimplex, the result is an instructive consequence of simple counting and induction. For the general case, induction applies. ]
In Table 1.4.1 there are two examples Wi : � 0"2 , i = 1 , 2, of Sper ner maps of � into 0"2 . Only the actions of the linear maps Wi on the vertices are described. If x E � x is in some unique 0, 1, or 2simplex 7 of � Hence x is a convex (linear) combination of the vertices of 7 . O"
O"
O"
O" ,
.
WI
w2
1�1 2�2
3�3
P�2
S�3
R�2 72 :
Q�3 (1, P, Q)
1�1 2�2 3�3 P�I S�I R�2 Q�2 (3, Q, S)
Table 1.4.1
r+
Section 1.4. Homotopy, Simplices, Fixed Points
29
Correspondingly, Wi (X) is assumed to be the same convex combination of the wiimages of those vertices. In each instance, T2 is a Sperner simplex, i.e., a simplex behaving as stated in Sperner's Lemma. Below the description of each map, a corresponding Sperner simplex T2 is indicated by its vertices. [ 1.4.34 Note. As the calculations for the two maps W I and W2 reveal, the respective associated Sperner simplices are different. However, for each map, further calculations show that each T2 is the only Sperner simplex: all the remaining five simplices fail to satisfy the defining conditions for a Sperner simplex. In general, the Tn in Sperner's Lemma is unique. The proof of uniqueness is omitted since the uniqueness of Tn is not essential for the work that follows. ] The penultimate step in the PROOF of Brouwer's Fixed Point The orem is the next consequence of Sperner's Lemma. Both the statement and PROOF of 1.4.35 display essential links between the combinatorics in Sperner's Lemma and the underlying topological structure of the con text. The combinatorial aspects may be termed finitisticvarious technical counting procedures are involvedbut no appeal is made to continuity or related topological notions. The topological arguments deal with the met ric properties of covers of a simplex. The essential fact proved in 1.4.35 is the following: If a cover consisting of closed sets is sufficiently redun dant, the intersection of all elements of the cover is non empty.
The statement above is a rather elaborate extension of the pigeonhole principle: If n objects are placed in fewer than n pigeonholes, at least one pigeonhole contains two objects. The argument relies on the metric properties of coverings.
n
1.4.35 L EMMA. I F an clef (xo , . . . , xn ) IS AN nSIMPLEX, :F clef = { Fk } k =o IS A SET OF CLOSED SUBSETS OF (jn, AND FOR EACH SUBSET { io , ... , ip} OF
=
p
n
k=O
k=O
{O, . . . , n} , (XiO > " . . ' Xip ) C U Fi k , THEN n Fk j. 0.
30
Chapter 1. Fundamentals n
PROOF. By hypothesis, X i E Fi, 0 :::; i :::; n,
an C U Fi , and every vertex of n
a barycentric subdivision is in some F i · If
i= l
n Fk = 0, then
k=O
is an open cover of an . From 1.4.14 it follows that for some r , the maximum diameter of any simplex in is less than the Lebesgue number l1(U) (v.1.2.36 1.2.40) . Each Y in is in some Fi, which contains Xi. For some Sperner map w, the map Y xi is w ( r) . Sperner's cle . . f · Lemma Imp1les that for some nslmp 1ex Tn = (Yo , ..., Yn ) .In an( r) ,
at ) (at) )
at ) 3 r+
The vertices Yi (relabeled as needed) may be assumed to be such that Yi E F i, 0 :::; i :::; n. Thus Tn n Fi j. 0 , 0 :::; i :::; n, and so diam ( Tn ) 2': l1(U) , a contradiction. 0 The trapezoids F 1, F 2 , and F3 in Figure 1.4.2 exemplify the closed 3
sets described in the LEMMA:
n F i is the shaded triangle.
i=l
1
2
D
B
3
Fl = lCD2 , F 2 = 2E F 3, F3 = l A B3
Figure 1.4.2.
31
Section 1.4. Homotopy, Simplices, Fixed Points
1.4.36 THEOREM. ( Brouwer's Fixed Point Theorem, again) IF X AND Bn ARE HOMEOMORPHIC AND f E C ( X, X ) , SOME x IN X IS A FIXED POINT FOR f: f(x) = x. PROOF. Owing to 1.4.8d) and 1.4.28, the truth of the result when X is the closed simplex
an cle=f [eo cle=f (0, 0, . . . , 0),e l clef ( 1, 0, . . . , 0) , . . . , en clef ( 0, 0, . . . , 1 )] =
=
implies the result for any X as described. If X = an and x E X, there are unique nonnegative Ak (X), 1 ::; k ::; n, such that n
n
and x = L Ak (x)ek. Then 1.2.15 implies that the functions k=O
are continuous. Since f (an ) C an , if x E an and f(x )
=
n
n
k=O
k=O
L Ak [J(x)] ek �f L Ilk (x)ek,
the functions Ilk, ° ::; k ::; n , are continuous. Hence the sets are closed. Furthermore, if {io , . . . , ip} C {O, . . . , n}, p
(eio ,..., eip ) C
U Fi k •
k=O
n
Thus, according to 1.4.35, F
�f n Fk j. 0.
k=O If x E F , then Ilk (X) ::; Ak(X), 0 ::; k ::;
n.
n
n
k=O
k=O
Since
1 = L Ilk (X) = L Ak (X), Ilk (X)
=
Ak (X), 0 ::; k ::; n , i.e., f (x) = x.
o
Chapter 1. Fundamentals
32 1.5. Appendix 1 . Filters
1.5.1 DEFINITION. FOR A SET X, A filter F �f { F} IN !fj(X) IS A SUBSET CONFORMING TO a)c) OF 1.2.8. A FILTER F' refines OR IS A refinement OF OR IS finer THAN A FILTER F IFF F' ::) F IN WHICH CASE ONE WRITES F' > F. WHEN F' IS FINER THAN F, F IS coarser THAN F' . The set of all filters in !fj( X) is partially ordered by > and for a given filter F there is, by virtue of any one of the equivalents Hausdorff Max imality Principle, A xiom of Choice, Wellordering A xiom, etc., [Melof Zorn's Lemma, a maximal filter U such that U > F. A maximal filter U is an ultrafilter. The filter consisting of the single set X is coarser than any filter. For x in X, the filter consisting of all the supersets of {x} is an ultrafilter. 
1.5.2 DEFINITION. A SUBSET 8 OF !fj(X) IS A FILTER BASE IFF: a) 8 j. 0; b) 0 1 8 ; c) THE INTERSECTION OF ANY TWO ELEMENTS OF 8 CONTAINS AN EL EMENT OF 8. 1.5.3 Exercise. The set of all supersets of a filter base 8 is a filter F. 1.5.4 Exercise. The intersection of all filters containing a filter base 8 is a filter F, the filter generated by 8. 1.5.5 Exercise. Every filter F is a filter base; F generates itself. 1.5.6 Exercise. For a map f : X Y: a) If 8 is a filter base in !fj(X), {f(B) } B EB �f f(8) is a filter base in !fj(Y). b) If C is a filter base in !fj(Y) , {I  I ( C) } CEe �f f  1 (C ) is a filter base in !fj( X) iff for all C, {I I (C) j. 0 } . When X is a topological space, a filter F converges to x in X iff every N(x) contains an element of F: F invades every N(x). Since N(x) is itself a filter, F converges to x iff F refines N(x). When F converges to x, x is the limit of :F. A filter F � {F} in !fj( X) is a diset with respect to the partial order induced by reversed inclusion among the elements of F: F' > F iff F' c F. A map n : F F n( F) E F ( c X) is thus a net corresponding to the filter :F. Conversely, when A is a diset, n : A A n(A) E X is a net, and B>. �f { n(p,) : p, > A } , {B>.h EA is a filter base that generates the filter Fn corresponding to the net n. The correspondence net + filter is an injective map. The correspondence filter + net is not a map since there can be more than one net corresponding to a given filter. [ 1.5.7 Note. Owing to the correspondences between nets and filters, any notion formulable in terms of nets is equally formulable in terms of filters. ]
r+
3 r+
3 r+
Section 1.5. Appendix 1. Filters
33
1.5.8 Exercise. A filter F converges to x iff every net n corresponding to F converges to x. 1.5.9 Exercise. If the filter F corresponds to the net n , then n converges to x iff F converges to x. 1.5.10 THEOREM. A TOPOLOGICAL SPACE X IS COMPACT: a) IFF EVERY ULTRAFILTER IN !fj(X) CONVERGES; b) IFF FOR EVERY NET n : A X THERE IS A CONVERGENT COFINAL NET n A' .
r+
l
PROOF. a) If X is compact and U �f {F} is an ultrafilter in !fj(X), each F is closed. If n F = 0, then for some finite subset {F1 , . . . , Fn } of U, n
n
FEU
n Fk c n Fk = 0, which is impossible for elements of a filter. k= l some x is in n F. For the filter N(x ) ,
k= l
Hence
FEU
{ N (x)
nF
: N ( x)
E N(x ) , F E U }
generates a filter V that converges to x. Furthermore, V > U and, since U is an ultrafilter, V = U. Conversely, if every ultrafilter converges, and E �f { E } is an fip set of closed sets, then E generates a filter F, contained in an ultrafilter U that converges, say to x. For N ( x) and Eo in E there is in U a U that is contained in N (x) and N ( ) ::) U n Eo j. 0. Since Eo is closed, x E Eo: x E n E. In other words, if a set E of closed sets is an fip set (v. 1 . 2. 25), x
EE£
the intersection of all the sets of E is nonempty. Hence X is compact. b) If X is compact and n : A X is a net, the corresponding filter F �f { B>. �f { n ( p, ) : p, > A } } is a subset of an ultrafilter U converging to some x in X. Hence x is in the closure of each element U of U, in particular x is in the closure of each B>. . Thus, for each N ( x) and each A in A, there is a U contained in N ( x) . In the nonempty set B>. n U there is an n (N) and A' > A. The set A' of all such N is cofinal in A and the cofinal net n I A' converges to x. Conversely, if U is an ultrafilter in !fj(X) and n is a net corresponding to U, by hypothesis there is a cofinal net converging to some x in X. Hence U converges to whence every ultrafilter converges and X is compact.
r+
x,
o
m
The following proof of Tychonov's Theorem ( 1 .2.30) is phrased terms of filters. PROOF. For A in A, the map 1f>. : X 3 x �f {xJL } JL EA x>. E X>. is contin uous. If U is an ultrafilter in X, 1f>. (U) is a filter base of a filter U>. in
r+
Chapter 1. Fundamentals
34
Xx . Since 7rA 1 (U).. ) contains U, U).. is itself an ultrafilter. Because X).. is compact, U).. converges to some x).. in X)... The filter U converges to the . X cle=f { X).. } )..EA m. X . pomt 0 1.6. Appendix 2. Uniformity
For a metric space ( X , d) there is in lfj (X 2 ) an associated filter base
B �f { B_ �f { (X, y) :
(x, y )
E X2 , d(x , y) < E } } _ >0 .
The operations o
:lfj (X 2 )
x
lfj (X 2 ) :1 (A, B),
r+ { (x, y) : 3(z){ ((x, z) E A) 1\ ((z, y) E B)} } , �f A B E lfj (X2 ) , and  1 :lfj (X 2 ) :1 A r+ { (x , y ) : ( y , x) E A } �f A I 0
permit a succinct description of two fundamental properties of the filter U generated by B, viz.: a) If U E U, then U I E U; b ) If U E U, for some V in U,
that c)
0
V V I c U.
For the particular filter U associated with the metric d it is true also
nU UEU
=
{ (x, x) : x
E X } �f 6.
The preceding discussion motivates the following 1.6.1 DEFINITION. FOR A SET X, A FILTER U IN lfj (X 2 ) IS A unifor mity IFF a) AND b ) OBTAIN. IF c ) OBTAINS AS WELL , U IS A Hausdorff
uniformity . A PAIR (X , U ) CONSISTING OF A SET X AND A UNIFORMITY U IS A UNIFORM SPACE, USUALLY DENOTED SIMPLY X. EACH ELEMENT OF U IS A VICINITY. Uniform spaces are discussed in [Bou, Tuk, We1 ] .
1.6.2 Example. When G is a topological group (v. Section 4.9) and N(e) is the filter of neighborhoods of the identity e of G, the set
U �f {UN �f { (x , x')
: XX,I E N } }
NEN(e)
Section 1.6. Appendix 2. Uniformity
35
is a uniformity for G and (G, U ) is a uniform space.
1.6.3 DEFINITION. FOR A UNIFORM SPACE ( X, U ) WHEN x E X AND U E U THE UINDUCED NEIGHBORHOOD OF x IS
Ux �f { y : (x, y) E U } . 1.6.4 Exercise. The system {UX } XE X is a base of neighborhoods of a UEU topological space ( X, T ) . When ( X, U ) and (Y, V) are uniform spaces, a function f : X Y is uniformly continuous iff for each V in V there is in U a U such that u
r+
r+
{(x, y) E U} '* { (f(x), f(y) ) E V} .
A net n : A X is a Cauchy net iff for each U in U there is in A a Ao such that {P > Ao} 1\ {I.l > Ao} } '* ( (n(A), n(I.l)) E U}. A net n : A eX is a uniform Cauchy net iff for every positive E there is in A a Ao such that
r+
Vx {{A > Ao} 1\ {I.l
>
Ao}} '* { I n(A) (x)  n( l.l ) (x) I
< E} .
r+
1.6.5 Exercise. If ( X, U ) and (Y, V) are uniform spaces and f X Y is uniformly continuous, f is ( T Tv ) continuous. 1.6.6 Exercise. If ( X, U ) is a uniform space compact in the Uinduced topology, (Y, V) is a uniform space for which V is a Hausdorff uniformity, and f E y X is continuous with respect to the uniformity induced topologies T and Tv, each of the following obtains. a ) The map :
u,
u
F :X
x
X
3 (a, b) r+ [J(a), f(b)] E Y
x
Y
is continuous. b ) If W E y, then W is open in Y x Y, V �f F  1 (W) is open in X x X, and the diagonal 6 in X x X is contained in V: 6 C V. c ) If V is not in U: •

:F cle=f { (U \ V) } UEU is an fip set of closed subsets of the compact space X x X (whence A �f n F i 0) ;
FEF
•
•
the diagonal 6 =
n
UEU
u,
is compact, and 6 n
(n ) FEF
F
=
0;
if (x, y) E A, each neighborhood of (x, y) meets 6 (whence (x, y) E 6 ) and (x, y) E ( X x X \ 6 ) , a contradiction: V E U. d ) The map f is uniformly continuous.
Chapt er 1. Fundamentals
36
A continuous map of a compact uniform space into a uni form Hausdorff space is uniformly continuous.
1.6.7 THEOREM. IF: a) (X, U) AND (Y, V) ARE UNIFORM SPACES, AND T AND Tv ARE THEIR UNIFORMITY INDUCED TOPOLOGIES , b) U
3
n : A A r+ I>. E C(X, Y) IS A NET, AND c) f E y X , AND FOR EACH VICINITY V IN V, THERE IS A Ao SUCH THAT {>. > Ao } ::::} {Vx { [J.x (x), f(x)] E V }}, THEN f E C(X, Y). The uniform limit of continuous functions is continuous.
Y = f(x) and N(y) is a neighborhood derived from a vicinity V in V, there is a vicinity Z such that Zl 0 Zl C V and there is a vicinity W such that W o W C Z  l . For some Ao , if A � Ao, then VxVx ' { [I>. (x) , f (x)] E W } 1\ { [I>. (x') , f (x')] E Z}, and for some N (x) de rived from a vicinity U, {x' E N(x) } ::::} { [I>.o (x), 1>.0 (x' )] E W}. Since W o W 0 Zl C Zl 0 Zl C V, it follows that [J(x) , f (x' )] E V. 0 For uniform spaces (X, U) resp. (Y, V) and their uniformityinduced topologies T and Tv , a set S �f {I>. L EA of elements in C(X, Y) is equicon tinuous iff for each vicinity V of V there is in U a vicinity U such that if (x, x' ) E U, for all A , [h(x), h (x')] E V.
PROOF. If x E X,
u
1.6.8 THEOREM. IF (X, U) RESP. (Y, V) ARE UNIFORM SPACES THAT ARE COMPACT HAUSDORFF SPACES IN THEIR UNIFORMITYINDUCED TOPOLO GIES AND THE NET n : A A r+ I>. E C ( X, Y) IS AN EQ UlCONTINUOUS SET OF FUNCTIONS, SOME COFINAL SUBNET n : A' >.' r+ hI CONVERGES UNI FORMLY TO AN f ( NECESSARILY IN C(X, Y) ) .
3
3
PROOF. For each x in X if Yx �f Y , then X Y = Xx E X Yx and 1.2.30 im plies Y x is compact. Consequently, 1.5.10 implies there is a cofinal subnet n : A' A' r+ fA' that converges to some f in y X .
3
If V is a vicinity in V, there is a vicinity Z such that Z  l 0 Zl C V and there is a vicinity W such that W o W C Z l . The equicontinuity of the functions in {fA' } A' EA' implies there is in U a vicinity U such that for all >.' , { (x, y) E U} ::::} { [JA' (x) , JA' ( y )] E W}. The set {NU (x) } x E X of derived neighborhoods is a cover of X and hence, for some finite set { Xm h �m � M ' U Nu (xm ) = X. If x E X, for some m, ( x , xm ) E U, whence for all l �m�M
>.' , [JA' (x), fA' (x m )]
[JIL ' (x) , fIL ' (xm )]
E
E W. By a similar argument, if A' > A� and 11/ > f.l�,
Z l .
The convergence of
Section 1.6. App endix 2. Uniformity
37
implies that for some x Iree (A�, f.1.�) , Consequently, if A ' > A� and f.1.' convergence is uniform: Iv � I ·
>
f.1.�, then [Iv ( )
X ,
fJL f (x)]
E V, i.e., the
0
1.6.9 COROLLARY. (ArzelaAscoli) IF (X, d) IS A COMPACT METRIC SPACE AND {In } nEN IS A UNIFORMLY BOUNDED E Q UlCONTINUOUS SEQUENCE IN C(X, JR.) , SOME SUBSEQ UENCE {Ink hEN CONVERGES UNIFORMLY ON X TO A g IN C(X, JR.) . PROOF. Although the PROOF of 1.6.8 applies directly, the following proof uses a diagonalization technique of independent interest. For each positive r in Q, U B (x , rt = X and since X is compact, xEX there are finitely many xi (r) , 1 � i � I(r) , such that
X=
U
l �i� I (r )
X
B ( i (r), r) .
Hence {xi (r)} o< r E\Ql, l � i� I ( r ) is a countable dense subset D �f {Pn }nEN of X. Since { In }nEN is uniformly bounded, there is a convergent subsequence
a convergent subsubsequence
etc. The diagonal sequence
converges at each point of D. The equicontinuity of {In } nEN implies that if x E X, for a suitable Pk , the first and third terms in the right member of
I grn (x)  gn (x) 1 � I grn (x)  grn (Pk ) 1 + I grn (Pk )  gn (Pk ) 1 + Ign (Pk )  gn (x) l , are small, and then for large m and n, the second term is small: {gn }nEN converges at each x in X, say to g(x).
Chapter 1. Fundamentals
38
If E
> 0, there is a positive 1] such that
Furthermore { Nj �f { x : d (x, pj ) < 1] } } is an open cover of X and l EN thus contains a finite sub cover {Nj } l < There is a k (E) such that if k (E) and 1 � j � J, then d [grn (Pj ) , gn (Pj )] < E. Thus, if x E X, for some j in [1, J] , x E Nj and if m, k (E), then .
m,n >
_
<J" l_
n>
d [grn (x) , gn (x)] � d [grn (x) , grn (pj )] + d [grn (Pj ) , gn (pj )] + d [gn (pj) , gn (x)] < 3E. It follows that gn � g, whence g is continuous. [ 1.6.10 Note. There are many variations and extensions of 1.6.8. Some of these are discussed in [I s Kel] .
0
,
Viewed as a subset of C( X, JR.) topologized by the metric
d : C(X, JR.?
3
(f, g )
r+ xsup I f(x)  g(x) l , EX
{fn } nEN is a precompact set, i.e., its closure is compact . The set { f d 'xEA resp. {fn } nEN of 1.6.8 resp. 1.6.9 is an example of a normal family F. The general context for a normal family is: • •
two sets X and Y, and some topology
y X ,.
T for a subset S of
a set F contained in S and such that F is compact (F is
Tprecompact) .
In several applications of this notion, X and Y are themselves topological spaces and S = C(X, Y).]
1.7. Miscellaneous Exercises
1. 7.1 Exercise. If X and Y are topological spaces and f E y X then: a) EX f is continuous iff for every x in X and every net A 3 converging to x, f 0 converges to f(x) ; b) the filter formulation of a) is valid; c) f is open iff for each x in X there is a neighborhood N (x) such that f[N(x)] is open. What are the filter and net formulations of c)?
n
n
:
), r+ n( ), )
Section 1 .7. Miscellaneous Exercises
39
X
1. 7. 2 Exercise. If a set N of subsets of a set is such that: a) each x of is in some Ux (an xneighborhood in N); b) if Ux and Vx are x neighborhoods in N, for some xneighborhood Wx in N, WX c Ux n Vx ; c) for each xneighborhood Ux in N and each y in Ux, some yneighborhood Uy in N is a subset of Ux, then N is a neighborhood base for a topology for 1. 7.3 Exercise. If g} c JR.x : a)
X
X.
{f, f V g cle=f max{f, g } = f + g +2 I f  g l ' f 1\ g cle=f mIn. {f, g } = f + g 2 I f  g l ' b) {f V g, f 1\ g} C C(X, JR.) iff {f, g} C C(X, JR.) ) ; c) f is continuous iff f is lsc and usc. 1. 7.4 Exercise. If X is a compact space and Y is a topological space, and f : X Y is a continuous bijection, then f  1 is continuous, i.e., f is bicontinuous . 1. 7. 5 Exercise. Some f in JR.lR is continuous and not open. 1. 7.6 Exercise. If { rn} nEN is an enumeration of Q and if :Z: E \r 1 , . . . , rn} : fn (X) �f { ol otherwIse a) lim fn (x) �f f (x) exists for each x in JR.; b) each fn is Riemann integrable on [0, 1] ; c) f is not Riemann integrable. .
ft
n + =
1. 7. 7 Exercise. If
if Q 3 x = E , p E Z, q E N, p l q = otherwise
q
1
f is continuous at x iff x tJ. Q.
1. 7.8 Exercise. A point a is a limit point of a set A (a E Ae) iff for each N(a) and each net n : A ft A \ {a}, n is frequently in N(a), v. 1.2.22 .
1. 7.9 Exercise. If X and Y are topological spaces, X is connected, and f E C(X, Y), then f(X) is connected. 1. 7. 10 Exercise. If X is a topological space, C is a connected subset of X, A c X, and C meets both A and X \ A, then C meets 8(A) . 1. 7.11 Exercise. a) If Q is a region in C and { a, b} c Q, there is a finite set {[Zn , zn+ dL < n < N of complex intervals, each contained in Q and such that Z I = a and ZN= b. Their union is a polygon: Q is polygonally connected. b) A similar result obtains for a region Q in JR.n , E N. n
Chapter 1. Fundamentals
40
[Hint: a) The set { z : z is polygonally connected to a } is open, relatively closed in Q, and nonempty.] 1. 7. 12 Exercise. If X is a second countable topological space, X contains a countable dense subset, i.e., X is separable. 1. 7. 13 Exercise. a) In a topological space X, the closure of a subset is the intersection of the set of all closed sets containing b) If X is a locally compact Hausdorff space and E K(X), then is the intersection of the set of all open sets containing c) If X is a second countable locally compact Hausdorff space, each in K(X) is the intersection of a countable set of open sets containing each compact set is a compact G/j . d) If X is a metric space, every closed set is the intersection of a countable set of open sets: each closed set is a G/j . 1. 7.14 Exercise. For N regarded a diset ordered by < and a net
K:
K
K K.
K
A.
A
as
n : N 3 k r+ n ( k ) E !fj(X),
x E n resp. x E n iff x is in infinitely many sets n ( k ) resp. x is in all but finitely many sets n( k) . 1. 7. 15 Exercise. For a topological space X, an a in X, the filter N( a), and an f in ffi.x , there is a corresponding net
n : N(a)
3
N(a) r+ n[N ( a )] E ffi.
such that lim f(x) resp. lim f(x) is
x=a
x =a
lim
N ( a ) EN ( a)
n[N ( a )] resp.
lim n[N ( a )] .
N ( a) E N
1. 7.16 Exercise. If X is a topological space and f E ffi.x ,
: X 3 x r+ lim f(y) resp. ¢ : X 3 x r+ lim f(y) y =x y= x
is lsc resp. usc. 1. 7.17 Exercise. (Weierstra:B) If (X, d) is a compact metric space and {xn }nEN C X, for some x in X and some sequence { nk} k EN'
1. 7.18 Exercise. If (X, d) is a compact metric space and U is an open cover of X, there is a positive rS such that for each x in X and some U in U, B(x, rSt c U.
Section 1.7. Miscellaneous Exercises
41
[Hint: Lebesgue's Covering Lemma (1.2.39) applies.] 1. 7. 19 Exercise. The product topology for a Cartesian product
X cle =f
X
I'
Er Xl'
of topological spaces Xl' : a) is the weakest topology with respect to which all the projections : X 3 x �f {Xl'} ft x).. E X).. are continuous; b) con sists of the set of all unions of finite intersections of sets of the form A 1 ( U ) ). E r, U E 0 (X).. ) . 11")..
11"
,
{n
}
1.7.20 Exercise. If X is a set, Y has the topology
set
sEa
F, Us E T
fs 1 (Us) : # ( a) E N, fs E
T, and F e y X , the
is a topology for X and
is the weakest topology with respect to which each f in 1. 7.21 Exercise.
F is continuous.
If (X, d) and (Y, D) are complete metric spaces and
F e C(X, Y), then F is normal iff": a) for each in X, { f(x) : f E F } is precompact in C(X, Y) ; b) F is equicontinuous at each x in X, i.e., if E > 0 then for some positive J and all f in F, { d (x, x' ) < J} ::::} {D [J(x ) , ! (x' )] < E } . x
K,
K
1. 7.22 Exercise. If is compact and f is usc resp. lsc on K, for some a in f(a) = sup f(x) resp. f(a) = inf f(x) .
xEK
xEK
K
1. 7.23 Exercise. (Tietze's Extension Theorem) If is a compact subset of a locally compact space X, and f E C(K, JR.) , for some F in
(
Coo X, JR.)
�f { f
Fi K= f .
[Hint: The assumption f(K)
K+ cle=f {
x :
1
f(x) 2': 3
C
[1, 1] is justifiable. Applied to
} resp. K clef { =
1.2.41 implies there is in Coo(X, C) an
h(x ) =
If(x) 
( )},
(
: f E C X, C) , supp (I) E K X
{I
h
 3 if x E 1 if x E 3
x :
1 f(x) :::;  3
such that
K K+
h (x) 1 :::; 32 ' on K I h(x ) 1 :::; 31 on
x.
}'
42
Chapter 1. Fundamentals
The argument above applied to f  II yields an h . Induction 00
yields a sequence {fn } nEW Finally, F � L fn exists and meets
n= 1
the requirements.]
r+
r+
1.7.24 Exercise. a) If A 3 ), f>. E usc(X) resp. A 3 ), f>. E Isc(X) is a net, and f>. ..l f resp. f>. t f, then f E usc(X) resp. f E Isc(X) . b) If X is a locally compact Hausdorff space and 0 :::; f E lsc( X) there is a net 00
" x ( n,n ' f is lsc r+ f>. E Coo(X, JR.) such that f>. t f. c) If f �f � +l J n= CX) but for no net A 3 ), r+ f>. E Co(JR., JR.) is f>. t f valid. A 3 ),
[Hint: a) { x
�f
n { x : f>. (x) � a } . AEA
f (x) � a } =
=
b) If S {
clef cl=ef ), r+ ] 1.7.25 Exercise. If X �f { (x, y) : x E JR., O :::; y E JR. } (the upper half plane) , a topology T for X is defined by the following base B: B is in B iff either for some positive r, B �f { (x, y) : (x  a ) 2 + (y  b) 2 < r2 :::; b2 } 2 2 2
or B = {(a, O)}U { (x, y) : (x  a) + (y  b) < b } . Thc space X is sep arable but not second countable.
1. 7.26 Exercise. a) If 1.2.19a)d) are taken as the axioms for a closure operation and a set is defined to be closed iff A = the set of comple ments of closed sets satisfies the axioms 1.2.1 for open sets. b) (S. Mrowka) The axiom d) and the added axiom B= A B imply a) c).
A
Au
A, Au u
T)
1.7.27 Exercise. A topological space (X, is a Hausdorff space iff: a) each filter contained in !fj(X) converges to at most one point; b) each net A X converges to at most one point; c) the intersection of the set of all closed neighborhoods of a point x is x itself. 1. 7.28 Exercise. If X is a locally compact space, y tJ. X, and
r+
�f Xu{y}, K E K(X) } u O(X) is a base of neighborhoods for a X*
then: a) { X* \ K : topology for X*; b) if X is a Hausdorff space so is X*; c) in the given topology, X* is compact (X* is the onepoint compactification of X) .
=
1. 7.29 Exercise. If X JR. and X* is the onepoint compactification of X, for no f in C (X*, JR.) is fl id : there is no continuous extension of sin : X 3 x sin x to X*, v. 3.7.19.
r+
x
=
Section 1. 7. Miscellaneous Exercises
43
Fe
1. 7.30 Exercise. If X is a complete metric space, C(X, q , and for each x in X and each f in sup If(x)1 < 00, then on some nonempty open JET subset of X, sup If(x) 1 < 00. xEV fEF O [Hint: For some n in N, x : sup If(x) 1 ::;; n "1 0.] JET
F,
V
}
{
1. 7.31 Exercise. If f E JR.lR , N 3 k > I, and for each y in JR., # [Jl(y)] = k, f is not continuous. [Hint: The intermediate value property of continuous functions applies.] 1. 7.32 Exercise. a) If
S then:
�f {xn } nEN C JR., L �f nlim x , l �f lim x n , E > 0, + CXJ n n + CXJ
# ({ n : Xn > L  E }) No , # ({ n # ({ n : Xn < l + E }) = No , # ({ n S· c [l, L] . =
Xn > L + E }) < No , X n < l  E } ) < No ,
What are l, L, S· , sup (S· ) , and inf (S·) when, for n in N, X n ( I) n n, Xn n (( 1) n  1) , Xn n ((I) n + 1) and ( 1 )'n (  1 )m Ym if n = 2m + I ?. , Zm   1 + , Xn  Zm Ym  1 + if n = 2m m m
�f
�f
�f �f �f { b) If n : A JR. is a net, L �f lim n(>.) resp. l �f lim n(>.) , and E > 0: bl) A EA A EA r+

_
_

for each f.l in A and some >. such that >. > f.l , n(>.) > L  E; for some f.l in A, if >. > f.l, then n(>.) 'I n(f.l); b2) for each f.l in A and some >. such that >. > f.l, n(>.) < l + E; for some f.l in A, if >. > f.l, then n(>.) A n (f.l) . For a subset S of JR., N (S)
�f { x
: x E S n (0, 00), for infinitely many n in N, nx E S } .
1. 7.33 Exercise. If S c [0, (0) and S is open and unbounded, then N (S) = [inf(S), (0 ) . [Hint: For m in N, if Sm { x : m x E S }, Sm is open, U Sm
�f
is dense in [inf(S), (0), and N (S) =
00
00
n U
n= l Tn = n
Sm; 1.3.6 applies.]
2
Integration
2.1. DaniellLebesgueStone Integration
The function I I : JR. 3 x r+ Ixl is in C(JR., JR.) , whence inf(x, O)
1
1
=
and sup(x, O) = "2 (x + Ixl) depend continuously on x. Therefore
"2 (x

Ix l )
L �f C([a, b] , JR.) is a function lattice, more particularly a vector space closed with respect to the operations inf (1\) and sup (v) : {f, g E L} ::::} {f 1\ g, f V g E L} . Furthermore, if fn ..l 0 , then fn � 0 (cf. 1.2.46 and 1.2.48) .
1
b
For the Riemann integral the map 1 : L 3 f r+ 1( 1 ) �f is a nonnegative linear functional, i.e., '
b
1 f(x) dx
{f 2': O } ::::} {1( 1 ) 2': O} (1 is nonnegative), 1(0'1 + (Jg) dx od(l ) + (J1(g) (1 is linear). =
(If f ::;; g then 1( 1) ::;; 1 (g) : 1 is a monotonely increasing functional.) Furthermore, 1 .2.46 implies
{fn ..l O}
::::}
(2.1.1)
{1 (In ) ..l O} .
The purpose of this Chapter is to develop a general theory of inte gration based on the paradigm above, i.e., a nonnegative linear functional 1 defined on a function lattice L and subject to the condition For a set X, L denotes a function lattice in JR. , i.e., a) x
(2.1.1).
{f, g E L} ::::} {J 1\ g, f V g E L}
(L is a lattice) ; b) when f, g E L and a E JR., then O' f E L; c) for a in JR., the conventions ±oo ± a = a ± 00 = ±oo, and a · ±oo 44
=
{
0 if a = 0 ±oo if a > 0 =t=oo if a < 0
Section 2.1. DaniellLebesgueStone Integration
45
are observed and when {f, g} C L
f + g : X 3 x r+
{� (x) + g(x)
iff f(x) + g(x) i ±oo + ( =F oo ) otherwise.
Hence L is a vector space and f + ( f) is the additive identity of L: f + ( f) == o. The following discussion takes place in the context of a set X , a func tion lattice L contained in x ffi. , a nonnegative linear functional 

I : L 3 f r+ I (f) E ffi.. Thus: a) {f 2': O} ::::} { I(f) 2': O}; b) when {a, b} C ffi. and {f, g} C L, then
I ( af + bg)
=
aI(f) + bI(g).
The crucial added assumption about I is:
A functional I for which the preceding obtain is a DaniellLebesgueStone functional, abbreviated as DLS functional. The next results lead to a function lattice L 1 such that L C L 1 C JRx and to an extension of I from L to a nonnegative linear functional
Various forms of abstract integration are special cases of the general context provided. The associated concepts and theories of measure and measura bility (v. Section 2.2) are derivable as well. The motivation for what follows is the improper Riemann integral. For example, when 0 < a and
{
a F(x) clef x  if x > 0 if X = 0 1 the determination of the improper Riemann integral F(x) dx (improper because F is unbounded and JRvalued on [0, 1]) is made by evaluating the 1 1 proper Riemann integrals F(x) 1\ n dx �f Fn (x) dx, n E N, and cal culating the result as n t oo . The sequence {Fn } nEN increases monotonely 1 which implies that nlim Fn (x) dx exists in JR. According as a < 1 or + CX) a 2': 1 the limit is in ffi. or ( in JR) . F : [0, 1 1
3 X r+
1 1 0
00
=
00
1
1
46
Chapter 2. Integration
Limits of monotonely increasing sequences of continuous ( hence Riemann integrable ) functions provide a larger class of functions to which an extension of the Riemann integral is definable. In the context of L , when L 3 In t I, the analogous procedure is to sup I (fn ) . The set of functions that are the lim I (fn ) define l(f) n+ CXJ n limits of monotonely increasing sequences drawn from L is denoted Lu . Since I maps L onto JR., if L 3 In t i E Lu , then
[=
�f
]
00 < I (fn ) < 00 and  00 < l(f)
::;;
00.
As 2. 1 . 2  2.1.4 show, if {In } nEN c L u and a 2': 0, then In + fm E L u , aln E Lu and, if In t I, then I E Lu . However, although F in the paradigm above belongs to the corre sponding Lu, F does not since no function continuous on [0, 1] lies below  F: Lu is not necessarily closed with respect to multiplication by nega
�f
tive constants. ( If X consists of a single element and L JR.x ( JR.) , then x Lu = JR. , which is closed with respect to multiplication by arbitrary real constants. ) Owing to the preceding observations, there is a temptation to extend L to the set of limits of all monotone sequencesnot only monotonely increas ing sequences. However, the goal of the DLS construction is achievedand more economicallywithout resorting to the more elaborate procedure. =
2.1.2 THEO REM. IF {In } nEN , {gm } mEN C L, AND In t I, gm t I, THEN
PROOF. Since In ::;; I
=
lim gm , m+ =
Hence lim I (gm ) 2': nlim + I (fn ) . The argument is symmetrical in the pair + CXJ {{I" } nEN , {gm }mEN } · D [ 2.1.3 Note. Thus if Tn
ex)
L 3 In
lim I (fn ) , t I and l(f) �f n+ =
Section 2 . 1 . DaniellLebesgueStone Integration
47
then 1( 1 ) ( in ffi:!) is independent of the sequence {fn } nEN . Hence 1 is a map from Lu to ffi: .] 2.1.4 THEOREM. FOR f af E Lu , AND 1( 1 + g) . AND semzhomogeneous; b)
=
AND g IN Lu AND a IN ffi.+ : �) f + g E Lu , 1( 1 ) + 1(g) , 1(af) = afJ , I.e., 1 IS ADDITIVE f 1\ g E Lu ; c ) V
{f � g}
'*
{ 1(1 ) � 1(g) } ; �
�
d ) IF Lu 3 fn t f, THEN f E Lu AND 1 (In ) t 1( 1 ) . PROOF. a) In L there are sequences {fn } nEN and {gn }nEN such that
1( 1 ) + 1(g)
=
fn t f, gn t g, fn + gn t f + g, 1 (In ) t 1( 1 ) , 1 (gn ) t 1( g), lim 1 (In ) + n�= lim 1 (gn ) = n�= lim 1 (In + gn ) n� =
=
1( 1 + g).
=
=
Since afn t af and 1 ( afn ) a1 (In ) , 1(af) a1(1 ) . 1\ 1\ b) If L 3 in t f, L 3 gn t g, and f(x) .::;: g(x) , then L 3 fn v gn t f v g · c) If L 3 fn t f and L 3 gn t g, then
{
fn (x) 1\ gn (x) fn (x) � fn+1 (X) = fn+1 (X) 1\ gn+1 (X) if fn+1 (X) .::;: gn+1 (X) , <  gn (X) '::;: gn+1 (X) = gn+1 (X) 1\ fn+1 ( :r ) if gn+1 (X) .::;: fn+1 (X) , whence fn 1\ gn � fn+1 1\ gn+1 . If fn 1\ gn t k, then k .::;: f 1\ g. If then for large n, k(x)
<
k(x)
<
f(x) 1\ g(x),
fn (x) 1\ gn (x) .::;: k(x), a contradiction: fn 1\ gn t f 1\ g.
=
limex) 1 (gn ) t 1(g) . Hence 1( 1 ) nlim + CXJ 1 (gn 1\ fn ) '::;: n+ d ) For n in N, L contains a sequence {gmn },.nEN such that
�f
Hence, if kn sup { % : 1 .::;: i, j '::;: n }, then L 3 kn :::; kn+1 .::;: fn+1 :::; f. For some k in Lu, kn t k and k :::; f. If m .::;: n, then gmn .::;: kn , whence for
Chapter 2. Integration
48
= limCXJ gm n :::; lim each m, and so I = k (E Lu) . Furthermore, since t k = I,
=
n+ kn
fm n+
ex)
k
:::;
I · Thus I = limCXJ 1 :::; k :::; I Tn
+ m
kn 1 ( n ) t 1 ( ) 1 (f), 1 ( kn) � 1 (fn) � 1 (f), whence 1 (fn ) t 1(f ) . The next development is the basis for the construction of L 1 , the cen �
k
=
k

�
�
D
tral aim of the discussion to this point. x
2.1.5 DEFINITION. FOR I E ffi. , WHEN { U
U E Lu , U 2': I } = 0,
00 .
1(f) �f WHEN { U : U E Lu , U 2': I } 1 0, 11 �f inf { 1(U) U E Lu , U I } . FURTHERMORE, l( f) [1( f) ] . THEOREM. FOR THE FUNCTIONAL 1 : a) {I � g } {{1(f) 1( g) } 1\ { l (f) � I(g) }} , {a 2': O} {1(af) a1(f) } , i.e., 1 AND ARE NONNEGATIVE AND semihomogeneous ; b) 1(f + g � 1(f) + 1( g) , resp. l (f + g H I) + l g , IF THE RI.9 HT =
2':

2.1.6
'*
)
'*
�
=
I
( )
) 2':
MEMBERS OF THE PRECEDING INEQUALITIES ARE DEFINED, i.e., 1 , IS ESSENTIALLY SUB ADDITIVE resp. I IS ESSENTIALLY SUPERADDITIVE; c) l ( f ) � d) { I E Lu} '* l (f) = 1 ) } ; e)
{
1(f) ; { ) 1(f
=
(f
{ � In}} { �1(fn)}; '*
{O � In } 1\ I �f
1( 1 ) �
PROOF. a) If I � g and Lu 3 U 2': g , then U 2': I , whence � 1(g) . If 1 2': } and ) = 1 ) . Hence 2': 0, then {Lu 3 2': f} {} {Lu 3
a
aV af 1( af) = a1(f) .
V
I (aV a(f)( V
b) If I � h E Lu , g � k E Lu , then I + g :::; h + k E Lu and )
1
1(f + g � (h +
k)
=
1 h) + 1(k) ,
(
49
Section 2.1. DaniellLebesgueStone Integration
whence 1 (f + g) :s; 1 (f) + 1 (g) if the right member of the inequality is de fined; the inequality l (f + g) 2: l ( 1 ) + l (g) follows from similar arguments. (When L C([O, IJ , ffi.) , I is the Riemann integral, and
�f
r+
f : [0, 1]
3
g : [0 , 1]
3
x { 00x1 If�f xx > 00 x { x00 2 if1· f x > 00 ' =
r+
X
=
then 1(f) = 00 and 1(g) = 00, whence 1(f) + 1(g) is not defined. ) c ) If 1(f) 00, the result is automatic. If 1(f) < 00, then =
1 (f) + 1 (  I )
is defined, whence
0 = 1 (0) = 1 [1 + (f)]
:s;
1 (f) + 1 (  I ) = 1(f)  [ 1(  I )]
=
1 (f)  l (f);
d ) Since f :s; f, if f E Lu, then 1 (f) :s; J(f) . If f :s; g E Lu, then
J(f)
:s;
J(g),
whence J(f) :s; 1(f) : 1(f ) = J(f) . Owing to the linearity of I and the last equality, if g E L, 1( g) = J( g)
=
I( g) = I(g).
Thus l (g) = 1(g) =  [I(g)] = I(g) . If f E Lu and L 3 gn t f, then 1 (gn ) = I (gn ) :s; J( f) and so
l (1 )
:s;
1(f) = J(f)
=
I nlim + (gn ) ex)
=
lim+ CXJ 1 (gn ) n
:s;
l (f) :
1(f) = l (f) = J(f) . e) If any 1 (fn ) = 00, the implication is automatic. If each 1 (fn ) is finite and E 0, then for some gn in Lu,
>
Moreover, by virtue of 2.1.4d ) , g
J(g) =
00
2:
n= 1
00
�f 2: gn E Lu and
J (gn )
n= 1 :s;
00
2: 1 (fn ) + E.
n= 1
50
Chapter 2. Integration
Furthermore, g 2':
00
I and hence 1(1) :::; I(g) :::; nL= 1 1 (In)
+ E.
D
[ 2.1. 7 Note. Although the implication
{ I 2': � In}
*
{
1(1 ) 2': 1
(t,ln) }
is valid, the implication
is not.
] x
I IN IS IN L 1 IFF 00 < (I ) 1 (1) < 00. FOR IN L 1 , J (I ) IS THE COMMON VALUE OF (I) AND 1 (I ) . THE I MAP J : L 1 I J ( I) IS THE DANIELLLEBESGUESTONE (DLS ) FUNC 2.1.8 DEFINITION. A FUNCTION
ffi.
=
I
3
1
r+
TIONAL. ( If I E Lu , then 1( 1 ) > 00. Hence, if I E Lu and 1( 1 ) < 00, then 00 < 1( 1) l( l ) = 1 ( 1 ) 1( 1 ) < 00, i.e., if E Lu and 1(1) < 00, then I E L1 . ) The properties listed in 2.1.6 for 1 lead to =
I
=
2.1.9 THEOREM. a) L C L 1 AND I; b) L 1 IS A VECTOR SPACE AND IS LINEAR ON L 1 ; c ) I E L 1 IFF FOR EVERY POSITIVE E AND SOME g AND h IN Lu , h :::; g AND I[g + (h) < E; d ) L 1 IS A FUNCTION LATTICE; e ) THE MAP IS A NONNEGATIVE (LINEAR) FUNCTIONAL ON L 1 ; f ) IF L 1 3 t AND FOR ALL n , :::; M < 00 , THEN E U AND t CONVERSELY , IF I E L 1 AND L 1 3 t THEN
JI L =
J
J (ln) J(I) . In I
]
I J:::;
J (In) In I , l J (In) J (I) . PROOF. a If I E L and gn clef I, E then L gn I , whence I E Lu: L Ln · Furthermore, 1(1 ) 1( 1 ) : Il L = I, whence 00 1 ( 1 ) = 1( 1 ) l(l) 00 : I E L 1 . A similar argument shows that JI L = I. t
C
)
= =
>
n
=
3
N,
>
t
Section 2.1. DaniellLebesgueStone Integration
51
L l , C E JR., then { l1((cfcl)) == inf1Lu(3h?:cl) f=J(ch)[c1(1)] = c infLu3h?:f J(h) = c1 (1) if c 2': 0 = c1 (1) = cl ( l ) if c < 0 ' whence cf E L I . If {f, g } e L I , then the preceding argument and the subadditivity of 1 imply 1 ( 1 + g) � 1 (1) + 1 (g) = J(I) + J(g), l(f + g) = 1 ( f g) �  1 (1)  1(g) = J(I)  J(g ) , 1 (1 + g) 2': l ( l + g) 2': J(I) + J (g) 2': 1 (1 + g), whence J is a linear functional on L l . c) If f E L l and E > 0, then for some g in Lu, g 2': f and E I(g) < J(I) + 2 . E Similarly for some h in Lu, f � h and I(h) < J(f) + 2 . Hence h :::; f :::; g and J[g + (h)] = J(g) + J(h ) :::; J (I) + J(I) + E = J(O) + E E. Conversely, if there are g and  h as described, then, as noted after the introduction of Lu, for any k in Lu, 00 < J(k) . Since I(g + (h) = I(g) + I(h) < E, both I(g) < 00 and I( h) < 00. Hence 1 (1 ) � l(g) < 00, 0 :::; 1 (1)  l ( l ) = 1 (1)  [  1 ( I)] , � J(g) + i(h) = J[g + (h)] < E. d) If Lu 3 h i :::; Ii :::; gi E Lu, i = 1, 2, then b) If f E


=





e L I and gi and hi , i = 1, 2, are chosen as in c) , then  ( h l � h2 ) h I �  h2 E Lu, J (g
When {h , h }
=
52
Chapter 2. Integration
1\ whence h h E L 1 . V e) Since I is nonnegative on L, I is nonnegative on Lu . Since JI J is nonnegative on L 1 . f) If g
00
L
u
=
I,
�f I  h , then g � 0 and g = L ( In+ 1  In) . The subadditiv
(In)n=1:s; M < 00 imply lim J ( In)  J ( ld , J ( In+1  In) n+CXJ l (g) n=1 1 ( 1 ) 1 (h + g) J ( ld + l (g) :S; nlim J (In) < 00 , H I) � ( In) J ( In) , ( I ) � nlim J ( In) � 1 ( 1) , whence I E L 1 and J (I ) nlim J ( In) . If I E L 1 , then, because J is nonnegative and 0 :s; I  In E L 1 , O :S; J (I  In) J (I)  J (ln) : J ( In) J ( I) < 00. The argument in the previous paragraph applies. ity of I, 2.1.6d), and the hypothesis J 00
=
:s; '" �
=
:s;
I
=
I
...... oo
=
...... oo
...... oo
=
:s;
D
[ 2.1.10 Note. The result 2.1.9f) is Lebesgue 's Monotone Con vergence Theorem as it applies to the functional J.] 2.1.11 DEFINITION. THE SET Lui CONSISTS OF ALL g SUCH THAT FOR CONTAINED IN Lu , gn ..l g, 00 < lim I(gn ) , SOME SEQUE NCE ...... AND 1 ( g d < 00. 2.1.12 Exercise. Lui C L 1 . [Hint: If g E Lui and gn are as described in 2.1.11, for each n,
{gn}nEN
n
oo
Lebesgue's Monotone Convergence Theorem applies.] Although any function considered in the preceding development is ffi: valued, some of the functionals introduced are JR.valued and others are ffi:valued. In sum: Any function in L, Lu , L 1 or Lui is ffi:valued. The functionals I and J are JR.valued. The functionals I and 1 are ffi:valued. •
•
•
Section 2.1. DaniellLebesgueStone Integration
53
2.1.13 THEOREM. AN I IS IN U IFF FOR SOME g IN Lui AND SOME NONNEGATIVE FUNCTION P IN L l , J (p ) = 0 AND 1 = g  p. PROOF. If I = g  p as described, then 2.1.9 shows I E L l . If I E L l , then 1 (1 ) E ffi. and for n in N, and some In in Lu , In 2': I and 1 ( 1n ) < 1 ( 1 ) Thus { gn h /\ . . . /\ In } nE is a sequence such
+ .!. .
�f
n
that for some g in Lui , gn ..l g. Moreover, I(g)
J(p ) = 0, and I = g  p.
=
N J(I) , 0 .:s: g  I cle=f p E L ,
2.1.14 Exercise. a) (Fatou's Lemma) If 0 .:s: In E L l , n E N, then
(
)
J lim In .:s: lim J ( In ) . n + CXJ n + CXJ b) If Iln l .:s: I h l ,
n
2': 2, then J
( nlim+CXJ In) 2': nlim+CXJ J (ln ) .
The functional J is an emphasizer. It makes inferior limits more inferior and superior limits more superior.
[Hint: a) Only if lim J (In ) <
n
...... =
00
is an argument needed. If
. In , k E n , and gkn clef h /\ . . . /\ h+n, t hen gk cle=f nmf >k 1').T
=
2.1.12 implies gk E L I , k E N, and gk t lim In . Furthermore,
n
...... =
whence Lebesgue's Monotone Convergence Theorem [2.1.9f)] ap plies. b) The mnemonic lim ( · ) =  lim ( · ) and 2.1.14a) apply.] 2.1.15 Exercise. (The Dominated Convergence Theorem) If
and nlim +CXJ In
�f I , then I E L l and nlim +CXJ J (In ) = J(I) .
[Hint: The results in 2.1.14 apply.]
Lebesgue's Monotone Convergence Theorem, Fatou's Lemma, and the Dominated Convergence Theorem are central features of the DLS functional J. Each result can be described crudely by the mnemonic: lim J J lim. =
1
D
Chapter 2. Integration
54
In particular, Fatou's Lemma is an extension of the fundamental in equality { { {fn} nE N C 1\ {fn ..l O } ::::} { I (In) ..l O } with which the devel opment began. 2.1.16 Exercise. When X � JR., is the set of JR.valued functions f such that supp (I) is compact and f is Riemann integrable on an interval [a, b] containing supp (I) , and I is the Riemann integral, if {fn } nE N C and fn ..l 0, then fn (x) dx ..l o.
L}
}
L
L
J
[Hint: Since fn t o , 2.1.19f) applies.] When X
2.1.17 Exercise.
I:
L
3
00
f r+ L f(n) , then: a)
�f N, L �f Coo(N, JR.) , and I is defined by U
is the set of (real) sequences {a }
n nEN n= 1 such that L l an l < 00; b) the Dominated Convergence Theorem implies n= 1 that if L an is an absolutely convergent series and is an autojection n= 1 (permutation) of N, then L a n) L a . n= 1 n= 1 2.1.18 Exercise. If {fn } n EN C C([O, 1] , JR.) , sup Ifn(x) l :::; Mn , and xE [0, 1j 00
00
00
7r (
=
00
11"
n
00
L Mn < 00,
n= 1
00
L1 ) . n= 1 For a set E, the characteristic function of E is (x) �f {I if x E E . :
then f [0, 1]
3
x r+ L fn (x) is continuous (and in x
E
0 otherwise
�f
L
2.1.19 Exercise. a) If X JR. and is the set of all real linear combi nations of characteristic functions of bounded rightopen intervals, then
L
N
N
 ) a n= 1 nX [an ,bn) n=L1 an (bn an is a D LS functional. b) If X �f JR., L �f Coo (JR., JR.) , and I is the Riemann integral, then I is a DLS functional. c) The L 1 created via a) is the same the L 1 created via b). I
as
:
3
L
r+
55
Section 2.2. Measurability and Measure
�f
2.1 .20 Exercise. a) If X JR.n and L is the set of all real linear com binations of characteristic functions of halfopen ndimensional intervals,
then I : L
3
� an [ X�=l[an , bn )] � an 11 (bn ft
x
N
�f
N
N

an ) is a DLS func
�f
tional. b) If X JR.n , L Coo (JR.n , JR.) , and I is the nfold iterated Rie mann integral, then I is a DLS functional. c) The L 1 created via a) is the same as the V created via b) . 2.1.21 Exercise. a) For an arbitrary set X, a in X, and the function sublattice L of JR.x , : L 3 f ft f( a), the Dirac functional at a, is a DLS functional. b) What are Lu and V ? 2.1.22 Exercise. a) For a set X in the discrete topology !fj(X) , Coo (X, JR.) is a function lattice Ld. b) The functional L : Ld 3 f ft L f(x) is a DLS
Ja
xE X
functional. c) What are (Ld) u and L�? (The vector lattice L� is usually denoted £ 1 (X) .) 2.2. Measurability and Measure
The works of Lebesgue and Caratheodory in the field of measurable func tions, measurable sets, and measures (of measurable sets) are natural de velopments from the DLS functional. In the DLS development of these concepts, measurability of functions is introduced first. It takes a useful form in terms of the following 2.2.1 DEFINITION. WHEN { a, b, c} c JR.,
(a 1\ c) V (b 1\ c) mid (a, b, c) E ffi.. CORRESPONDINGLY, FOR FUNCTIONS f, g, h (IN JR.X ) AND x IN X , mid (a, b, c)
�f (a 1\ b)
V
mid (I, g, h ) (x)
�f
�f mid [J (x) , g (x) , h (x)].
[ 2.2.2 Note. Although mid (a, b, c) E { a , b, c} , mid (I, g, h) is not necessarily in {f, g, h } , e.g., if X
=
JR., f(x)
=
lx i , g(x) = I , h(x)
=
3

Ixl .]
2.2.3 Exercise. a) mid ( a, b, c) (a V b) 1\ (a V c) 1\ (b V c) ; for functions f, g, h, if g :s: h, then g :s: mid (I, g, h) :s: h. b) A function lattice is mid=
closed. c) If L C jR x and L is closed with respect to addition and multi plication by scalars, then L is a function lattice if some nonzero constant
56
Chapter 2. Integration
function is in L, each element of L is bounded in absolute value, and L is midclosed, i.e., {{f, g, h} c L} ::::} {mid (I, g, h) E L}. [Hint: c) If If I :::; M then If I = mid (M, f, f) .] For a set X, if a E ffi. \ {O} and {J, g, h , p , fn }
2.2.4 Exercise.
then: a)
{ mid (af, g, h)
(
=
� , �) } ,
a mid f' p + mid (I, g , h) = mid (p + f, p + g, p + h) , {h :::; h } ::::} {mid (h , g, h) ::;; mid ( h , g, h) } , mid ( h � h , g, h ) mid (h , g, h) � mid ( h , g, h) , =
mid (nlim fn, g , h ) + CXJ b) If g(x) :::; h(x) , then
=
x ffi. ,
(2.2.5) (2.2.6) (2.2.7) (2.2.8)
mid (In , g, h) . nlim + CXJ
{
C
(2.2.9)
h 1\ f(x)  g 1\ f(x) h(x)  g(x) if mid (I, g, h) (x) = h(x) (2.2.10) = mid (I, g, h) (x) f(x) . � (x)  g(x) ifotherwise c) The functional mid is not additive: for some {f, g, h, p , q} contained in x JR , mid (p + q, g, h) i mid (p, g, h) + mid (q , g, h) . =

2.2.11 DEFINITION. FOR A FUNCTION LATTICE L CONTAINED IN ffi.x AND AN f IN ffi.X , f IS DLS measurable, i.e. , f E D(L) , IFF
{L 3 g :::; h E L} ::::} {mid (I, g, h) E L} . A motivation for 2.2.11 derives from consideration of the functions if x E � f : [0, 1] 3 x r+ { I otherw Ise o and when 0 < a, F [0, 1] 3 x r+ F(x) d�f { �:
a
if x > 0 if 0 X =
considered in the development leading to 2.1.2 . Although f is bounded, it is not Riemann integrable (every upper Riemann sum is I , every lower Riemann sum is 0). If g(x) � 2x and h(x) � 3x, then g ::;; h, g and h are Riemann integrable, but mid (I, g, h)
Section 2.2. Measurability and Measure
57
is not Riemann integrable. Thus if L is the function lattice consisting of functions Riemann integrable on [0, 1] , f fails to belong to L not because f is unbounded but because the part of it that is trapped between g and h behaves badly with respect to Riemann integration: f is not Riemann measurable.
On the other hand, F is unbounded and yet, for the L where F is in troduced, if L 3 g ::;; h E L, then {mid (F, g, h) , mid (F, g, h) } c L so that ±F may be viewed as Riemann measurable. In what follows, for the particular function lattice L l , V ( L 1 ) is denoted
V.
[ 2.2.12 Note. Owing to 2.2.4, the spaces are midclosed. In particular, l C V.]
L
Lu, Lui, L l , and V
V E L l }}. E V} E LI } {J E V} {} { { 0 ::;; g E L l } { mid (I, g, g) E L l } } .
2.2.13 THEOREM. a) THE SET IS A FUNCTION LATTICE AND A monotone c) 1\ { I f I class of functions. b) {J {} {{f '*
PROOF. a) By virtue of (2.2.8) and 2.1.9d) , and L l 3 g ::;; h E L l , then for n in N,
V is a lattice.
If h , 12 E V
cPin �f mid [Ii , n( l g l + Ihl ) , n( lgl + I h l ) ] E L l , i = 1, 2, cl cPn ef ml· d ( cP l n + cP2 n, g, h) E L 1 , g ::;; cPn ::;; h, and cPn t mid (h + 12, g , h) . =
I Then 2.1.9f) implies mid (h + h , g, h) = nlim + oo cP" E L : h + 12 E V. If a i 0, (2.2.5) implies a h E V: V is a function lattice. If L l 3 g ::;; h E L l and V 3 fn t f, then mn �f mid (In , g, h) E L I and 00 < J(g) ::;; J (mn ) ::;; J(h) < 00. The conclusion follows from (2.2.9). If V 3 fn ..l f a similar proof applies. Thus V is, by abuse of language, lim, lim, and limclosed. b) The result follows from the formula: mid (I,  I f I , If I ) f · c) '* : L l is a vector space. {=: : If L l 3 P ::;; q E L l , (2.2.6) implies =
(
p+q mI. d q p q p ' f,  + 2 2 2
Hence f + p+2q
)
=
(
)
mI. d f + p+2q , P , q .
E V. Since L 1 e V, a) implies f E V.
D
58
Chapter 2. Integration
2.2.14 THEOREM. A NONNEGATIVE f IS IN V: a) IFF FOR ALL g IN L 1 , f /\ g E £ 1 ; b) IFF FOR ALL h IN L , f /\ h E L 1 . PROOF. a) Since L 1 C V, if 0 :s; f E V and g E L 1 , then g E V and 2.2.13a) implies f /\ g E V . Since I f /\ g l :s; I g l E L 1 , 2.2.13b) implies f /\ g E L 1 . Conversely, if 0 :s; f and {g E L 1 } ::::} f /\ g E L 1 , the conditions
f /\ h E L 1 , and g /\ h ( = g) E L 1 , whence (f /\ g) (f /\ h) (g /\ h) mid (f, g, h) E L 1 :
imply f /\ g E L 1 ,
V
V
f E V.
=
b) If 0 :s; f E V and h E L , then h E L 1 a) implies f /\ h E L 1 . Conversely, if 0 :s; f and for all h in L, f /\ h E L 1 , then, since L 1 is a lattice, 2.1.9f) implies that if k E Lu and I(k) < 00 , then f /\ k E L 1 and that if q E Lui, then f /\ q E L 1 . If g E L 1 , then, according to 2.1.13, for some q in Lui and some nonnegative r in L 1 , g = q  r and J (r) = o. Since g :s; q, as in (2.2.10), :
q
/\ f (x )  g /\ f(x) q( x )  g (x ) r (x ) 2': 0 f (x )  g (x ) 2': 0 =
{
=
o
if q (x ) :s; f (x ) if g (x ) :s; f (x ) if f (x ) :s; g (x )
Owing to (2.2.15), 0 :s; f /\ q  f /\ g :s; r. Since J (r)
=
:s;
q (x ) . (2.2.15)
0,
1 (f /\ q  f /\ g) :S; 1 (r) = J (r) = 0, i.e., f /\ q  f /\ g E L 1 . However, f /\ q E L 1 and so f /\ g E L 1 . Thus a) implies f E D 2.2.16 DEFINITION. A SUBSET 5 OF !fj(X) IS A ring resp. aring IFF 5 o :s;
V.
IS CLOSED WITH RESPECT TO THE FORMATION OF SET DIFFERENCES AND FINITE resp. COUNTABLE UNIONS, i.e.,
C s } ::::} {AI \ A2 E S} , {N E N} ::::} � An E 5 resp. 'v An E S. N 1 A aRING 5 IS A aALGEBRA IFF X E S. { {An } nEN
{
}
For any set X and a subset E of !fj(X) , aR(E) is, the aring generated by E, i.e., aR(E) is intersection of all arings containing E.
59
Section 2.2. Measurability and Measure
X is a topological space, 5,(3(X) �f a R[O(X)] is the set of all X. When f E JR.x , a E JR., and is one of <, :S;, >, 2: , 1, Eo (f, a) �f { x : x E X, f (x) a }. When
Borel subsets of
0
=,
0
(Hence
Eo/(f, 0) supp (f).) When 5 is a aring in !fj(X ) an f in JR.x is by definition 5measurable iff r 1 (5,(3) n Eo/(f, 0) c 5. ( The presence of Eo/(f, 0) in the formulation is related to the possibility that X tJ 5, i.e., that 5 is not a aalgebra. ) When f E y X , T is a aring in !fj(Y) and E o/(f, 0) is meaningless, f is (5, T)measurable iff f l (T) C 5. In particular, when X is a topological space and 5 = a R[O(X)] [= 5,(3(X)], an f that is (5, T ) measurable is Borel =
measurable.
[ 2.2.17 Note. It is convenient to observe the convention whereby sansserif fonts, e.g., 5, denote subsets of !fj(X) and cursive fonts, e.g., S, denote subsets of JR.x or x JR. . For example, when 5 is a aring ( cf. 2.2.16 ) contained in !fj(X), the set of functions that are in JR.x or JRx and are 5measurable is denoted S.]
[ 2.2.18 Remark. Since X E F(X), 5,(3(X) = a R[F(X)].] 2.2.19 Exercise. In !fj (JR.n ), 5,(3 (JR.n ) is the aring generated by: a ) the
set of all open balls; b ) the set of all closed balls; c) the set of all halfopen intervals; d ) the set of all cubes. A set is a DLS measurable subset o X iff X E E D . The set of DLS measurable subsets of X is denoted D.
f
E
2.2.20 THEOREM. THE SET D IS A aRING AND IF 1 E D, D IS A ALGEBRA. PROOF. For C �f U A n , from the equations X ( AUB )
=
nEN X A V X B , X (A\B)
=
X (AUB)  X B , X C
=
a
V X ( A n)' nEN
and 2.2.13 it follows that D is a aring. When 1 E D, the formula X ( X\A )
implies D is a aalgebra.
=
1

XA
D
60
Chapter 2. Integration
f1
fE
2.2.21 THEOREM. I F JR.X , THEN [S,(3(JR.)] C D IFF FOR EACH REAL a, D. PROOF. If [S,(3(JR.)] c D, for each real a,
Edf,O') E f1
df, O' ) r 1 (00, 0') E D. Conversely, if each Edf, a ) is in D and if a < b, then f 1 ([a, b)) = Edf, b) \ Edf, a) E D. Since S,(3(JR.) aR ({ [a, b) : a < b, a, b E JR. } ) , r 1 [S,(3(JR.)] c D. E
=
=
D 2.2.22 Exercise. The assertion in 2.2.21 remains valid if E< is replaced by E<:: , E> , E� , or Eo/ (v. the last paragraph of 2.3.5 for a discussion of
E=).
E D,
f E D IFF { a E JR.} ::::} { E< (f, O') E D } . E D, it follows from 2.1.9b) and 2.1.9d)
2.2.23 THEOREM. I F 1 THEN PROOF. If then, since 1 that for in N,
f E D, n n(O'  f) E D, l /\ n(O'  f) E D, and [I /\ n(O'  f)] v O E D. Furthermore, = lim [1 /\ n(O'  f)] 0, whence 2.1.9f) implies j, o: ) n+= X
V
E< (
Edf, O') E D.
f f 0 f /\ 0 �f f++ + f  , whence f f f  (f  ). For k
Conversely, if JR.x , then = V + is the difference of the two nonnegative functions: and in N, if r::k is the characteristic function of
fE
n
=
n"2n k 1 cle then � r:: =f r:: t f + · For every a , � k 
k= O
0' < 0 . 0 Hence, if E< (f, a ) E D for every a , then f + E D. A similar argument applies to  j, hence to f: f = f + + f  E D. D + 2.2.24 Exercise. If I f I :s; M < 00, then f;t: � f and f;; � f  · 2.2.25 Exercise. If 5 is a aring in !fj(X) and f E JR.x , then f E S iff for each a in JR., Edf, a) n Eo/ (f, 0 ) E 5: f is Smeasurable iff for each a in JR., Edf, a) n Eo/ (f, 0) E S. if if a 2':
61
Section 2.2. Measurability and Measure
On D there is defined the JR.valued set function:
E L1
[0, 00].
if X E otherwise
{En} nEN is a sequence of pairwise disjoint subsets in D and E �f U nEN En , then E E D and " �X tX . n= l Lebesgue ' s Monotone Convergence Theorem ( 2.1.9f)) implies Thus ,AD) c
Furthermore, if
N
En
E
IL (U nEN En ) = f IL (En ) , n=l i.e., IL is a nonnegative countably additive set junction, that is to say, a measure. The triple (X, D, IL) is a measure space, i.e., a set X, a aring or a aalgebra D contained in !fj(X), and a measure IL : D [0, 00]. 2.2.26 Exercise. a) For a measure space ( X, D, IL) , 1L(0) 0 and if D 3 A C B E D, then IL(A) :::; IL (B). b) If {En}n EN C D, En ::) En+1 ' n En 0, and IL (E1 ) < 00, nEN then IL (En) ..l o . 2.2.27 Exercise. If 5 is a aring and ¢ : 5 [0, (0 ) is a finitely additive set junction, then ¢ is count ably additive iff r+
=
=
r+
[Hint: For n in N,
E1 = (E1 \ E2 ) l:J . . . l:J (En  1 \ En) l:JEn nEN U (En \ En+ d , n ¢ (Ed  L ¢ (Ek \ Ek + d = ¢ (En ) .] k= l For a measure space (X, 5, IL), a set A in 5 is a null set iff IL(A) = o . The set N �f { N : N C A E 5, IL(A) � O } consisting of subsets of null sets in 5, gives rise to the completion 5, consisting of all sets of the form (E \ Nd u (N2 ) , E E 5, N1 , N2 E N. For F �f (E \ Nd u ( N2 ) in 5, ji(F) �f IL (E). =
Chapter 2. Integration
62
5 X, 5, Ii)
5 5 5, 5
2.2.28 Exercise. If is a aring resp. aalgebra, then is a aring resp. aalgebra and ( is a measure space. (When and f.l) are complete.) For two functions f, in differs from zero only on f � iff f a null set. More generally, the notation a.e. is used to indicate that a statement is true almost everywhere, i.e., except (at most) on a null set. For example, lim f = f a.e. or f �. f means lim exists a.e. and lim f � f. A function f in is a null function iff = O. =
g Ll ,
n4 � n
n+= n
g
g
Ll
(X, 5,
n4� fn J( l f l )
n
2.2.29 Exercise. a) The relation � is an equivalence relation. b) If f is a null function, E ffi.x , and :s; f, then E and is a null function. c) An f is a null function iff f == O. d) The aring D is complete; e) D is a aalgebra if f E ::::} fE [Hint: For b) , 0 :s; :s; 0 applies. For c) , if :s; f :j:: 0, one may assume f V O :j:: O. Then for n in N,
g
g Ll
Igl
{ L} { 1 /\ L}. I( l g l ) I( l g l) J(f) E
n �f {
X
: fVO>
g
=
�}
applies. For d) , b) applies.] The quotient set 1 / � of �equivalence classes is usually identified itself. (When the identification is not made, 1 / � is sometimes with denoted £ 1 . In all the results that are of interest in the remainder of this and £ 1 is of no consequence. Thus, for book, the distinction between simplicity and ease of presentation, the distinction is henceforth ignored:
L
Ll
Ll
L l = £1 . )
2.2.30 Exercise. The map
L
Ll L l :1 ( f, g) d(f, g) (� J( l f  gl) �f I l f  gi l l is a metric for L l . 2.2.31 Exercise. In the context of the DLS construction, L is dense in L l metrized by d as described in 2.2.30. [Hint: The criterion in 2.1.9c) for membership in L l applies.] 2.2.32 THEOREM. METRIZED BY d AS IN 2.2.30, L l IS A COMPLETE METRIC SPACE. PROOF. For a Cauchy sequence {fn } nEN in L l , there is a sequence d:
x
r+
Section 2.2. Measurability and Measure
63
I lfnk  fnk , ll l
such that if k 2': 2, then < 2 k . Lebesgue's Monotone Con vergence Theorem (2.1.9f) ) implies that for some in
g Ll ,
K
I fn, I + L I fnk  fnk _ ' I t g .
[
k= 2
]
lim fnK f: f E D f, K4= lim fn, + t ( Ink  fnk_,) = K4= k= 2 and since I fnK I ::;; I g l , the Dominated Convergence Theorem (2.1. 15) implies f E L l and lim Ilfn K  fil l o . Consequently, for any in N, K4= Thus for some
=
=
n
Ll ,
Since is a Cauchy sequence in for large n and large K, I + II is small, i.e., n4= lim D = o. is a set of pairFor a measure space A, when wise disjoint elements of A and L C JR., the linear combination
{fn} nEN
Il fn  fi l l (X, p,), {En} < an � n� N l< n N { N N L an x En �f s is a simple function. When L anp, (En ) is meaningful, n= l n= l N Ix s(x) dp,(x) �f � anp, (En) [= J(s)]. If 0 ::;; f E D,
Ix f(x) dp,(x) �f sup { Ix s(x) dp,(x) f D, f f
f
:
s simple and 0 ::;; s ::;; f } .
For in V 0 + 1\ 0 and, when the right member below is mean ingful, i.e., when at least one of the terms in the right member is finite, =
Ix f(x) dp,(x) �f Ix (I V O)(x) dp,(x)  Ix (1 1\ O)(x) dp,(x). An f in D is integrable iff 00 < Ix f(x) dp,(x) < 00. The set of integrable l functions is denoted
U(X, p,). For f in L (X, p,) and E in 5,
[ 2.2.33 Note. The distinction between L l (X, p,) and ,C l (X, p,), the set of equivalence classes with respect to the relation � defined according to {f � g} {} {Ix I f  g l dp, = O } , is usually ignored.
Chapter 2. Integration
64
L 1 (X, p,) 1 L
The notation emphasizes the involvement of the measure used in the DLS construction suggests p" whereas the notation that there is no a priori measure used to define The next results sort out the relations between and
1 1L (X, p,) LL.1 .]
p, IS THE MEASURE DEFINED ON D VIA THE L L, AN f IN D IS IN L 1 (X, p,) IFF f E L 1 , IN WHICH CASE Ix f(x) dp,(x) = J(f) . PROOF. According to 2.2.13b) , {J E L 1 } {} {(f E D) 1\ ( I f I E L 1 ) }. Since f f+ + f  and I f I = f +  f  , the discussion may be confined to the set of nonnegative functions. If 0 :s; f E L 1 , then, since L 1\ 1 C L, as in the PROOF of 2.2.23, there is a sequence {fn } nEN of nonnegative simple functions such that fn t f · Lebesgue's Monotone Convergence Theorem implies 2.2.34 THEOREM. (Stone) IF FUNCTIONAL AND 1\ 1 C
J
=
Ix f(x) dp,(x) � Ix fn (x) dp,(x)
J (fn ) t J(f) . If Ix f(x) dp,(x) > J(f) , for some simple function s, O :s; s :s; f and J(s) > J(f), in denial of the nonnegative character of J: Ix f(x) dp,(x) = J(f) . If o :s; f E L 1 (X, p,), for some simple functions { s n} nEN ' Sn t f and Ix sn(x) dp,(x) = J (s,,) t Ix f(x) dp,(x) < Consequently, 2.1.9f) implies nlim += Sn �f s E L 1 and J(s) Jrx f(x) dp,(x). On the other hand, the preceding paragraph implies Ix s(x) dp,(x) = J(s) and so Ix f(x) dp,(x) Ix s(x) dp,(x). If f  s is not 0 a.e., there emerges the contradiction 0< r [J (x)  s(x) ] dp,(x) :S; Jr [J(x)  s(x)] dp,(x) D JE> S,O) =
00 .
=
=
(f
x
(X, S, �) IS A MEASURE SPACE AND L IS THE SET OF SIMPLE FUNCTIONS s SUCH THAT Ix s d� E JR., THEN I : L 3 s r+ I(s) �f Ix S d� E JR. 2.2.35 THEOREM. IF
Section 2.2. Measurability and Measure
65
IS A DLS functional, i.e. , A NONNEGATIVE LINEAR FUNCTIONAL SUCH THAT { s n ..l o} ::::} { I (Sn) ..l o}. PROOF. If En def { X : S n (X) > 0 } , then En ::) En+ 1 , and n En 0. Fur=
thermore,
� (Ed < 00, whence 2.2.27 implies � (En ) ..l O.
nEN Since
=
sup S 1 (X) < 00, I (s n ) ..l o. D xE En s n (x) ::;; xEX In the context of 2.2.35, the DLS functional I gives rise to I, 1, J , D, D, and, via 2.2.34, J . dp,. The next discussion deals with the relations between 5 and D resp. � and p, resp. L 1 (X, O and L 1 (which, according to 2.2.34 is L 1 (X, p,)). If E E 5, then X E E D, whence 5 c D. If 0 ::;; f E L 1 ( X, �), there is a sequence { s n } nEN of simple functions such that 0 ::;; S n t f and Ix S n d� t Ix f d�. and sup p"
On the other hand, Lebesgue's Monotone Convergence Theorem for the DLS functional J derived from and the associated measure implies f C Since D is always complete, J (f) = f if is not complete, �D. However, since is dense in both (metrized according to
Ix
5
d�
=
Ix
5
I 1 dp, : L (X,�) L 1 .
p,
L 1 (X, �)
L
L 1 , it follows that modulo null functions, L 1 (X, �) L 1 L 1 (X, p,). The previous developments offer justification for largely ignoring, in the remainder of the book, the distinctions among U(X, �), L 1 , and L 1 (X, p,). and
=
=
They also establish that Daniell's approach and Lebesgue's approach to integration (v. 2.2.50) are logically equivalent. As circumstances dictate, for exploring a problem one approach might appear superior to the other. For example, as formulated by Kolmogoroff [Kol] , probability theory lends itself to Lebesgue's ideas (v. 2.2.50). On the other hand, the elegant presentations of integration on locally compact groups [Loo, Nai, We2] and the Riesz Representation Theorem (2.3.2 below) are more readily un derstood from the perspective of Daniell's theory. [ 2.2.36 Note. The basic results, Lebesgue's Monotone Conver gence Theorem (2. 1.9f) ), Fatou's Lemma (2.1.14a) ) , and the
Chapter 2. Integration
66
Ix .]
Dominated Convergence Theorem (2.1.15) are valid for the func tional J and hence for the functional
::;; f E D: a) for the map D 3 E r+ ¢(E) �f L f(x) dp,(x) ,
2.2.37 Exercise. If ° ¢
:
Ix
Ix
(X, D, ¢) is a measure space; b) if ° ::;; g E D, then g d¢ gf dp" a relation that may be summarized by the equation d¢ = f dp" v. Section 5.8. [Hint: For a) , Lebesgue's Monotone Convergence Theorem ap plies. For b) , the general result follows from the treatment of the special case when g is the characteristic function of a set E in D :
g
=
=
XE ·]
[0, 1] and, for f in C([O, 1], JR.) , I (I) is the Riemann integral 1 f(x) dx of f, then D a R[K ([0, 1])] = 5,6 ([0, 1]), the aalgebra of Borel sets in [0, 1]. If ° ::;; a < b ::;; 1, p,([ a, b]) b  a. [ 2.2.39 Note. Although D is complete (v. 2.2.28 ) , the following discussion concludes that D � 5,6([0, 1]) and that 5,6([0, 1]) is not 2.2.38 Exercise.
1
If X �
::)
=
complete.]
[0, 1),
2.2.40 Example. When a E the Cantor set deleting from a sequence of open intervals:
[0, 1]
•
•
•
•
1
10 , 20 centered at  ; 2 Ill , 11 , 2 1 centered at the midpoints of the two components of
In 1 , . . . , In, 2 n centered at the midpoints of the 2n components of
[0, 11 \ etc.
Co: is constructed by
['Q (�l;" ) 1 ;
67
Section 2.2. Measurability and Measure
The intervals in any group In 1 , . . . , In,2n are of equal length, say an ,
1
00
= nL 2n an . It follows that Co: is a closed set and =O f.l (Co:) = a. If an 3 n , then a = ° and there emerges the Cantor set : En = ° or 2 The Co. A direct calculation shows that Co = 00 En 00 1 En map Co 3 L r+ L "2 n + 1 carnes Co onto [0, 1] ( v. 1.2.31). Hence n =l 2 n #([0, 1]) 2': #= l(Co) 2': #([0, 1]): # (Co) Because f.l (Co) = ° and D is complete, !fj (Co) c D. Furthermore, 2' :::; # ( D) :::; #[!fj (ffi.)] = 2'. Thus #(D ) 2'. On the other hand, as the next lines show, # [5,(3 (ffi.)] Hence some subset of Co is not in 5,(3(ffi.) : 5,(3(ffi.) is not complete and, to boot, D � 5,(3(ffi.) . Since every singleton set is in 5,(3([0, 1]), # (5,(3) ([0, 1]) 2': Further more, 5,(3([0, 1]) = a R ({ (a, b) : O :::; a :::; b :::; l , a, b E Q}) �f a R (Eo), i.e., 5,(3([0, 1]) is generated by the countable set Eo. For every ordinal number a in (0, Q), if Eo: is the set of all unions of count ably many set differences drawn from U E", then 5,(3([0, 1]) = U E o: . However, an > an + l ,
and  a ==
:;;:
}.
{��:
•
= c.
=
= c.
c.
,,<0:
o: < n
# ({ a
a
}) :::;
c,
whence = c. Customarily, the complete aalgebra in the context of 2.2.38 is denoted subsets of By the the set of same token, the measure is usually denoted A , and the measure space is A).
# [5,(3([0, 1])] 5A([0, 1]), ([0, 1] ' SA ([0, 1]),
f.l
D Lebesgue measurable
[0, 1]. o:
[0, 1) there is the Cantor function cP defined as follows: cP ( ) if E h,2n , 1 :::; k :::; 2n  1 . Thus far cP is defined only on [0, 1] \ C . However, if E Co:, then lim cPo:(x) �f cPo:(y) ={
2.2.41 Example. For a in o: x
o:
2k  1 �
x + y x E [O , l ] \ Ca
x
o:
y
exists. The function cPo: is continuous, monotonely increasing, and
68
Chapter 2. Integration
2.2.42 Exercise. a) <Pa ([O, 1] )
=
[0, 1] ; b) there is a measure space
([0, 1] ' SA ([0, 1] ) , f.L) such that for [a, b) contained in [0, 1] , f.L([a, b))
then '1jJa : [0, 1]
r+
�f
).
{<pa [[a, b)] }; c) if
[0, 1] is a homeomorphism.
2.2.43 DEFINITION. FOR A SET X, AN outer measure
f.L* : �(X) 3 E r+ f.L * (E) E [0, 00] IS A SET FUNCTION THAT IS countably subadditive, i.e.,
MONOTONELY INCREASING, i.e., {A C B} ::::} {f.L* (A) :::; f.L* (B) } , AND SUCH THAT f.L* (0) = O. A SET H CONTAINED IN �(X) IS hereditary IFF
{{ E E H } !\ {F c En ::::} {F E H } . WHEN A c �(X), H (A) , THE hereditary aring of A, IS THE INTERSECTION OF ALL HEREDITARY aRINGS CONTAINING A. FOR A MEASURE SPACE (X, 5, f.L) ,
f.L *
:
H (S)
3 E r+ inf { f.L(F)
E c F E 5}
IS THE outer measure induced by f.L on H (S ) ;
f.L * : H (S) 3 E r+ SUp { f.L(F) : E ::) F E S } IS THE inner measure induced by f.L on H (S ) . FOR A IN H (S ) , A C IN 5 IS A measurable cover OF A IFF C ::) A AND {5 3 B e e \ A} ::::} {f.L(B) = O}; A K IN 5 IS A measurable kernel OF A IFF K c A AND {5 3 B c A \ K} ::::} {f.L(B) = O}. 2.2.44 Exercise. a) In the context of the DLS construction,
Section 2.2. Measurability and Measure
69
is an outer measure. b ) For a measure space (X, 5, p,): bl ) if C 1 and C2 are measurable covers of an in H (5),
A
A
b2) if A E H ( 5), some K in 5 is a measurable kernel of A; b3 ) if E H (5) and is contained in a countable union of sets of finite measure (A is p, * afinite) , some C in 5 is a measurable cover of 2.2.45 Example. In the context of 2. 1.22, the measure corresponding to the functional L on the aring D is counting measure:
A
A.
v :
{
5 "3 E t+ #(E) if #(E) E N . 00
otherwise
Contained in D is the subaring 5 consisting of (/) and the set of all finite or countable subsets of X. If # (X) > No, A C X and #(A) > No, no C is a measurable cover of For a set X and an outer measure p, * , a subset E of X is Caratheodory measurable iff for all A in !fj(X)
A.
p,*
(A)
=
p,* (A n E) + p,*
(A \ E)
(2.2.46)
(E splits every set A p, * additively. ) The set of Caratheodory measurable sets is denoted C. ( As 2.2.48 reveals, the condition (2.2.46) implies that p,* I is count ably additive. ) 2.2.47 Exercise. A set E is in C iff c
{{P c E} !\ {Q
C
X \ E}} '* {p,* (Pl:!Q) = p,* (P) + p,*(Q) } .
2.2.48 THEOREM. THE SET C IS aALGEBRA, (X, C, p,* ) IS A MEASURE SPACE, AND C IS COMPLETE.
When E in (2.2.46) is E1 \ E2 , there emerges
p,* (P) + p,*(Q) = p,*(P) + p,* (Q n E2 ) + p,* (Q \ E2 ) = p,* [Pl:! (Q \ E2 )] + p,* (Q n E2 ) = p,* (Pl:!Q.)
Chapter 2. Integration
70
E E2 E l:.! ( E2 \ El ), E E El l E2 l l 2 f.l* [(A n Ed l:.! (A n E2 ) ] = f.l* {(A n Ed l:.! [(A n E2 ) n Ed } + f.l* {[(A n Ed l:.! ( A n E2)] \ Ed f.l* (A n Ed + f.l* ( A n E2 ) , it follows that f.l* [A n ( El l:.!E2 ) ] + f.l* [A \ ( El l:.!E2 ) ] = f.l* [(A n Ed l:.! (A n E2 ) ] + f.l* [(A \ Ed \ E2] f.l* (A n Ed + f.l* (A n E2 ) + f.l* [(A \ E1 ) \ E2] ' ( ) On the other hand, f.l* (A \ E1 ) f.l* [(A \ Ed n E2] + f.l* [ ( A \ Ed \ E2] f.l* (A n E2 ) + f.l* [(A \ Ed \ E2 ] , whence the last two summands in the right member of ( ) may be replaced by f.l* (A \ Ed . There emerges f.l* [A n (Ell:.! E2 ) ] + f.l* [A \ ( El l:.!E2 ) ] = f.l* (A n Ed + f.l* (A \ Ed = f.l* (A) : El l:.!ESince 2 E C.C is closed with respect to the formation of set differences, in
Since the previous conclusion implies that U = to show U E C, it suffices to assume n = 0. Then for any set A, since =
=
*
=
=
*
duction shows that for N in N,
SN �f
2 �n � N ( En \ End �f lU�n � N Fn E C , l �Un � N En = El l:.!U Fn E aR (C) .
For N in N and A in !fj(X), since each Fn is monotone character of f.l * imply that
\
in
C, induction and the
f.l*(A) = f.l* (A n SN ) + f.l* (A SN) , f.l* (A n SN) = f.l* (A n SN n FN) + f.l* ((A n SN) FN) = f.l* (A n FN ) + f.l* (A n SN  d N Nl f.l = f.l* (A n FN ) + L ( ) L f.l* (A n F ) ,
\
* A n Fn = n= l =l n N f.l*(A) = L f.l* (A n Fn ) + f.l* (A \ SN) , n= l 00
n= l
n
Section 2.2. Measurability and Measure
Hence
71
Soo E C; if A = Soo the preceding calculations imply p,*)
i.e., (X, C, is a measure space. The definition of Caratheodory measur ability implies: a) if E C, then X \ E C; b) if = 0, then C. The monotone character of implies that C is complete. D
E p,*(A) AE p,* 2.2.49 THEOREM. IF 1 E D, AN E IN !fj(X) IS DLS MEASURABLE IFF E IS CARATHEODORY MEASURABLE WITH RESPECT TO THE OUTER MEASURE p,* OF 2.2.44. PROOF. If D is DLS measurable and p,* ( A ) = 00 the subadditivity of 1 implies p,* (A) = p,* (A n E) + p,* (A \ E). If p,* (A) < 00 and E > 0, for some g in Lu, g 2: X A and 1
i.e.,
E
(X A) + E 2: I(g ) ;::::: I(g /\ 1) = I (g /\ X D + g /\ X ( X \D) ) 2: I (g /\ x D ) + I (g /\ X ( X\D ) ) 2: I ( X ( An D) ) + I ( X ( A \D) ) ,
DLS measurability implies Caratheodory measurability. When C is Caratheodory measurable its DLS measurability is estab /\ g (v. lished by showing that i.e., that for all g in 2.2.14a) ) .
Xc E D,
Case 1:
I (x c) < 00. If E > 0 , for some h in Lu,
L, Xc E L 1 ,
Xc and I( h) < (Xc) E < 00 . Since 1 E D, L 1 '3 1 /\ g 2: Xc' Each of a)e) below follows from its prede cessors or from the statements made thus far: I
h 2:
D; XB E D;
a) 1 /\ h E b) B � E:2 (1 /\ h, 1) D; c) h 2: d) I I(h) < 00, whence e) 2:
(X ) :::; XB B Xc '
E
XB E L 1 ;
+
Chapter 2. Integration
72
Since X E L 1, B is DLS measurable and hence Caratheodory measur able. The foflowing equations and inequalities are validated, i.a., because: both B and C are Caratheodory measurable; •
•
•
•
X B E LI '. I is subadditive; for all sets A , I( X ) = /l* (A) (by definition): +
/l*(B) = /l* (Bn C) /l*(B \ C ) , I (X B) = I (X B 1\ xc) + I (X ( B\ C) ) = I (xc) + I (X B  xc) , I (xB  x c) S; I (xB ) + I ( x c) = l (x B )  1 (x c) , = 1 (X B  x c ) · Xc = X B  (XB  Xc) E L 1 , i.e., C E D. Case 2: I (X c ) = 00 Since the range of X c is the set {O, I}, it suf fices to show: {{g E L } 1\ {O S; g S; I}} ::::} { X c 1\ g E L 1 }. If A = E> (g, O), Thus X B  Xc E L 1 ,
.
owing to 2.2.22, A E D. Case 2a: X A E L 1 . Then A is DLS measurable, hence Caratheo dory measurable and
Thus C n A is Caratheodory measurable and since
0 < X (CnA) < X A E L 1 '

I (x (CnA) ) < 00 . The conclusion in Case 1 shows that whence X c 1\ g E L 1 . Case 2b: X A E
(D \ L 1 ) . For E > 0, if A. � E> (g, E), then
Hence X A, E L 1 . The conclusion in Case 2a implies that Cn A E D. Setting E at .!, n
n
E N, yields the result.
D
73
Section 2.2. Measurability and Measure
12K f(x) dx
[ 2.2.50 Note. The theory of integration, due to Lebesgue , permits meaning to be ascribed to 00
0
when
f (x) �f L an cos nx + bn sin nx, n =l
and
00
L I an cos nx + bn sin nx I < 00 for all x, even, as well might
12K f(x) dx
n =l be the case, when fails to be Riemann integrable. Such a function is integrable when the symbol is viewed as the Lebesgue integral.
f
f
Lebesgue's idea for a partition of the domain of a function in was contrapuntal to Riemann's. A Riemann partition of . d by pOlnts ' a cle=f Co < C l < . . . < Cn clef b and con[ a, b] ' determlne sists of the sets
ffi.[a, b]
IS
=
a partition made without regard to the function
f.
Lebesgue partitioned the domain [a, b] of f by direct reference to f via its range. If p > q > r, the sets Epq �f Ed f,p) \ K;(f, q), Eqr �f Edf, q) \ Ed f, r) are disjoint. If f is sufficiently restricted, the sets Epq are in the aring S,B ' Whereas the length of Epq is not necessarily meaningful without modification of the notion of length, Lebesgue developed a theory that consistently assigns a value to the size of any Epq in S,B and indeed to any set E in S,B ' Successive elaborations
by Lebesgue, Daniell, Stone, and others led to the development presented above and to the following discussion.] 2.2.51 DEFINITION. FOR AN ELEMENT
E OF !fj(X ) , A PARTITION
74
Chapter 2. Integration
OF E IS A SET OF PAIRWISE DiSJOINT SETS SUCH THAT = E. IN THE CONTEXT OF A MEASURE SPACE (X, S, p,) , WHEN E E S, THE PARTITION P IS MEASURABLE IFF EACH E S. THE PARTITION P IS FINITE IFF #(r) E N.
U y ErPy
Py
2.2.52 Exercise. For the measure space (X, S, p,) , if 1 is measurable, o � E L 1 (X, p,), P � is a finite measurable partition of X ,
f {Pk h�k�n n and sp(f) � L inf f(x)p, (Pk), then k= 1 XEPk Ix f d sup { sp P a finite measurable partition of X } . p, =
:
fn Un }n EN f f n n+ = f f ),), 11 f(x) d),(x ) 1 1 nlim + = 10 fn (x) dx 10 f(x) d), (x) O. 2.2.54 Exercise. For f as in 2.2.53, in C ([0, 1] , JR.) there is a se quence {gn} n EN such that: a) nl�� gnl lQ!n [o , j exists and is f l lQ!n [O j ; b) ,1 1 1 each gn is nonnegative; c) sup gn (x) 1; d) 1 gn (X) dx > 0, E N; e) 0 O�x� 1 1 1 nlim + = 10 gn (X) dx = 10 f(x) d), (x ) , cf. 1.3.5.
2.2.53 Exercise. a) Each in 1.7.6 is Riemann integrable; b) is a bounded sequence; c) lim �f exists; d) by contrast with 2. 1.9 and 2.1. 15, is not Riemann integrable; e) is Borel measurable and for the measure space ([0, 1] ' S,B, exists and =
=
=
n
[ 2.2.55 Note. As shown next, the results in 2.2.53 and 2.2.54 show that the DLS construction can provide a proper extension of the original DLS functional.
1 1 . dx is used to define the (DLS ) functional I : C([O, 1] , JR.) '3 f 1 1 f(x) dx E JR., the DLS exten sion (I) yields the measure space ([0, 1] , S),, )') and the (Lebesgue) integral 1 1 . d)'(x). The function f in 2.2.53 is Lebesgue inte
When the Riemann integral
r+
grable but not Riemann integrable.]
Section 2.2. Measurability and Measure
75
L
2.2.56 Exercise. If f E l ([0, 1] , ),) and E > 0: a) for some positive M, if A M �f E« l f l , M), then ), ([0, 1] \ A) < E and
b) for some lsc h, h 2': f and and
11 f

k d),(x )
11 h

f d)' ( x ) < E; c) for some usc
k, k �
f
< E.
L
2.2.57 Exercise. For a set X and a function lattice contained in JR.x , the smallest monotone class of functions containing i.e., the Baire space associated to is a function lattice. [Hint: For f in if M (f) consists of all g such that
B(L)
L,
L, B(L)
{g, f + g, f V g, f 1\ g}
c
B(L),
M (f) is a monotone class.] 2.2.58 Exercise. The set function if X E E
Ll
otherwise
Ix
is a measure and if f E I , then (f) = f ( x ) df.lo ( x ) . Furthermore, if f.l l : D '3 E r+ f.l l (E) E [0, 00] is a measure and for all f in l ,
L
J (f) =
J
L
Ix f (x) df.l l (X ),
then for all E in D, f.lo (E) � f.ll (E) . 2.2.59 Exercise. a) The set R([O, I] of functions f that are Riemann integrable on [0, 1] is a function lattice. b) If X �f [0, 1] ' �f e([O, 1], JR.) ,
L
and when f E c) If R([O, 1])
L, 1(f) �f 1 1 f(x) dx (Riemann integral) , then L e R e L l .
::)
{fn }nEN and fn + 0, then
1 1 fn (x) dx + o.
[Hint: If f E R([O, 1]), for a sequence { sn }nEN of stepfunctions, Sn t f a.e.(),) . If s is a step function, there is a sequence { cn }nEN of continuous functions such that Cn t s.]
76
Chapter 2. Integration
2.2.60 Exercise. For the Dirac functional rSa in 2.1.21, what are D, D, and the corresponding measure? 2.3. The Riesz Representation Theorem
An important consequence of the preceding development is the following theorem of F. Riesz. The result applies, when X is a locally compact Hausdorff space, to nonnegative linear functionals defined on Coo (X, JR.) , the set of continuous functions f such that supp (I) is compact. In Chapter 4 the result is extended for continuous Cvalued linear functionals defined on Coo(X, C) . For a locally compact Hausdorff space X, the class of Baire sets S b ( X ) is the intersection of all aalgebras contained in !fj(X) and including, for each f in Coo(X, JR.) and each a in JR., the set Edf, a). Hence if 0 denotes any one of � , > , 2: and a E JR., then b (X) contains Eo (I, a). However,
S
is a compact Gli , whence compact GliS.
Sb (X) is the aalgebra generated by the set of
2.3.1 Example. If #(X) > No, O(X) �f !fj(X) (the topology of X is the discrete topology ) , and X* �f Xl.,J {y} is the onepoint compactification of X ( cf. 1. 7.28 ) , each neighborhood of y is closed ( and, by definition, open) . Each finite subset of X is a compact Gli , whereas the set {y} is compact but is not a Gli . An infinite subset A of X is not compact, since its elements regarded as sets constitute an open cover V of A, and no finite sub cover of V exists. Thus the aalgebra generated by the compact Glis of X * is the aalgebra (X) generated ( in X * ) by the finite subsets of X. The construction given in 2.2.40 for implies, in the current instance, that {y} tic (X) . Thus (X* ) (X* ) : the sets and need not be the same.
Sf
Sf
S,B
SG8 S,B Sb "1
S,B
Sb
2.3.2 THEOREM. ( F. Riesz ) FOR A LOCALLY COMPACT HAUSDORFF SPACE
X AND A NONNEGATIVE LINEAR FUNCTIONAL
1 : Coo (X, JR.) '3 f r+ 1( 1 ) E JR., FOR SOME MEASURE SPACE (X,
Sb, p,) AND ALL f IN Coo(X, JR.) ,
1( 1 ) =
Ix f(x) dp,(x).
Section 2.3. The Riesz Representation Theorem
77
PROOF. If Coo (X, JR.) '3 In + 0, since supp UI ) is compact and supp UI ) ::) supp Un ) , n E N, Dini's Theorem (1.2.46) implies In � O. Urysohn's Lemma (1.2.41) im plies there is in Coo (X, JR.) a such that 0 :::; 1 and 2': I on supp ( II ) . If m E N, for some no, 0 :::; In (x) < !!.. if n > no, whence m
g
g :::;
g
1Un ) :::; 1 (�) = � 1(g) . Hence nlim 1 Un ) = O. Since 1 Un ) 2': 1 Un +I), 1 Un ) + 0: 1 is a DLS func+ =
tional. By abuse of notation, 1 /\ Coo (X, JR.) c Coo (X, JR.) , whence [2.2.29d)] the set D of DLS measurable sets is a complete aalgebra that includes, for each in Coo(X, JR.) and each a in JR., Edl, a ) : D ::) Sb(X) . D 2.3.3 Exercise. If X is a locally compact Hausdorff space,
I
I
K E K(X), and K < E Coo(X, JR.) , then for
n
in N: a) E>
K c g� u, 1) =
n On ·
nEN
(
n+ 1  l, l n
) cle=f On E O (X); b) On E K(X);
c)
In sum, every compact set K is contained in a
compact Cij . [ 2.3.4 Remark. When X is a locally compact space, KGij is the set of all compact sets each of which is a Cij . Some define Sb(X) to be the aring generated by KGij: Sb(X) �f aR (KGij) and some define S,(3(X) to be aR[K(X)]. The reader is encouraged to explore the relations among these definitions and those used in this book.]
D
2.3.5 Example. In 2.1.22, if X �f JR., consists of all functions that supp U) is empty, finite, or countable while
{J E Ll } {}
ttI:
{
UE
D} /\
{�I I(X) I } } .
I
such
< 00
Consequently, JR. D. The function JR. '3 x r+ x E JR. is such that # [supp U)] > No, whence although for each a in JR., E= U, a) = {a} E D (cf. 2.2.22). 2.3.6 Example. In its customary topology, again JR. is a locally compact �f I(x) dx (the Riemann Hausdorff space. If I E Coo(JR., JR.) and integral of Urysohn's Lemma implies that aR[K(JR.)] = D.
I tt D,
I),
1U)
l
Chapter 2. Integration
78
2.3.7 Exercise. In 2.3.2: a) if E E 5,6,
p,(E) = inf { p,(U) : E c U, U E O (X) } ; b) if p,(E) < 00, then p,(E) = sup { p,(K) : K c E, K E K(X) } . The conclusions a) and b) above motivate the following terminology. 2.3.8 DEFINITION. FOR A MEASURE SPACE (X, 5, p,) , A SET E IN 5 IS outer regular (inner regular) IFF
p,(E) = inf { p,(U) : E C U E O (X) } , (p,(E) = sup { p,(K) : E ::J K E K(X) } ) . AN E THAT I S BOTH OUTER REGULAR AND INNER REGULAR I S regular. WHEN EVERY E IN 5 IS OUTER REGULAR (INNER REGULAR) (REGULAR) , (X, 5, p,) AND P, ARE OUTER REGULAR (INNER REGULAR) (REGULAR) . 2.3.9 Exercise. In 2.3.2, if X �f [0 , 1] in its customary topology and I is a) p, = A (cf. 2.2.40 ) ; b) if E E 5,6([0 , 1] ) and the Riemann integral r ira.l] x E JR., then :
X
+E
clef { x + Y =
: Y E E } E 5,6([0 , 1]), A (E) = A ( X + E )
(5,6([0 , 1]) and A are translationinvariant) ; c ) if {E, F } c 5,6, ° < A (E) . A(F) < 00, and x E JR., then f (x ) � A[E n (x + F ) ] = d) E

l
X F (Y  x) X E (Y) dy and f is continuous;
F �f { x  y : x E E, y E F } , contains a neighborhood of zero.
[Hint: If E is an interval, b) and c) are valid. For d), c) applies.]
[ 2.3.10 Note. In the context of 2.3.9, the measure space is [JR., 5), (JR.)), A] and A is Lebesgue measure. The sets in 5), [JR.)] are
the Lebesgue measurable sets. Corresponding definitions apply for the notions of Lebesgue measurable functions, Lebesgue integrable functions, Lebesgue integrals, etc.] 2.3.11 Exercise. (VitaliCaratheodory) In the context and notation of 2.3.2, the conclusion in 2.2.56 obtains. [Hint: Urysohn's Lemma ( 1. 2.41 ) applies.]
Section
2.4 .
79
Complexvalued Functions
2.4. Complexvalued Functions
Little difficulty and much advantage follow from admitting Cvalued func tions to the discussion. From this point forward, functions with numerical ranges are to be assumed as Cvalued unless the contrary is stated. The image "(* �f "((lR.) of the curve "( : lR.
is { z
Izl
=
'3 t r+
1  t 2 . 2t C 1 + t2 + l 1 + t2 E
( 2.4. 1 )
1, z i  I } �f '][' \ {  I }. The function fJ : lR. '3 t r+ fJ(t) cle=f
2 it 2 dx o
1+
X
( 2. 4.2)
is continuous and strictly monotonely increasing. For some (finite) number, denoted 7r , +lim fJ( t) = ± 7r: "( is rectifiable and t ± oo length of "(
�f £("()
=
27r .
The inverse of fJ is the function t : ( 7r , 7r ) '3 fJ r+ t(fJ) �E ( 00, (0 ) :
t(O)
=
t(fJ) = 00, lim t(fJ) = 00, t(±7r) �f ±oo . 0, lim Ot7r O.j.7r
(2.4.3)
The two trigonometric functions,
1  t(fJ) 2 1 + t(fJ) . : [ 7r , 7r] '3 fJ r+ 2t(fJ) sm 1 + t(fJ) 2 ' cos : [ 7r , 7r] '3 fJ r+ '':2 :
(2.4.4) (2.4.5 )
are infinitely differentiable on ( 7r , 7r ) and
cos' fJ =  sin fJ, sin' fJ = cos fJ.
The formal
fJ 2 fJ4 1 +  ···' 2! 4! fJ3 fJ5 fJ  +  · · · 3! 5!
( 2. 4 .6 ) (2.4.7)
converge for all fJ in C. The remainder formulCE associated with the Maclau rin polynomials for cos fJ and sin fJ show that ( 2.4.6 ) resp. (2.4.7) represent
Chapter 2. Integration
80
cos fJ resp. sin fJ on (7r, 7r) and define cos fJ and sin fJ throughout C. Com bined, they yield
(i fJ ) n cos fJ + i sin fJ = 1 + � � n= l n!
and the definition
�f exp(ifJ) (Euler's formula)
00 fJn exp(fJ) cle=f 1 + L ,.
(2.4.8)
n= l n.
The right member of (2.4.8) converges throughout C. On JR., the series (2.4.6)(2.4.8) represent three infinitely differentiable functions mapping JR. into JR.. (By virtue of the argument in 5.3.2, they are infinitely differentiable throughout C. ) Direct calculations using (2.4.8) show exp(u + v ) = exp(u) exp(v) ,
(2.4.9)
whence if e � exp(I), successively, exp(n) = en , n E N, exp(m) = e m , m E Z, exp r = eT, r E Q.
(2.4. 10)
Owing to the continuity of exp, the definition e O �f exp( fJ) , fJ E C, is con sistent with the formulre in (2.4.10) . If fJ E JR., I e iO I = ( cos 2 fJ + sin2 fJ) "2 = 1. By virtue of (2.4.3) , 1
whence, for
k
k
e 7ri = cos 7r + i sin 7r =  1,
in Z, e 2 7ri = 1. If fJ E C, then k
{ COS(fJ . (fJ sm
+
+
(2.4.11)
2k7r)
k 2 7r )
} { cossm fJfJ } . =
.
If ¢ E JR., for a unique in Z, fJ � ¢ + 2k7r E (7r, 7rJ . If cos ¢ = 1, then cos fJ = 1, i.e., t( fJ ) = fJ = 0, ¢ E 2Z7r . If Z � x + iy and e Z = 1, then x = 0 and e iy = 1 whence e Z = 1 iff Z E 2Z7r i. (The last conclusion is alternatively deducible from the formula fJ 2 fJ2 + fJ6 fJ 2 + . . . cos fJ = 1 1 1 _
(2 (
_
12
)
6!
(
_
56
)
)
applied when fJ E [ 7r, 7rJ .) 2.4.12 Exercise. The least positive period of both cos and sin is 27r. [Hint: sin fJ =  cos' fJ.J 2.4.13 Exercise. a) exp l lR is a strictly monotonely increasing function; b) the function inverse to the exponential function exp on JR. is the logarithmic function In : (0, (0 ) '3 Y r+ In (y) E JR., i.e., exp 0 In (0, 00) '3 Y r+ y, In 0 exp : JR. '3 x r+ x ; :
81
Section 2.4. Complexvalued Functions
lim exp ( x ) c) lim exp(x ) = 00, X400 X400
eZ
=
0; d) if z = x + iy,
(cos y + i sin y ) ;
= eX
e) for z in C \ {a} and some unique fJ in (  7r , 7r] , Z = exp(ln I z l + ifJ); e) exp' = exp o
�f
[ 2.4. 14 Note. For ,,( as in (2.4.1), if z x + iy E "( * , there is in (  7r , 7r ) a unique 8(z) such that e i8 ( z ) = z. Hence, if e
Izl { cl sgn (z) f Z �
O
if z :;to O otherwise
Hence: a) z · sgn (z) = I z l ; b) sgn is continuous on C \ {O}; c)
{
if z :;to O I sgn ( z ) 1 = I otherwise. o
The unit circle { z : I z l = I } ( "( * u {  I } ) is the same as 'lI'. If z E (C \ {O} and z = I z l e i iJ ,  7r < fJ � 7r, the halfline =
[0
clef { =
w
:
w =
re'· 0 , r 2: 0 }
meets 'lI' in exactly one point, which is sgn (z) . If U is open in C, u n 'lI' is relatively open in 'lI' and sgn
1
 1 (U) = { SS Ucl=ef{O}{ z
[0 n U n 'lI' :;to (/) }
if 0 tic U otherwise
�f
Hence sgn (U) is either open or the union of an open set and a sin gle point: sgn is (aR[O(C)] , aR [O(C))]measurable. When z x + iy and sgn (z) u(x, y) + iv(x, y) , then u, v E D. For X, L, and I in the DLS development, a function
�f
f : X '3 x r+ f ( x ) = SRf ( x ) + i':Sf(x)
�f u(x) + iv (x) E C
is defined to be DLS measurable, Dmeasurable, or Caratheodory measur able iff u and v are: f E L I iff both {u, v } e L l , in which event,
Ilf ll l
�f J( l f l ) .
82
Chapter 2. Integration
For a measure space
(X, S, Il), by definition,
{ I �f u + iv E S } {} {{ u, v } C S} ,
{J E L 1 (X, Il) } {} { III E L l (X, Il) } . In the circumstances , smce ma:x{lul, I v l } � ';u2 + v 2 � lui + l v i , I E resp. I E iff
L 1 (X, Il)
J(lul) + J(lv l ) < 00 resp.
Ll
lx , u , dll + lx , v , dll < 00 .
Unless the contrary is stated, henceforth, functions will be assumed to be Cvalued. 2.4.15 Exercise. The map JR. '3 x r+ exp(27rix) E 'lI' is a continuous open epimorphism of the additive group JR. onto the multiplicative group 'lI'. (In particular, if k E Z, then exp((2k + 1)7ri) =  1.)
}
2.4.16 Exercise. If z i 0: a) Arg (z) �f { (J : (J E JR., z = I z l e i O i 0 ; b) . . (J l  (J2 E Z. #[Arg (z)] = No ; c) { (J l , (J2 } C Arg (z) Imphes 27r 2.4.17 Exercise. If 7r < (J � 7r: a) cP : C \ [0 '3 z r+ B[sgn (z)] is a non Arg (w). constant continuous map; b) cP (C \ (0 ) c
U
w EC\(e
2.4.18 THEOREM. a) IF THE CURVE "( : [0, 1] '3 t r+ "((t) E C IS NONCON STANT AND 0 tt "(* , FOR SOME NONCONSTANT cP, cP o "( :
[0, 1) '3 t r+
U
Arg (w  o )
wey'
IS CONTINUOUS. b) A MAP '1jJ DEFINED ON ,,([[0, 1)] AND SATISFYING
'1jJ 0 "( : [0, 1) '3 t r+
U
wE"!'
Arg (w  0 )
(2.4. 19)
IS CONTINUOUS IFF FOR SOME m IN Z, '1jJ 0 "(  cP 0 "( = 2nm.
PROOF. a) For the curve ;y �f "(  0, if ° = to < t l < . . . < tn
=
1 and
is sufficiently small (and positive), the arc ;Y [tk l , t k ] is contained in some C \ [0 where 2.4.17 applies: there is a cPk such that cPk o ;Y is continuous on [tk  l , tk ] . Owing to 2.4. 16c) , for some m l in Z, cPl o ;Y (t t ) = cP2 o ;y (t t ) + 27rm l , for some m2 in Z, c/J2 o ;Y ( t 2 ) = cP3 o ;y ( t2 ) + 27rm2 ,
Section 2.4. Complexvalued Functions
83
etc. Thus if if t E [0, t I ) if t E [t l , t2 ) if t E [tn  I , 1) then 1> is continuous on [0, 1) and if t E [tk  l , tk ), then
Since the correspondence ;Y(t)
++
l'(t) is bijective, the equation
defines the required ¢. b) If m E Z, then 27rm + ¢ 0 l' is continuous on [0, 1) and satisfies (2.4.19). Conversely, since the map '1jJ 0 l' ¢ 0 l' is continuous, the result in 2.4.16c) implies that '1jJ 0 l' ¢ 0 l' is 27rZvalued. Hence '1jJ 0 l' ¢ 0 l' is a constant. [] 2.4.20 Exercise. In the context of 2.4.18: a) If r5 is sufficiently small (and positive), then ¢ 0 1' is monotone on each interval [t k l , tk) ' b) For A ¢ 0 1'(0) . respect to a, . d ( a ) cle cle . A =f hm ¢ 0 l' ( t ) =f , the . dex of l' wzth 'Y �l 27r is in Z. c) On each component of C \ 1' * , ind 'Y ( a) is a continuous (hence constant) function of a. d) Only one component of C \ 1' * is unbounded. e) If a lies in the unbounded component of C \ 1'* , then ind 'Y(a) = 0. [Hint: e) If E > ° and 10'1 is large enough, then in the notations of 2.4.18, sup 1> o ;y < E . ] t E [a,l ) When z E C \ {O}, the count ably infinite set In(l z l ) + Arg (z) is de noted Ln (z). 2.4.21 Exercise. a) If 1' : [0, 1] 3 t r+ l'(t) E C is continuous and ° tt 1'* , for some ,£ defined on 1'[[0, 1)], ,£ 0 l' : (0, 1) 3 t r+ L n (w) is continE 'Y uous. b) If z i ° and eW = z, then w E Ln (z) . c) If t in [0, 1), then e£o 'Y ( t ) = l'(t). [Hint: The argument in the proof of 2.4.18 applies.] There are profound connections between the map h, a} r+ ind 'Y (a) and basic topology, e.g., the Jordan Curve Theorem , Brouwer degree of a map, etc. A dense but useful reference here is [Sp] where an extensive 

, Ill


m
l I
wU
'
Chapter 2. Integration
84
bibliography is offered. The discussion provided above for the basic topics of this subject is adequate for the current and later purposes of this book. 2. 5 . Miscellaneous Exercises
A A
{
¥A} 2.5.1 Exercise. a) If the set of indivisible elements of aring 5 is finite and U is their union, �f { A \ U : A E 5 } is a aring and no element of is indivisible. b) If is infinite, #(5) No. c) If 5 is infinite, then is infinite and #(5) No. In sum, if 5 is a aring, then #(5) i No. [Hint: If A i some nonempty Sset B is a proper subset of A.] An element in a aring 5 is defined to be indivisible iff every proper 5subset B of is 0, i.e., 5 '3 B ::::} {B = 0 } . 5
5
>
5 '3
I
I
5
>
0,
2.5.2 Exercise. For a group G, a function lattice L contained in JR.G , a DLS functional 1 : L r+ JR., an I in JR.G , and an a in G, if the left a translate of I is I[a] : G '3 x r+ I(ax), and for each a in G and each I in L, 1 (f[a] ) = 1( f ) , then: a) for all I in D, I[a] E D; b)
1 { f E L 1 } ::::} { { f[a] E L } !\ {J (f[a] )
=
J (f) } } ;
c) {E E D} ::::} {{aE E D} !\ {tt (aE) = Il(E) }} (cf. 2.3.9) . 2.5.3 Exercise. If I E L 1 ( X, Il) and n+ lim= Il (En ) = 0, then
r nlim +CXJ }En
ill dll = o.
[Hint: If I is a simple function the result is a consequence of the nonnegativity of Il. For the general case the density of the set of simple functions in L 1 (X, Il) applies.] 2.5.4 Exercise. If n : A '3 ), r+ n(),) E !fj(X) is a net, then lim X ' ) = xn and lim Xn ( A' ) .AEA .A EA n ( A
=
X . !l
2.5.5 Exercise. An I in JR.[O.1 ] Riemann integrable iff: a) for some finite M and all x , II (x) I � M and b) the Lebesgue measure of the set Discont ( f ) of discontinuities of I is zero: ),[Discont (f)] = O. 2.5.6 Exercise. If 5 is a aalgebra contained in !fj(X ),
h : JR.2 '3 (x, y) r+ h(x, y) E JR.
Section 2.5. Miscellaneous Exercises
85
is continuous, and Ii : X '3 x r+ Ji (x) E JR., i = 1, 2 , are Smeasurable, then H �f h (II , h ) is Smeasurable. [Hint: a) The set E< (h, a) is an open subset U of JR.2 ; b) E< (H, a) consists of all x such that [II (x) , h (x)] E U; c) U is a union of (count ably many) pairwise disjoint halfopen rectangles
2.5.7 Exercise. If { 1, J} C D, then sgn (I) E D. [Hint: The set E �f E= (I, 0) is in 5 and I (X \ E) c C \ {O}; C \ {O} '3 w r+ sgn (w ) is continuous, whence sgn (I) is measur able on X \ E.] 2.5.8 Exercise. If 5 is a aring contained in !fj(X), S is the associ ated set of Smeasurable functions, and I E S, there is in S a (J such that
I(x) == II(x) l e iO ( x) .
2.5.9 Exercise. a) If 5 is a aring contained in !fj(X), S is the associated set of Smeasurable functions, and I E S, then I II E S. b) The converse of a) is false. c) {J E L 1 } {} {{J E S} !\ { I I I E L 1 } } .
[Hint: For b) , if E E !fj(X) \ S ,(3 (JR.) , then
I
X E  X (lR\E)
I
==
1.]
Ix
2.5.10 Exercise. What is the result of applying the DLS procedure to the lattice L �f L 1 (X, p,) and the functional I L '3 I r+ I dp,? :
2.5.11 Exercise. If R is a ring of sets and is monotone, i.e.,
{ { {En }nEN C R } !\ {En C En + d }
'*
{ { {En }nEN C R} !\ {En ::J En + d }
'*
{U {n
nEN
nEN
} }
En E R , EN E R ,
then R is a aring. 2.5.12 Exercise. For a aring 5, if {In } nEN is contained in the corre sponding set S of Smeasurable functions, each of lim In , lim In , and n n (when it exists) lim In is in S.
n+ =
+ CXJ
+ CXJ
2.5.13 Exercise. If: a) (X, S, p,) is a measure space; b) X is totally finite, i.e., X E S and p,(X) is finite; c) p,* is the induced outer measure; d)
Chapter 2. Integration
86
(X, C, ll) is the measure space for the aalgebra of Caratheodory measurable sets, then C = 5 (the completion of S). [Hint: If E c F E 5 and p,(F) = 0, then p,* (E) = 0: 5 c C. If A E C, then for sequences {An } nEN and {Bn } n EN contained in S, E C An ::) An +l , n E N, E ::) Bn C Bn +l , n E N, and
fI ( A) = p,* (A) = nlim + p, (An ) = nlim + CXJ p, (Bn ) .] ex)
2.5.14 Exercise. In 2.5.13 the conclusion remains valid if X is the countable union of sets of finite measure, i.e., if X is totally afinite. 2.5.15 Exercise. For a curve "( : [0, 1] '3 t r+ "((t) E C and an a not in "(* : a) for some positive J, o inf h(t)  0' 1 2: J and :s;t:S; 1
b) if 0 = t l < t2 < . . . < tn = 1, sup tk  tk l < J, and :S; k :S;n
2
then for some m in Z,
e
n
� 2)'h = 2m7r; c) ind ')'(O') = m. k= 2
2.5.16 Exercise. If f E JRlR , f OR.) C [00, (0 ) , and f is usc: a) f is Lebesgue measurable; b) A(E) < 00 implies either f ( x) dx E lR. or, by
Ie
Ie
abuse of notation, f(x) dx = 00. Corresponding statements are valid when f(lR.) C (00, 00] . 2.5.17 Exercise. If X is a set and {En} nEN C !fJ(X), then: a) There are clef clef 1 1· m  En = E resp. 1·1m En = E such th at sets 
n+ oo
n+ =
lim
n+CXJ b) lim En = { x
X E = XE n
_
and lim
n+CXJ
XE = XE . n 
=
n U
X
is in infinitely many En } =
X
is in all but finitely many En } =
mEN n=m
En , =
U n En·
mEN n= m
87
Section 2.5. Miscellaneous Exercises
c) lim En C lim En . d) If 5 is a aring and { En } n E N C 5, then n n + CXJ
+ CXJ
lim En E 5 and lim En E 5 . n+ =
2.5.18 Exercise. If (X, 5, p,) is a measure space and £ is the set of simple s dp, E JR., how does the completion of £ with refunctions s such that
Ix
spect to the metric J : £ 2 '3 {J, g} r+ J ( f, g)
�f
L l (X, p,)?
Ix I f

g l dp, compare with
2.5.19 Exercise. (Egorov) If: a) (X, 5 , p, ) is totally finite; b)
x, fn (x) f(x); x ])
c) for each + and d) E > 0 , there is in 5 a set E such that p,(X \ E) < E and fn l E � f i E (v. [GeO]). For each in JR., nlim = 0 yet if 00 > A ( E ) > 0, on JR. \ E, + = X ( rn , n +l then X ( rn , n +l fi o.
[Hint: If ENm �f
P,
(ENm ) < ET m .]
] ) (x)
9 { x : Ifn (x)  f(x)1 2: � }, for large
n N
N,
The symmetric difference Al1B of two sets is (A \ B) U (B \ A) . 2.5.20 Exercise. If (X, 5, p,) is a measure space: a) 5 is closed with respect to the formation of symmetric differences; b) for elements A and B of 5, the relation {A rv B} {} {p,(Al1B) = O} is an equivalence relation. The rvequivalence class containing A is denoted A� .
�f 51 rv of equivalence
2.5.21 Exercise. For rv as in 2.5.20, the set 5� classes, the map
is welldefined, i.e., independent of the choice of the representatives A resp. B of A� resp. B� . Furthermore, p is a metric in 5�. 2.5.22 Exercise. In the context of 2.5.20 and 2.5.21, if
X �f JR., 5 �f 5 ,(3 ( JR.) ,
P,
�f
A
,
then (5; p) is not a complete metric space. Its pcompletion is D, i.e., 5is pdense in 5.
88
Chapter 2. Integration
2.5.23 Exercise. If X is a set, S C !fj(X) , and M
then aR(S) =
U
�f { S a
: S a C S, # (S a ) � No } ,
aR (S a ) .
[Hint: The right member of the preceding equation is a aring.] 2.5.24 Exercise. If E E S),(ffi.) , then I : ffi. '3 t r+ I(t) � >. ([0, t) n E) is continuous. The previous assertion is valid if [0, t) is replaced, for any a in [00, (0) by any of [a, t) , (a, t), (a, t] , [a, t] , or by any of the last four when a and t are interchanged. If g : ffi. '3 t r+ g( t) E ffi. is Lebesgue measurable and t in the first sentence is replaced by g(t) is I : t r+ >. { [a, g(t) ] n E} Lebesgue measurable? 2.5.25 Exercise. a) For measure spaces (X, S, f.ln ) , n E N, such that f.ln � f.ln +l , f.l � sup f.ln is a measure. b) For (ffi., S)" >.) if f.ln �f .! >., n E N, n n . not a measure. t hen f.l cle=f . f f.l IS ,
m
n n
2.5.26 Exercise. If I E (Lu n ffi.X ) , for some nonnegative p in Lu and some II in L, 1 = p + II ·
[Hint: If L '3 In t I, 2.5.27 Exercise.
In � 0, while
l in
00
L Un  In I) E Lu .]
n=2
For some sequence {In } nEN contained d>' t
00 .
2.5.28 Exercise. If En �f [n, (0) , n E N, then: • · •
S ), '3 En ::) En +1 ; . n En = 0 ; nEN >. (En ) == 00 ;
cf. 2.2.26.
m
Ll (ffi., >.) ,
3
Functional Analysis
3.1. Introduction
For a set X, there are various important subsets of CX , e.g., L 1 , L 1 (X, p,), C( X, JR.) , etc. Each of these is an JR.vector space or a Cvector space and is endowed with a topology related to its manner of definition. Thus L 1 and L 1 ( X, p,) are metric spaces, whereas C ( X, JR.) inherits a topology from CX viewed a Cartesian product. In short, each is a paradigm for a topological as
vector space .
3.1. 1 DEFINITION. A topological vector space (TVS) (V, T) ( OR SIMPLY V) IS A CVECTOR SPACE ENDOWED WITH A HAUSDORFF TOPOLOGY T SUCH THAT THE MAPS V
x
V '3 (x, y) C x V '3 (z, x)
r+
x + y E V, r+ zx E V,
FOR VECTOR ADDITION AND MULTIPLICATION OF VECTORS BY SCALARS ( ELEMENTS OF C ) ARE CONTINUOUS. THE ORIGIN ( THE ADDITIVE IDEN TITY ) OF V IS DENOTED O. WHEN SOME NEIGHBORHOOD BASE FOR T CONSISTS OF CONVEX SETS, V IS A locally convex topological vector space (LCTV S ) . The class of locally convex topological vector spaces includes the class of normed spaces, i.e., the class of those vector spaces V for which there is a norm, namely a map II II V '3 x r+ Ilxll E [0, (0 ) such that: a) Ilxll = 0 iff x = 0; b) Ilx + yll � Ilxll + Ilyl l ; c ) for z in C and x in V , Ilzxll = Izl . Ilxll . 3.1.2 Exercise. If (V, II I I ) is a normed space, then :
Ilx  yll 2': I llxll  Ilyll l · When ( V, II I I ) is a normed space, d : V 2 '3 (x, y) r+ Ilx  yll is a metric for V. When (V, d) is complete, V is a Banach space. THEORE M 2.2.32 implies L 1 and, for any measure space (X, S, p,), L 1 (X, p,) are Banach spaces. 89
90
Chapter 3. Functional Analysis
1
When < p < 00, the set LP resp. LP( X, p,) consists of the DLS mea surable resp. Smeasurable Cvalued functions I such that
1
In Section 3.2 it is shown that if < p < 00 and, according to the convention adopted, each null function is regarded 0, II li p is a true norm and that LP and LP(X, p,) are complete with respect to II li p : each is a Banach space. For a topological space X, CX contains: as
a) Coo (X, q , the set of continuous functions I for which supp (f) is com pact (v. 1.7.23 ) ; b ) Co (X, q consisting of those continuous functions I such that for each positive E, K.(f) �f { x : I I(x) 1 2: E } is compact. For I in Coo (X, q or in Co (X, q , 11111 00 � sup II(x) 1 < 00.
xEX
3.1.3 THEOREM. WITH RESPECT TO II 1 1 00 , Co (X, q IS A BANACH SPACE. PROOF. The verification of the norm properties a) c ) for II 1 1 00 is straight forward. If {fn }nEN is a Cauchy sequence in Co (X, q , then for each x in X , limoo In (x) exists. {fn (x)} nEN is a Cauchy sequence ( in q , whence I(x) �f n+ If E > 0, since I ll oo convergence is uniform convergence, for some N and all x, { m , n > N } ::::} { llm (x)  In (x) 1 < E } , whence
mlim + oo Ilm (x)  In (x) 1
=
II (x)  In (x) 1
� E.
In short, In � I, i.e., nlimoo III  In ll oo o. ( The preceding argument is + valid well if {fn } nEN is a Cauchy sequence in Coo(X, q : for some I, In � I· However, as shown in 3. 1.5, Coo (lR., JR.) is not I ll ao complete. ) If E > 0, S. de =f { x : I I(x) 1 2: E }, and m 2: 2, then for some nm and all n greater than nrn , Sc C E �f Knm . By defix : I ln (x) l 2: =
as
{
nition, each Knm is compact and S. C
(1  �) }
n n
Knm � K , which is also
compact. On the other hand, if x E K , m 2: 2, and n 2: nm , then
Section 3.1. Introduction
whence
91
I I(x) 1 2': ( 1  �) E and x E Sf: Sf
=
K. Thus
I E Co( X, q .
D
[ 3.1.4 Note. When X is compact,
Coo( X, q Co( X, q = C( X, q .] =
� X[ . sink27rx ' n E N, then JR. and In (x) de=f k� ] + k k l , 00 =l . sin 27rX def = I(x) although { In }nEN c Coo (X, q and In (x) + � X [k k+ l ] k , k =l I tt Coo(X, q: Coo( X, q need not be a Banach space with respect to the norm I 1 00 ' 3.1.6 Example. The LCTVS Coo(JR., q is a li ll I dense subset of L l ( JR., >. ) and Coo(JR., q ¥L l (JR., >. ). Hence Coo(JR., q is not li ll I complete. The following construction, of independent interest, validates the pre ceding statements and provides an explicit I Il l Cauchy sequence contained in Coo (JR., q and for which the I I I limit is in L 1 (JR., >. ) \ Coo (JR., q. 3.1.5 Example. If X =
U
""'
For n in N,
In : JR. 3 X ft nx
n(1 o
x
1
1
if  < < 1n n  '1f 0 < X < 1
1
x)

n 1 if l   < x <  1 n
otherwise
Coo(JR., q and In 11�1 X [0, 1 ] . If a < b, there are real constants a , (3 such that if gn (x) �f In (ax + (3) , then gn 11�1 X [ ] . It follows that Coo(JR., q is I Il l dense in L l (JR., >. ). For a in (0, 1 ) and an enumeration { h h EN of the intervals deleted in the construction of the Cantor set Co: (v. 2.2.40) there are real conde stants ak, (3k such that if In k(X) =f In (akx + (3k), then supp (Ink) = h 00 and Ink 11�1 X h ' If gn �f Link , then {gn } nEN is a I Il l Cauchy sequence k=l contained in Coo(JR., q and if its I Il l limit is g, then X [O , l ]  g X( Ca ) is not a null function and is not in Coo (x, q . is in
a ,b
=
92
Chapter 3. Functional Analysis
3 . 2. The Spaces
LP, 1 � p �
00
Henceforth, L 1 denotes some L 1 (X, p,) or some L 1 derived from a DLS func tional I. As noted earlier, L l (X, p,) and L l are, for appropriately related p, and I, essentially the same. Similarly S and D differ only by a set of null functions. For p in [1, (0), LP { I : I E S, III P E L I } and, when
I E LP, IIIII � = II III P Il l '
�f
3.2.1 LEMMA . ( Young) IF: a) ¢ IS A STRICTLY MONOTONELY INCREASING CONTINUOUS FUNCTION DEFINED ON [0, (0) ; b) ¢(O) = 0; c) '1jJ ¢ 1 ; d)
lx ¢(t) dt, AND \II (y) �f lY '1jJ(s) ds;
�f
AND e ) { a, b} C [0, (0) ; THEN ab �
PROOF. In the context, the roles of ¢ and '1jJ resp.
{ (x, y) : x E [0, a] , 0 � y � ¢(x) } U { (x, y) : y E [0, b] , 0 � x � '1jJ(y) } , whence ab �
p The numbers p, p' form a conjugate pair. p' = . p1 y  axis
I I
I
Figure 3.2.1.
(a,
p
p'
D =
1:
Section 3.2. The Spaces
LP, 1
:::;
p :::; 00
93
3.2.2 COROLLARY. IF p E ( 1, 00 ) , f E LP, 9 E LP' , THEN fg E L 1 AND Il fg ll l :::; Il f ll p . Ilgllp" (HOLDER'S INEQUALITY)
�
PROOF. The measurability of fg is a consequence of 2.5.6. If ¢(x) f xp 1 , then '1jJ(x) plies
=
p
1
x ¢ I (X) = x  and
If Il f llp = Ilgllp'
1
I f(x) g (x) 1 :::; =
1,
=
xp'  . Thus 3.2.1 imp'
I f(x W + I g(x W' p p'
(3.2.3)
integration of both members of (3.2.3) yields
If either f or 9 is 0, the conclusion is automatic. If neither f nor 9 is 0, f 9 . . cle 1 1 whIch case the they may be rep1aced by F cle =f 1 1 ' 1 1 resp. G =f 9 ' I f Ip P previous argument applies to F and G. D m
3.2.4 COROLLARY. EQUALITY IN HOLDER'S INEQUALITY OBTAINS IFF
II f lip . Ilgllp'
=
0 OR Il f llp . Ilgllp' i' 0 AND I
:j:�� I :::�;; ' �
PROOF. Trivialities aside, if, for all x in a set of positive measure,
for some n in N and all x in a set
E of positive measure,
Owing to the criterion for equality in Young's inequality (3.2.1), for some
I 1 ' lg 1 < I f(xW + I g(xW . Integration positive E and all x in f Il llp ' Ilgll p' Il f llp Ilgllp' over of both members of the inequality above yields
E
E,
f(x)
(x)
'
D
3.2.5 COROLLARY. IF f, g E LP, THEN f + 9 E LP AND
Il f + g llp :::; Il f llp + Ilg llp ( MINKOWSKI'S INEQUALITY ) . EQUALITY OBTAINS IFF FOR SOME NONNEGATIVE CONSTANTS A , B, NOT BOTH ZERO, Af � Bg .
94
Chapter 3. Functional Analysis
�f
PROOF. Since I f I P , Igl P E L l , a vector lattice, h I f l P V Ilgl P E L l , whence I f + gl P :::; 2h E L I . Since I f + g l P :::; I f + g l P  l . I f I + I f + g l P l . Igl , inte gration of both members of the last inequality, the identities relating p and p', and Holder's inequality imply Il f + gll� :::; Il f + gil ;' ( II f lip + Ilgllp ) · Division by Il f + g il; ' leads to Minkowski 's inequality. If equality obtains, I f + gl P == I f + gI P  l ( l f l + Ig l ) , i.e., .E.
.E.
If + gl
�
I f I + Igl
and the elementary properties of C ( v. 1.1.4, 1.1.5) imply that for some nonnegative constants A, B, not both zero, Af == Bg . D � [ 3.2.6 Remark. For notational consistency, when p 1, p' 00 Furthermore, the discussions, when p = 1, of the appropriate ex tensions of Holder's and Minkowski's inequalities and the criteria for equality in them, take slightly different forms. First, when f E 5 , =
Il f ll oo
�
{ inf { m : 00
.
I f I :::; m a.e. } if { m : I f I :::; m a.e. } j. (/) ' otherwIse
and Loo � { f : f E 5 , Il f ll oo < oo } Young's inequality no longer applies when p = 1 since xp  l and x r+ xp  l is not strictly monotonely increasing.] .
==
1
If f E L l and 9 E L oo , then
If f, g E L 1 , then
Thus, when p = 1, both Holder's and Minkowski's inequalities are valid. If Ig (x) 1 j::. Ilgll oo , for some positive E and all x in a set of positive measure, Ig (x) 1 < Ilgll oo  E. Hence
E
r I f gl dll = r + r I f gl dll JX \ E JE < Il g ll oo r I f I dll + ( 1lglloo  E) r I f I dll E
Jx
<
JX \ Il f ll l · ll g ll oo .
JE
Section 3.2. The Spaces LP,
1 :s; p :s; 00
95
When p = 1, equality obtains in Holder's inequality iff
x E If gl IfI Igl , I f (x) g(x) 1 I f(x) 1 I g (x) 1 I l f g il l Ilfll l I l g ll l ' I f(x) g(x) 1 I f(x) 1 I g (x) l . A, B, A(x)f(x) B(x)g(x). B(x) f(x) E¥(fg, A(x)B(x) A(x) g(x) When p = 1 equality obtains in Minkowski's inequality iff almost f(x) > O. everywhere on E¥ (fg , O), g(x)
If + :f:. + for some positive E and for all in a set of positive measure, < E. Thus, as in the cal + + culation of the preceding paragraph, < Hence, if + + equality obtains in Minkowski's inequality, � + + It follows (v. 1.1.4) that there are nonnegative functions, not both zero, such that almost everywhere, Almost every= where on = 0), > 0 and > o.
E lR SERVES 3.2.7 THEOREM. IF p E [1, (0 ) , d : (U) 2 '3 r+ AS A METRIC AND (LP, d) IS A COMPLETE METRIC SPACE. PROOF. The conventions about LP together with Minkowski's inequality and the criteria that it be an equality, imply that ( LP , d ) is a metric space. When E 5 , and p E [0 , (0 , for some e in 5 ,
(f, g) Il f  g l p
)
f
(cf. the discussion following 2.4.14). The map
is a DLS functional that generates a functional J and a corresponding space £ 1 . A sequence S �f in LP is a lipCauchy sequence iff S c £ 1 and S is a JCauchy sequence in £ 1 . Since £ 1 is Jcomplete, v. 2.2.32, LP is complete. D 3.2.8 Exercise. The conclusion in 3.2.7 is valid when p = 00. [Hint: Off the null set
{In} nEN
I
E �f (u { x : I h (x) 1 > Ilfkl l cx, ) } kEN U { x : I fm (x)  fn (x) 1 > Il fm  fnl l cx, } u {m,n}CN" {fn} nEN is a uniform Cauchy sequence.]
(
)
,
96
Chapter 3. Functional Analysis
3.2.9 Exercise. For the map
v1J7) i
a) 11 1 11 2 � 2': ° and equality obtains iff 1 = 0; b) (f, g ) for w , z in C, (w + zg , h ) = w ( f, h ) + z (g , h ) ; d)
= (g , I); c)
is a metric for L2 ; e) (L 2 , d) is a complete metric space. 3.2.10 Exercise. If S) is a vector space, and an inner product E
( , ) : (S) ) 2 '3 {x, y } r+ (x,y) is such that a) e) of 3.2.9 obtain, for all X,y in S), l (x, y ) 1 � II x l 1 2 · ll y I1 2 , Il x + y l1 2 � II x l12 + Il y 112 .
C
Equality obtains in (3.2. 1 1 ) or (3.2. 12) iff for some
Ax = By.
(3.2 . 1 1 ) (3.2.12)
A, B, not both zero,
[Hint: For (3.2 . 1 1 ) , trivialities aside, 3.2.9a) d) imply that the quadratic polynomial p ( z ) �f ( zx + y, zx + y) is nonnegative. For (3.2. 12) , (3.2. 1 1 ) applies in the calculation of (x + y , x + y ) . [ 3.2.13 Note. The inequality (3.2. 1 1 ) is Schwarz's inequality; (3.2.12) is the triangle inequality.
]
]
When (x, y ) = 0, x and y are orthogonal or perpendicular: x . l y. When S C S), S1 consists of all vectors y such that for each x in S, y . l x. A subset S of S) is orthogonal (0) iff whenever X, y E S and x j. y, then x . l y. The set S is orthonormal (ON) iff it is orthogonal and for each x in S, Il x ll = l . 3.2.14 Exercise. a) An ON set is linearly independent. b) If S) j. {O}, the set oN of all nonempty orthonormal subsets of S) is a poset with respect to the order < defined by inclusion: S1 < S2 iff S1 C S2 . For an orthonormal set S �f {x>.} >'E!\ and an element x in S), the set
Section 3.2. The Spaces
LP, 1 :s; p :s; 00
iff ¢ C '1jJ. c) If x E S) and a>.
�f (x, x>.) , then II x l1 2 2':
set 5 is <maximal iff for each x in S), II x l1 2 = the net n :
L
L
L
>'E/\
97
l a>. 1 2 ; d) The ON
l a>. 1 2 iff for each x in S),
>'E/\ a>. x>. converges to x.
>'E [Hint: For c) , Schwarz's inequality applies to
(L , L ) >'E
(x, x>. )
>'E
(x, x >. ) .
For d), c) applies to prove that n(A) converges to some Y in S) (even if 5 is not maximal) and that x y E 5 1 . For e) , d) and the maximality of 5 apply.] [ 3.2.15 Note. The customary name for S) is Hilbert space. The results c) resp. d) in 3.2.14 are Bessel 's inequality resp. Parseval 's equation. A maximal orthonormal set is often called a complete orthonormal (CON) set. Hence, if {x>.} >'E/\ is a CON and 
v (A ) �f { !(A)
if # (A) < No otherwise
( v is counting measure), S) engenders a measure space
(A, �(A), ) v
so that S) and L 2 (A, ) are isometrically isomorphic.] v
3.2.16 Exercise. If T �f {xn } l < n < N < N o is a linearly independent subset of S), the algorithm represented by the formulre
produces an orthonormal set 5 �f { Yn } l
98
Chapter 3. Functional Analysis
[ 3.2.17 Note. The algorithm in 3.2.16 is known
Schmidt process .]
3.2.18 Example. For L 2 (A, v), the set { e), � X P }
[
as
the Gram
} ),E!\ is a CON set.
3.2.19 Exercise. If X is a finite set endowed with the discrete topology,
p E 1 , 00 ] , and v is counting measure defined on !fj(X ) , as sets, C(X, q , Co (X, q , coo (X, q ,
and LP(X, v ) , 1 � p � 00
,
are all eX . As Banach spaces they have various norms. If f E eX , then Il f lloo = sup I f(x) 1 when f is regarded as a member of any of the first three x EX
or LOO (X, v) ; when 1 � p < 00 Il f llp = ,
pair of these Banach spaces, the map
( L I f (X ) xE X
W
1
p.
If V, W is any
id : V '3 f ft f E W is a normbicontinuous bijection. 3.2.20 Exercise. For a set X, a countable subset E of !fj(X) , and S � a R ( E ) , if (X, S, p,) is afinite, then L 2 (X, p, ) is normseparable. Any CON set in a normseparable Hilbert space is finite or countable. 3.2.21 Exercise. For (X, S, p, ) , if X E S, p,(X) 1, and f E L2 (X, p,), then =
L [f(x)  L f(y) dP,(Y)] 2 dp,(x) = L [J (XW dp,(x)  [L f(x) dp,(x)] 2
3.2.22 Exercise. a) If
�f {O, 1}, S �f !fj(X) , ° < p < 1, p,({1}) �f P �f 1  p,({O}), and f cle=f X { l } ' then X
ml �f L f(y) dp,(y) p, m2 � L [J(y)] 2 dp,(y) p, L (f(y)  md2 dp,(y) L [J(y)] 2 dp,(y)  [L f(y) dP,(Y)] 2 , m2  mi = p  p2 O. =
=
=
=
>
Section 3.2. The Spaces LP, 1 :::; p :::; 00
b) If
99
X �f {o, l} n �f { y : y �f (Yl " " ' Yn ) } ' Y7 = Yi , S �f �(X), L ° < p < 1, lAy ) �f f.ln ( Y l , . . . , Yn) = pL:=1 Yk (1 pt  :=1 Yk , f(y ) �f L�=nl Yk , 
then
Ml �f irx f(y ) df.ln (Y )
1
=
�n
tk ) Yk df.ln (Y) :2 [tk=l 1 Y% df.ln + Lk#l 1 YkYI df.ln (Y)] =
r
X
= � np = p, n
M2 �f x [J(y )] 2 df.ln (Y ) = x X 1 [ p2 = 2 np + (n2  n ) p2 ] = ;;P + p2  ;; , n r [J(y )  Md 2 df.ln (Y ) = M2 M� = p(l n p) :::; � 4n . ix _
Hence
1 < . n
4
(3.2.23)
3.2.24 THEOREM. ( The WeierstraB Approximation Theorem ) E AND E > 0, THERE IS A polynomial function SUCH THAT
IF
f C([O, l],lR) B sup { I f(x)  B(x)1 0 :::; x :::; I } �f I l f  B l oo < E. :
PROOF. There is a positive 15 such that
{I x  y l < r5 } { I f(x)  f(y) 1 < � } . 4 oo If n > sup {r5 4 , 1 l }, then xk (l  xt k 1 implies � k=o I f(x)  Bn (f)(x) 1 �f f (X) f ( �) � x k (l  x) n  k '*
��
t( ) l I � ( ) k (l  x) n  k l ) ( ) f f(x X � �)] ( [ I� =:
=
:::; I L I*  xl
100
Chapter 3. Functional Analysis
2': 1
in the second summand of the right member above,
I I (x)  Bn (f)(x) 1 � � 2 1111100 � (�) xk (l  xt  k +
<:
+
E
� "2 + � �2
11111 :",
� ( �n�:) ' (�) X' (l  x)n' 2 1111100 . n 2l . 4ln [v. (3.2.23)] 11111 :", . 2n 2
i 2 11 f ll � +
I I  Bn(f) 1 00 E. Thus I is uniformly approximable
< If < � , then 2 2n2 by one of the Bernstein polynomials
D
E
I E C([a, b] , lR) , THERE IS A 111  B lloo < E. 3.2.26 COROLLARY. THE FUNCTION I I [ 1, 1] '3 x r+ I x l IS UNIFORMLY
3.2.25 COROLLARY. IF a < b, > 0, AND POLYNOMIAL FUNCTION SUCH THAT
B
:
APPROXIMABLE BY POLYNOMIALS. An alternative approach to the HolderMinkowskiSchwarz inequalities flows from the following discussion of convex functions and their elementary properties. 3.2.27 DEFINITION. A FUNCTION ¢ : (a, b) '3 WHENEVER ° < t < 1 AND a < p < q < b,
x r+ ¢(x) E lR IS convex IFF
¢[tx + ( 1  t)y] � t¢(p) + (1  t)¢(q) .
(3.2.28)
3.2.29 THEOREM. IF 00 � a < b � 00, THEN ¢ IN lR( a , b ) IS CONVEX IFF ¢(p)  ¢(q) FOR q IN (a, b) , THE MAP (a, b) \ {q} '3 r+ INCREASES MONO pq TONELY, i.e., IFF THE SLOPE OF THE LINE (PQ) THROUGH P [p, ¢(p)] AND Q [q, ¢(q)] DOES NOT DECREASE WHEN P MOVES RIGHT.
P
�
�
Section 3.2. The Spaces
LP, 1 .s p .s 00
101
yaxis
a
p
q
Figure 3.2.2.
p
xaxis
PROOF. In English, the condition (3.2.28) says that on any subinterval of (a, b), the graph of y ¢(x) lies below the chord through P and Q. Thus, if ¢ is convex, as in Figure 3.2.2, then =
slope(PQ) = slope(RQ) 2': slope(P'Q) , slope(PQ) .s slope(PR) = slope(P P') ,
(3.2.30)
which, for the two possibilities, p < p' .s q and p < q .s p' is the burden of the assertion that the slope increases as the determining point P moves right. Conversely, if the slope of (PQ) increases as P moves to the right, e.g., q  p' when p < , < q, and t cle=f , then qp
P

° < t < 1, p' = tp + (1  t)q, ¢(q)  ¢(p ) < ¢(q)  ¢ (p') , qp q  p' ¢[tp + (1  t)q] .s t¢(p) + (1  t)¢(q). D 3.2.31 Exercise. If ¢ : (a.b) r+ lR is convex and a .s p < r < s < q .s b, for some constant L(p, q) , I¢(s)  ¢(r) 1 .s L(p, q) l s  rl . A function ¢ convex on (a, b) is Lipschitzian on every subinterval of (a, b) . Hence a convex function ¢ is absolutely continuous on every subinterval of the (open) interval that is its domain, i.e., if E > 0,
Chapter 3. Functional Analysis
102
for a positive J, if a n
< P In < ql < < q2 < . . . < Pn < qn < b and P2
k= l k= l [ 3.2.32 Note. In 3.2.27 the hypothesis that ¢ is defined on an open interval (a, b) is essential; e.g., if ¢(x) �f
{O1
�f x E [0, 1) , 1f x = 1
then ¢ is convex on [0, 1] but ¢ is not continuous when x = 1.] 3.2.33 Exercise. a ) If ¢" exists on (u, v), then ¢ is convex if ¢" > ° on (u, v). b ) If ¢ is convex, then ¢" 2': ° on (u, v). c) exp is convex on lR. 3.2.34 Exercise. a) If tk 2': 0, 1 .s k .s K, K
K
L tk k= l
=
1, and {a d := l C lR,
then min a k ; b ) If ¢ is convex, then, via matheak .s " � tkak .s l max l < k 
(t, tkak) .s t, tk¢ (ak) ' 
mati cal induction, ¢

3.2.35 THEOREM. ( Jensen's inequality ) IF: a) ¢ IS CONVEX ON (a, b) ; b ) (X, S, tt) IS SUCH THAT X E S AND tt(X) = 1; c) f E L l (X, tt ) ; d ) f (X ) C (a, b) , THEN ¢ f dtt .s ¢ o f dtt.
< <
(Ix )
Ix
PROOF. If a z b, then 3.2.29 implies that the righthand derivative resp. lefthand derivative i.e., ' ¢( t)  ¢( z) resp. D  ( z ) D + ( Z ) cl=ef l1m tJ·z t z A'f'
A'f'
A
exists and �f sup D  ¢(p) .s inf D+¢(q) z
A
< <
cl=ef l'1m ¢( t)  ¢( z) , tz ttz
� B.
¢(t) 2': ¢(z) + m(t  z),
Ix
i.e. , the graph of any supporting line lies below the graph of ¢. If z �f f dtt, a x b, and t = f(x), then {t, z} C (a, b) and
< <
m (f(x)  z) + ¢(z) .s ¢ [J(x)] .
(3.2.36)
103
Section 3.3. Basic Banachology
Since ¢ is continuous and f is measurable, integration of both members of (3.2.36) is permissible. Furthermore, p, (X) = 1 and z is fixed, whence ¢(z) dp,(x) = ¢ (z) and Jensen's inequality follows. D
1
[ 3.2.37 Remark. The conclusion in 3.2.34b) is an elementary instance of and a motivation for Jensen's inequality. The PROOF above is an efficient replacement for a tedious limiting argument based on approximations of f by simple functions.]
The inequality (3.2.3), the heart of the proof of Holder's inequality, is a consequence of the convexity of expo Indeed, if f(x)g(x) i 0, for some If(x) P Ig(x W' a and b in JR, exp ( a ) = I f(x) 1 and exp(b) = Ig(x) l . Thus , + p p' the right member of (3.2.3) , is the convex combination exp(ap)
+ �� p The convexity of exp implies
(
ap . bP' If(x) 1 . Ig(x) 1 = exp p p' 

)
=
exp (bp') . p'
exp ( a + b) .s
exp(ap) exp (bp') . + p p '
Jensen's inequality applied when ¢ �f exp shows that if f is measur able and JRvalued and exp( (0 ) �f 0, then exp f dp, .s exp ( f ) dp,.
(1

)
1
Uk �f exp [J ( ak ) ] , and p, (ak ) �f Ok , 1 .s k .s n, then L Ok 1 and the preceding inequality reads II U�k .s L Ok Uk : k= l k= l k= l
If X �f {a l , . . . , a }, n
=
n
n
n
Geometric means do not exceed arithmetic means. 1 1 In particular, if n = 2, 0 1 =  , and 02 = ; , again the essence of p p (3.2.3 ) follows.
3.3. B asic B anachology
In many parts of mathematics, the study of a set 5 is carried out in part by the study of a wellchosen set of maps of 5 into a concrete and better understood structure Y. The more the maps in can and do respect the structure of 5, the more likely is to reveal the nature of 5. A map in respects the structure of 5 if respects both the alge braic and topological character of 5. More specifically, if 5 is a topological vector space V, is useful if it is a vector space homomorphism of V into
m M m
M M
M m
104
Chapter 3. Functional Analysis
m
m
m m m m
m m
c: ( ax + by) = a (x) + b ( y) ( E [V, C] ), whereby is respectful of the algebraic structure; and is continuous: if n : A '3 A r+ n(A) E 5 is a net and n(A) converges to x, then [n ( A ) ] converges to (x ), whereby is respectful of the topological structure. Such an is a continuous linear functional. The set of all continuous linear functionals on V is its dual space or dual denoted V', [V, ej c or simply [V] c . When the elements of V are de noted . . . , the elements of V' are denoted . . . and when E V and V', the number ) Thereby a in V reveals it ( ) is written ( E self as an element of (V' ) ' �f V". In analogy with the situation in 5) , when 5 cle f {VA AE /\ C V resp. 5' cle=f {VA AE/\ C V , v,
'
v
=
'
v
}
m
'
v
v, v
5 1
cle=f { , cle=f {
v
51
' :
v
:
.
' v ,
v
m
v
, '} V, E V, , ( VA , V' ) = 0 } , E V, ( V' V'A ) = 0 } .
v
_
W
3.3.1 DEFINITION. THE TOPOLOGICAL VECTOR SPACES V, CONSTI IFF: a) V C TUTE A dual pair {V, AND C V'; b) FOR resp. ) = 0 IFF = 0' resp. = O. FIXED, ( [ 3.3.2 Note. The set of not necessarily continuous linear maps of V into C is denoted V* by some. In extensive treatments of the subject of topological vector spaces, e.g, [Kot, Sch] , the dis tinction between [V, ej , and [V, ej c ( = V') is explored at some length. By definition, V' C V*. The Hausdorff Maximality Principle im plies that a vector space V contains a maximal linearly indepen dent subset H �f {xAh E /\' usually called a Hamel basis. When the topological structure of V is sufficiently rich, e.g., when V is an infinitedimensional Banach space such as L 1 ( [0, 1] , A), a Hamel basis for V leads to the conclusion V' ¥ V*, [Ge3, GeO] . '
v
'
v, v
W}
'
v
W'
v
W
v
The results in Section 3.2 imply that if p E [1, (0 ) and V = LP , then Lpi C V'. Later arguments (v. Section 4.3) show that in mildly restricted circumstances: a) if p E [1, (0 ) , then (LP) ' = LP' ; b) if X is a locally compact Hausdorff space, corresponding to each continuous linear functional F operating on one of Co (X , q or Coo (X, q , there is a complex measure space (X, 5 (3, p, ) such f dp,.] that for each f in Co (X, q or Coo (X, q , F(f) =
Ix
Section 3.3. Basic Banachology
105
T I II ) I I) T x K I T (x) 1 .s K ll x ll . T n+CXJ xn x n+CXJ (xn  x) K T K Xn li T (xn ) 1 Il xnll l i T CI ::II ) I l i T (Yn ) I ever, I Ynl 1 1, whence }��� � = 0, whereas l i T (�) I > yn, in con tradiction of the continuity of T at O . D [ 3.3.4 Note. In the context above, the norm of T is II T I �f inf { K : I T(x) I .s K l x l } . ]
ARE BANACH SPACES, A IN AND (F, 3.3.3 THEOREM. IF (B, [B, F] IS CONTINUOUS IFF EITHER: a) IS CONTINUOUS AT 0; OR b ) FOR SOME IN [0, (0 ) AND ALL IN B, = iff lim PROOF. Since lim 0, the linearity of implies the truth of a) . For b ) note that if some as described exists, is continuous at o . Conversely, if no such exists, for each n in N, there is an such that >n or, equivalently, �f > n. How=
==
Exercise. Exercise. T 1 . x "l } I T I = sup { I T(x) I xi i sup = { II T(x) 11 O .s I xi i .s I } = sup { I T (x) 1 : Il x l = I } , x') I : I l x I · I l x' I "I ° . { (T(x) = sup I } . xi i I I l x'll
3.3.5 With respect to the norm defined in 3.3.4, the set [B, F]c of continuous elements of [B, F] is a Banach space. 3.3.6 For in [B, F]c, o
:
The results that follow are at the heart of Banachology. The first is formulated in terms of a seminorm, i.e., for a vector space V, a map
p : V '3 x r+ p(x) E [0 , (0 ) such that p(x + y ) .s p(x) + p( y ) and, for z in C, p(zx) = I z l p (x). 3.3.7 THEOREM. (HahnBanach ) IF p IS A SEMINORM ON A VECTOR SPACE V, W IS A subspace OF V, m E [W, q , AND FOR ALL IN W,
w
1 m (w) 1 .s p (w) ,
THERE IS IN [V, q AN
m SUCH THAT FOR ALL x IN V, I m(x) 1 .s p(x)
106
Chapter 3. Functional Analysis
AND ml w = m: m IS AN EXTENSION OF m. PROOF. The argument is carried out in three steps: a) It is assumed that m(W) c JR and for some y in V, W ¥ U = { w + zy : w E W, z E JR } . b ) One of the equivalents of the Hausdorff Maximality Principle is used to remove the restriction on U. c ) The restrictions m(W) c JR and z E JR are removed. a) For all w in W, some a in JR, and all z in JR, the formula m(w + zy)
�f m(w) + za
defines a linear functional on U and m l w = m. The condition I m( u) 1 :::; p( u ) for all u in U places a demand on a, since Im(w + zy ) 1 :::; p(w + zy) for all w in W iff for all nonzero z and all w in W, Im(zw + zy) 1 :::; p(zw + zy) iff ( after division by I z l ) I m(w)  al :::; p(w  y) = p(y  w) iff m(w)  p(y  w) :::; a :::; m(w) + p(y  w) .
(3.3.8)
q)]
(A similar argument appears in the PROOF of 3.2.35 where the existence of an m in sup D� ¢(p), inf D + ¢ ( is used. ) z
[
m( u)  m( v ) = m( u  v ) :::; p( u  v ) = p( u y+y v ) :::; p( u  y) +p( v  y), m( u )  p( u  y) :::; m( v ) + p(y  v ), sup [m( u)  p( u  y)] :::; inf [m(v ) + p(y  v )] . uEW
yEW
Since ml w = m, at least one a satisfying (3.3.8) exists. b ) The set S of proper superspaces of W to which m can be extended correctly is nonempty since S includes U. With respect to consistent in clusion as order, i.e., U1 < U2 iff U1 c U2 and the extension of m to U2 coincides on U1 with the extension of m to U1 , S is a poset. The Hausdorff Maximality Principle applies and provides a maximal extension M of m. If the domain of M is not V, the discussion in a) implies a contradiction of the maximality of M: M is defined on all V. c) If m(W) c C and r(w) � � [m(w)] , on W, Ir(w) 1 :::; Im(w) 1 :::; p(w) .
Section 3.3. Basic Banachology
107
The arguments in a) and b) apply to r since a vector space over C is automatically a vector space over JR.. If R is the extension of r to V, then
M : V '3 x ft R(x)

iR(ix) =
sgn [iVf(x )],
p (x) .
D
maps V into C and M is an extension of iii . If x E V and e iQ then e iQ M(x) = I M(x) 1 {= � [e i Q M(x)] } and I M(x) 1
=
M (e i Qx)
=
R (e iQx) :s; p (e i Qx)
=
3.3.9 COROLLARY. IF B IS A BANACH SPACE AND 0 i x E B, THERE IS IN B' AN x' SUCH THAT Ilx' ll = 1 AND (x, x') i O. PROOF. The set W �f { zx : z E C } is a subspace of B. The formula iii : W '3 zx ft iii ( zx) �f zllxll E C defines a linear functional on W and iii ( x)
Ilxll i o . With I II serving as
=
p in 3.3.7, it follows that for some x' in B', x' l w = iii and for each y in B, Ix'(y) 1 :s; Ilyll · The result in 3.3.6 implies Ilx' ll :s; 1, and since (x, x') = Ilxll, it follows that Ilx'll = 1. D 3.3.10 COROLLARY. IF W IS A CLOSED SUBSPACE OF A BANACH SPACE B AND x tJ. W, FOR SOME x' IN B', (x, x') = 1, x'(W)
=
{O } ,
i.e., x' E Wi ; IF d �f inf { Ilx  wll : w E W } , THEN Ilx' ll :s;
�.
PROOF. The subspace Y �f { zx + w : z E C, w E W } is closed. The for mula iii : Y '3 zx + w ft iii ( zx + w) �f z defines a linear functional on Y and iii ( W) = { O } . If z i 0, Ilzx + wll = I z l · x + I z l d, whence l iii ( zx + w) 1
=
Izl
:s;
l 7 1 2':
� Il zx + wll ·
(3.3.11)
Hence iii E [W, Clc , and the HahnBanach Theorem (3.3.7) implies for some x' in B', X' l y = iii , (x, x') 1, x' E Wi. Moreover (3.3.11) implies =
Ilx'll
:s;
�.
D
108
Chapter 3. Functional Analysis
. ExercIse.
x, as in 3.3.10, I x' I = "d1 : [Hint: For some sequence {wn} nEN contained in W, I l wn  x ii + d and 1 m (wn  x ) 1 = 1 :::; I l x' l · l xn  w l · ] 3.3.13 Exercise. If {vn} nEN is a linearly independent set in a Banach space the statements: 3.3.12
For
X l cle=f V I , for some x�, (x , x�) = 1, n l Xn � Vn  L (Vk , X�_ l ) Xk , k= l for some x�, x� span [(X l ' . . . ' xn _ d ]1 , and (xn , x�) = 1, 1
E
engender an analog of the GramSchmidt process and lead to a biorthogonal such that for n in N , pair
({xn} nEN ' {x�J nEN )
m, if m = n clef Urn (Kronecker ' s functi. on ) , n otherwIse and for which span ({Vn} nE N ) span ({xn} nEN ). 3.3.14 Exercise. A proper closed subspace M of a Banach space B is .
=
s:
=
nowhere dense in B. COROLLARY 3.3.10 implies that for any Banach space B, B' is not only nonempty but is equipped with a ready supply of separating elements that distinguish any nonzero from ° and hence any two elements of B from one another: if i y, for some in B', i 0, i.e., ias defined in 3.3.4, B' is a Banach 3.3.15 With respect to space. Although topological completeness plays no role in the HahnBanach Theorem, topological completeness is an essential ingredient in the next results.
x Exercise.
x
x'
(x  y , x') II I
(x, x') (y , x').
3.3.16 THEOREM. (The Open Mapping Theorem) IF B AND F ARE BA NACH SPACES, T E [B, AND T(B) F, THEN T IS OPEN: A CONTINU OUS LINEAR SURJECTION ( A CONTINUOUS epimorphism) OF ONE BANACH SPACE ONTO ANOTHER IS OPEN . PROOF. To show T is open, it suffices to show that for some positive p, T [B(O, It] contains some B(O, pt . Since B = B(O , nt , it follows that F = T [B(O, nt] · Be
F] c ,
U nEN
=
U nEN cause F is complete, F is of the second category, i.e., F is not the union
Section 3.3. Basic Banachology
109
(0,
of countably many nowhere dense sets. Thus some T [B no t] is not nowhere dense, i.e., T [B no t] is somewhere dense: for some positive R and some y, T [B no )O] contains B(y, Rt �f W. Since for some x, T(x) = y, if I l x ll = D, then
(0,
(0,
T [B (O, no + D)O] :::) T [x + B (O , no )O] :::) B(O, Rt , T [B(O, 1) ° ] :::) B
(0, no R+ D ) ° �f B(O, r t .
(The technique just used may be described as translation/scaling.) If E E (0, 1) and Z E B(O, rt, for some X l in B(O, I t , (whence Z  Z l E B(O, Er t ) . For some X2 in B(O, E t ,
(0,
(whence Z  Z l  Z2 E B E2 r) 0 ) . Induction yields sequences {xkh EN {z kh EN such that for k in N, I l xk ll
< Ek l (whence Xk E B (O, Ek  I ) O ) , ( ) clef Z k E B (0 , Ek I r) ° ,
T Xk
=
li t. I < z
Zi
E k r.
f Xk E B ( 0, 1 � E ) and, owing to the continuity of T, 0 T (x( ) = z. Thus T [B (0, ] :::) B(O, r t . Since T is linear, ) � 1 E
Hence x( �f
k=l
T [B(O, It] :::) B[O, (1  E)r] O
�f B(O, p) .
D
3.3.17 THEOREM. FOR BANACH SPACES B AND F, IF T E [B, F]c , T I EXISTS, AND T(B) = F, THEN T I E [F, BJ c . A continuous linear bijection between Banach spaces is bicontinuous.
PROOF. The graph 9 �f { {x, T(x) } : x E B } of T is a subset of B x F. Normed according to the formula II {x, T(x) } 1 1 � I l x ll + I IT(x) ll , B x F is
110
Chapter 3. Functional Analysis
a Banach space and 9 is a closed subspace, hence is itself a Banach space. ( )} ft Moreover, for : 9 '3 exists. The Open Map E F, ping Theorem (3.3.16) implies is open, whence is continuous. Since : 9 '3 ( )} ft E B is (automatically) continuous (v. 1.2.15 and 1.7.19), and since is continuous. = D
f {x, T x
71"1
f 1 l f
{x, T x T(x) f x 1 1 1 71"1 o f T , T
3.3.18 COROLLARY. (The Closed Graph Theorem) IF B AND F ARE BA NACH SPACES, E [B, F] , THEN E [B, F] c IFF THE GRAPH 9 ( ) � 9 IS CLOSED IN B F NORMED AS IN THE PRECEDING PROOF.
T
T
X
T
PROOF. If 9 is closed, it is a Banach space. The map ft
f : B '3 x [x, T (x) ] E 9 is in [B, 9] , f  1 exists, and f  1 E [9, B] . If f 1 [xn , T xn Xn
Xn x, f l n�= f 1 f 71"2 : 9 '3 [x, T(x) ] T(x) E:F is also automatically continuous, T ( = 71"2 f) is continuous. Conversely, if T is continuous, and [xn , T (xn ) ] converges to some (x , y ) in B F, then Xn converges to x and the continuity of T implies T ( xn ) converges to T(x): y = T(x), i.e., ( x , y) E 9 , 9 is closed. D then is continuous. The = ( ) ] = and, since lim Open Mapping Theorem implies is open, whence is continuous. Since ft
0
x
The context for 3.3.18 is the theory of Banach spaces. In a more general context of topology there is the following analogous result. 3.3.19 If X and Y are compact topological spaces, f E y X , the graph of f, 9(1) � (x, y) : y = f(x) } is closed iff f is continuous. That a Banach space is a set of the second category is essential in the PROOF of
Exercise.
{
3.3.20 THEOREM. (BanachSteinhaus) IF B AND F ARE BANACH SPACES,
AND FOR EACH
x IN B, sup I T), (x) 11 < 00, FOR SOME POSITIVE M, sup l i T), I :::; M . ),E /\
),E /\
Section 3.4. Weak Topologies
111
{
PROOF. If, for n in N, En �f X
sup I I T), (x) 1 1 :::; n
} then ,
B
=
U
En . nEN Thus some En is somewhere dense and, by virtue of the translation /scaling ),EA
device used in the PROOF of the Open Mapping Theorem (3.3.16), for some positive r, El is dense in B( O, r t . Hence, for all x in El , sup I I T), (x) 1 1 :::; 1. ),EA
If I l z ll :::; r, some sequence { xn} nEN contained in El converges to z . For each >', (3.3.21 ) II T), (z ) 11 :::; l i T), (xn ) 1 1 + II T)' II · ll z  xn l l ·
For large n, the second term in the right member of (3.3.21) is small, whereas the first term in the right member does not exceed 1: l i T), I I :::; � . r When the translation /scaling is reversed, the required assertion follows. D 3.3.22 DEFINITION. A SEQUENCE S �f {Xn} nEN IN A NORMED (VECTOR) SPACE V IS summable IFF summable IFF
00
00
L xn E V.
n=l
THE SEQUENCE S IS absolutely
L I l xn ll < 00.
n=l
Exercise.
3.3.23 a ) A normed vector space V is complete iff every abso lutely summable sequence is summable. b ) The result in a) offers another proof that LP is a complete metric space. [Hint: If: If {xn} nEN is a Cauchy sequence, for some subsequence {Xn k } kEN ' { Xn k+1  Xnk } kEN is absolutely summable. A modifi cation of the argument in 3.2.7 applies. N
Only if: The partial sums Sn �f L Xn form a Cauchy sequence. ] n=l
3.4. Weak Top ologies
Section
The results in 3.3 deal with the uniform or norminduced topology for the set [B, F]c of continuous linear operators between the Banach spaces B and F. For some important invpstigations, other topologies are more useful. For a Banach space B, the sequence B, B', (B') '
�f B", . . .
112
Chapter 3. Functional Analysis
is meaningful. For any fixed x in B, (x, x') is a continuous function on B' and thus x may be regarded as an element of B". 3.4.1 Exercise. If B is a Banach space, the map � that identifies each x in B with its correspondent in B" is an injection, and for each x in B, 11�(x) 11 = I l x l l · 3.4.2 Exercise. If B, F are Banach spaces, T E [B, F] and for each x in B, I I T ( x) 11 = I l x l l (T is an isometry), then T E [B, F] c and T (B) is a closed subspace of F. (Hence � (B) is a closed subspace of B".) Owing to the last two results, whenever convenience is served, no dis tinction is drawn between B and �(B). 3.4.3 DEFINITION. THE BANACH SPACE B IS reflexive IFF �(B) = B". 3.4.4 Exercise. The Baill\ch space B is reflexive iff B' is reflexive. (hence, . ). · ff B" , B'" , . . . are refleXlve For an infinit�dimensional Banach space B and its dual B' there are two important topologies different from those induced by their norms. I
3.4.5 DEFINITION. FOR THE DUAL PAIR {B, B'} OF BANACH SPACES, (B, B') resp. (B', B) IS THE WEAKEST TOPOLOGY SUCH THAT EVERY x' resp. x IS CONTINUOUS ON B resp. B'. THESE TOPOLOGIES ARE THE weak ' resp. weak! TOPOLOGIES FOR B resp. B'. THE NOTATIONS BW resp. ( B' ) W ARE USED TO SIGNIFY B resp. B' IN ITS WEAK resp. WEAK ' TOPOLOGY. 3.4.6 Exercise. a) For a Banach space B the set a
a
N (0; x� , . . . , x�; E ) � { x : x:
E B', E > 0, I (x, x; ) I
< E, 1 :::; i :::; n } ,
a
is a convex (B, B')neighborhood of O. Dually,
a
is a convex (B', B)neighborhood of 0'. Furthermore, each such neigh borhood is circled, i.e., if ), E C and 1 )' 1 :::; 1, then )'N c N. b) The set NW resp. NW ' of all sucR (B, B')neighborhoods resp. (B', B)neighborhoods is a base of neighborhoods at 0 resp. 0'. c) The sets x� , . . . , x� and X l , . . . , Xn may be chosen to be linearly independent without disturbing the conclusions in a) . d) With respect to these topologies B and B' are LCTVSs. 3.4.7 Exercise. For a Banach space B, the weak resp. weak' topology for B resp. B' is weaker than the norminduced topology. The weak resp. weak' topology is the same as the norminduced topology iff dim (B) E N. a
a
Section 3.4. Weak Topologies
113 (J"
3.4;8 LEMMA. IF B IS A BANACH SPACE, FOR THE TOPOLOGIES (B, B') AND (B', B), {B, B'} IS A DUAL PAIR AND EACH MEMBER OF THE DUAL PAIR IS THE DUAL OF THE OTHER. PROOF. Since B and �(B) may be regarded as indistinguishable, a) in 3.3.1 is satisfied. If x' E B' and (x, x') 0, then x' 0' by definition. If x i 0 the HahnBanach Theorem (3.3.10) implies that for some x', (x, x') ;f:. 0, whence b) in 3.3.1 is also satisfied. If m E (BW ) ' , since (B, B') is weaker than the norminduced topol ogy, m is normcontinuous, whence (BW) ' c B'. On the other hand, if x' E B' and E > 0. for any x in N (0; x'; E ) , I (x, x') 1 < E, whence (J"
==
=
(J"
)
' If m E ( (B') W ' there is a weak' neighborhood I,
such that {x . . . , xn } is linearly independent and if y ' E N, then 1m (y') 1 < 1.
( )
rSy, rSy' E N, whence m < 1, 20' 2 0' sup I (Xk , y') I . In particular, if y' E [span (x l , . . . , xn )] i ,
For any y', if a = sup I (Xk , y' ) I , then l �k�n
i.e., m (y') <
� l �k�n u
then m (y') = 0. For the biorthogonal pair (cf. 3.3.13) {Yk } 1 9 � n ' { YD I �k�n asso ciated with {Xl , . . . , xn } , if z ' E B', then z' =
Since
v
'
�
(Yk , Z') Y� +
( � z'
}
{
( Yk ' Z') Y� '
m ( z ' ) = m ( u') + m ( ) v
.
)�
u' + v
'
,
E [span (X l , . . . , Xn ) ] i ,
n
Consequently, m may be identified with
L O'kYk ·
k=l
D
In a topological vector space V, a set S containing {O} is absorbent iff for each in V there is a nonzero t such that tv E S. v
Chapter 3. Functional Analysis
114
3.4.9 Exercise. For a topological vector space V and a neighborhood N of 0: a) N is absorbent; if a j. 0, then aN is a neighborhood of 0; AN
U
E,
1 >' 1 < 1
<E
is circled; for some positive if I r l then r N C N; at 0 there is a base of circled neighborhoods; b) A(x, N) �f { a : 0' 2': 0, x E aN } j. 0; if N is circled and
PN : V '3 x r+ inf { a : 0' 2': 0,
x
E aN } ,
then: bI) { {a E A(x, N ) } 1\ {;3 > a}} '* {;3 E A(x, N) } ; b2) for some N, {x j. O} '* {PN (X) j. O}; b3) PN (O) = 0 ; b4) for t in C, PN (tX) = I t l pN (X); b5) if N is convex, then PN (X + y ) :::; PN (X) + PN (Y ) ; c) if PN is a function for which bI)b5) obtains, then { x : PN (X) I } is a convex, circled, absorbent set; d) V is a LCTVS iff for some set {P>. } >. E /\ of functions conforming to b I) � b5), the set N �f { N>. : N>. � { x : P>. (x) 1 } A E A is a base of neighborhoods at 0, and for each v in V \ {O}, there is a A such that
<
}
<
v tJ N>. .
E
{ �, �}
[Hint: b5): If > 0, for some positive a and ;3 , 0' PN (X) :::; a and ;3  :::; PN (Y ) :::; ;3 . Furthermore,
E<
X+Y
0' + ;3
E
e N,
a ;3 = __ � + __ � E N.
0' + ;3 0'
0' + ;3 ;3
<
Hence PN (X + y) ::; 0' + ;3 PN (X) + PN (Y ) + 2 E. ] [ 3.4.10 Note. The function PN is the Minkowski functional as sociated to the neighborhood N [Kot, Sch] . Owing to c)�d) , PN is a seminorm. If A(x, N) = 0, by definition, PN (X) = 00. If, for each x and each N, A(x, N) j. 0 , PN is a norm.] a
3.4.11 LEMMA. IF B IS A BANACH SPACE, THEN �(B) IS (B", B')DENSE IN B". PROOF. Otherwise for some weak' neighborhood N � N (0") �
{ Y" : l (x� , Y" ) 1
< E, 1 ::; k :::; n }
Section 3.4. Weak Topologies
115
x" in B", (x" + N) n �(B) = 0. For w" in L �f span [x" + �(B)] , ' ' ( x ) r+ a I. S a lmear map f L mto the map m : L '3 w" cl=ef ax Since (x" + N) �(B) = 0, [x" + �(x)] 2': 1. Since N is circled, and �(B) is a vector space, if x E B and a j. 0, then and some
"
n
i.e.,
c +"
\Lrr .
0
PN
I m (w") 1 :::; (w"), an inequality that remains valid when a = 0. PN
The HahnBanach Theorem (3.3.7) implies that there is a linear func tional m : B" '3 r+ m ( ) E C such that u
"
"
u
"
"
"
"
Furthermore, if 15 > ° and E r5N, then PN ( ) :::; 15, whence m( :::; 15. In short, m is (B", B')continuous. However, 3.4.8 implies that for some z ' in B', m ( ) = (z ' , ) Since m[�(B)] it follows that z ' = 0', whence m is the zero functional. However, m (y " ) = 1, a contradiction. D
1
u
"
u
 v") 1
 v
a
"
"
u
u
.
u
 v
=
{a},
3.4.12 LEMMA. (Alaoglu) IF B IS A BANACH SPACE, THEN B (0', 1) IS (B', B)COMPACT. a
I) } [0, I x i Ix. { (x,x') : x' K XXE B lx x' (x') {(x, x')}xEB (O, I ) K x,
x
PROOF. For each in B, E B (0', ]� Ty C is compact in the product topol chonov's Theorem implies �f �f E is, by ogy T. The map (J B (0', 1) '3 r+ (J virtue of the HahnBanach Theorem, injective and (B', B) T continu ous, whence on (J [B (0', 1 )] �f Y, (J l is T (B', B) continuous. For y in B, and in C, the maps :
a
a
a �x, y : K '3 {aX}xE B r+ ax +y  ax  ay E C, 1]n ,x : C K '3 (a , {aX}xEB ) r+ O'ax  anx E C, are continuous. Hence, for each map, the inverse image kx ,y resp. kn ,x of { a } is closed. Thus, [B (0', 1)] = Y (n kx , y ) n (n kn ,x ) is a X ,Y n Ix x
(J
=
a
closed, hence compact set. It follows that B (0', 1) is (B', B)compact. D 3.4.13 THEOREM. THE BANACH SPACE B IS REFLEXIVE IFF B(O, l) IS WEAKLY COMPACT.
116
Chapter 3. Functional Analysis
]
PROOF. If B is reflexive, then �(B) = B", � IB(O, 1 ) = B (0 " , 1), and the topology inherited by �[B(O, 1)] from the weak' topology of B" is the same as the weak topology of B(O, 1). By virtue of 3.4.12, B(O, l) is weakly compact. Conversely, if B(O, 1) is weakly compact, the (B, B')  (B", B') continuity of � implies �[B(O, 1)] is (B", B')compact. However, 3.4.11 implies �[B(O, 1)] is (B", B')dense in B (0 " , 1 ) , whence a
a
�[B(O, 1)
a
a
] = B (0 " , 1) ,
which implies �(B) = B".
D
T
3.4.14 DEFINITION. FOR BANACH SPACES B AND F AND IN [B, Fl c THE ADJOINT IS THE UNIQUE ELEMENT OF [F' , B' e SUCH THAT FOR = EACH IN B F' , [ 3.4.15 Remark. The notation for the adjoint is consistent with the notation for the dual space. Some writers use instead of correspondingly they use instead of
(x, y')
T'
[x, T' (y')] [T (x) , y']. T' V' T*
X
V';
]
T'.]
V*
T' ]
3.4.16 Exercise. a) The statement in 3.4.14 is meaningful, i.e., exists and is unique. b) c) If a , b C C and C [B, F e , then (a s + = as' + Hint: a) The Closed Graph Theorem (3.3.18) applies. b) 3.3.6 applies.]
[
bT)'
I T'I I I T I . bT'.
=
{ }
{S,T}
3.5. B anach Algebras
Gelfand [Gelf] introduced the notion of a normed ring, known today as a Banach algebra. It combines the concepts of Banachology and algebra to form a discipline with many useful developments. Only the outlines of the subject are treated below. Details are available in [Ber, HeR, Loo,
Nai,
Ri) .
Some Banach spaces, e.g. , function algebras such as Co (lR., C ) , form the context for introducing not only addition and scalar multiplication of their elements but also a kind of additiondistributive product of elements. The basic aspects of this development are treated below. 3.5.1 DEFINITION. A BANACH SPACE A THAT IS ALSO A Calgebra IS A Banach algebra IFF FOR AND IN A AND IN C:
a
b z I l ab l :::; I l al l b l ; z(ab ) = (za)b ; I l z al = I z i l al ·
Section 3.5. Banach Algebras
117
Example.
3.5.2 When X is a locally compact Hausdorff space, the Banach 11 space Co (X, q �f A, normed by 00 , is a commutative Banach algebra with respect to pointwise multiplication of its elements; A has a multiplica tive identity iff X is compact, in which case = l .
I
e e 3.5.3 Example. For a Banach space B, the set [B] c cl�f A of continuous endomorphisms (of
B), normed according to the discussion in 3.3.4, is a
Banach algebra with respect composition of its elements; A is commutative iff dim :::; 1; the identity endomorphism id is always the identity for A.
(B)
3.5.4 Exercise. If the Banach algebra A contains a multiplicative identity e such that ea axae11 a: a) e is the only identity; b ) renormed according I to I l x l ' �f sup l 11 , A is again a Banach algebra and I l e l ' = 1; c ) for some #0 l a positive K and all x, K l x l :::; I l x l ' :::; I l x l · Thus I I and I I ' are equivalent norms: If A contains an identity e , I l e l may be taken 1. ==
==
as
3.5.5 DEFINITION. WHEN A BANACH ALGEBRA A CONTAINS AN IDENTITY Ae cle =f A; WHEN A CONTAINS NO IDENTITY, IS A SYMBOL SATISFYING tJ. A AND Ae cle E C, E A } . WHEN A CONTAINS NO =f { + IDENTITY AND {Zie + Xi } i= 1 , 2 C Ae ,
e,
e
ze x : z
x
e
+ Z2e + X2 �f ( Zl + Z2 ) e + ( Xl + X2 ) , ( Z2e + X2 ) � Zl Z2e + ZlX2 + Z2 X l + XIX2 ,
Z l e + Xl (Zl e +
xd
I l ze i I z i I X I .
+X � + AND Ae IS NORMED ACCORDING TO If A = Ae and = is a left inverse of b and b is a right inverse of
ab e, a
a.
Exercise. For a Banach algebra A : a) Ae is a Banach algebra; b) x Oe + x E Ae is an isometry. 3.5.7 Exercise. If A = Ae and and are left and right inverses of x, clef x 1 and every eft Inverse . . . . then (nght mverse ) f x IS X  1 . 3.5.8 Example. When is counting measure, the classical Hilbert space (v. Section 3.6 ) is L2 (N, v) �f £2 consists of all vectors a �f (al , a2 , . . . ) 00 3.5.6 the map A '3 u = v
=
r+ 
v
1
of complex numbers such that
u
v
L l an l 2 < 00.
n= l
0
The set [p2 L of continuous
a ter 3. Functional Analysis
Ch p
118
endomorphisms of A is, with respect to composition of endomorphisms as product, a noncommutative Banach algebra. The maps T £2 '3 (al ' a 2 , . . . ) ft (a 2 ' a3 , . . . ) , S £2 '3 (a I , a 2 , . . . ) ft a I , a2 ' . . . ) . :
:
(0,
are continuous endomorphisms. Furthermore, T S = id but ST i id . Thus T[S + (id  ST)] id : both S and S + (id  ST) are different right in verses of T: absent commutativity, right inverses need not be unique. Since S'T' id ' = id and T'S' i id a similar argument shows left inverses need not be unique. introduces the In Ae , the identity = + expression + If has a right inverse, it may be written as and 0: is meaning = or + + ( ful even when A contains no identity. =
=
y  exy. (eex)(x e  y)y e  (x y  xy ) x e  y, (e  x)  y) e x  xy x y  xy =
x
3.5.9 DEFINITION. FOR AND Y ELEMENTS OF A BANACH ALGEBRA A, 0 Y �f X + Y WHEN 0 Y = 0 , Y IS A right adverse OF (AND IS A left adverse OF
x
 xy.y
).
x
x
x
3.5.10 THEOREM. THE BINARY OPERATION 0 IS ASSOCIATIVE. IF U AND ARE LEFT AND RIGHT ADVERSES OF X , THEN U = cle =f X , THE ad verse OF AND PROOF. The associativity of 0 follows by direct calculation. If U 0 0 = 0 , then
V
=
x
°
V
xxo xOx. x x � XO u = u u (x ) = (u x) and u xu  x = ux  x. Hence xOx xux  x2 = xxo. D l 3.5.11 Exercise. a) The adverse XO exists iff (e  X) exists in Ae. b) 00 If I l x l < 1, then XO exists and XO =  L xn . c) In Ae, n= l (x o y )(e  x) = (e  x)(y ox); =
V
00
=
0
0
v
=
0
=
0
v
=
0 0
v
=
v
d) If XO exists, for any z, 0
0
(z  XC) (e  x) = z x and (e  x) (z  XC) x z. e) If XO and yO exist, then x y is advertible and (x yt = yO xo. f) In Ae , if l i e  x i < 1, then X I exists. g) In Ae, if X  I exists, for some positive r, y  l exists if I l y  x i < r. 0
=
0
In Ae the set H of invertible elements is nonempty and open.
0
Section 3.5. Banach
Algebras
1 19 00
e + 2:)e  x) n converges and n= 1 [e  (e  x)] (e+ � (e  x) n ) = e. 1 g) If I vi i < I ' f) implies e + X I V is invertible.] Ix I [ 3.5.12 Note. The result g) has a counterpart for adverses, v. 3.5.14.] [Hint: f) The series
Exercise.
3.5.13 a) In Ae the set of H of invertible elements is: a group relative to the operation of multiplication in A. b) The map EH H '3 ft is a bicontinuous bijection, i.e., an auteomorphism.
x X I
3.5.14 DEFINITION. FOR A BANACH ALGEBRA A, THE SET adv (A) CON SISTS OF THE ADVERTIBLE ELEMENTS, i.e., THOSE FOR WHICH THERE IS A ( UNIQUE ) LEFT AND RIGHT ADVERSE. 3.5.15 LEMMA. FOR A BANACH ALGEBRA A, adv (A) IS AN OPEN SUBSET (cf. 3.5.11g) ) AND ° adv ( A) '3 ft IS AN AUTEOMORPHISM. :
g gO 1 PROOF. If x E adv (A) and I l y  x i < then 1 + I l xo l ' XO (y  x)  XO(y  x) , y o XO (y  x)  (y  x)XO, xI l o y l � I l y  x i (1 + I l xO I ) < 1, I l y x0 1 � I l y  x i (1 + I l xO I ) < 1, whence XO and y XO are advertible. If is the adverse of XO y then XO is a left adverse of y. If z is the adverse of y xO, then XO ,z is a right adverse of y. Thus y is advertible: adv (A) is open. If x E adv (A) and x + h E adv (A), then I l h  hxo l � I l hl (1 + I l xO I ) , and if I l h l is small, then u �f h  hxo E adv (A). Furthermore, 3.5.11 implies (x + h) o xo = u, (x + ht XO uO, (x + h) °  XO UO  xO uo , 0Y =
w
0
0Y
0
0
0
=
w
=
=
0
0
0
0
Chapter 3. Functional Analysis
120
I l u l � I l h l ( l + I l x l ! ), when I l h l is small enough, I l u l < 1. Thus I l h l (1 + I l xO I ) 2': I l u l = I l uo  uuo l 2': I l uo l  I l u l I l uo l , I l u° I < 1 l I uI llul l < 1 I l hI ll h(1l (1+ +I l xI Ol xI !O) I ) ' whence the map ° : adv (A) adv (A) is continuous. The symmetry of the definition of x y implies that ( uot u and thus adv (At = adv (A) . Since
ft
0
=
D
3.5.16 THEOREM. IF x IS AN ELEMENT OF A BANACH ALGEBRA, THEN lim n EXISTS. IT IS DENOTED sr (x) AND THERE OBTAINS THE
n+= I l xn l
1
1
nEN I l xn l n .
Il xl ;
FURTHERMORE: a) 0 � sr (x) � b) EQUATION sr (x) = inf FOR z IN C, sr (zx) z ST (x) ; c) sr (xy) = sr (yx) , AND FOR n IN N, sr (x ) = [sr (x)t; d) IF xy = yx, THEN sr (xy) � sr (x)sr (y) .
n
Il
=
1
E>
PROOF. If 0, for some m in N, � sr (x) + p, q in N, n = pm + q, 0 � q < m, whence
pm n
I l xm l m
E. If n E N, for some
As n + 00,  + 1, whence 1
1
1
n inf I l xn l n � lim I l xn l n , nEN n+CXJ n+CXJ I l l i.e., n+= lim I l x l i;. exists and is sr (x) . _
lim x n � sr (x)
=
Items a)d) follow by direct calculation.
D
3.5.17 DEFINITION. FOR x IN A BANACH ALGEBRA A, THE spectrum of x IS sp(x) �f { z : (x  e )  I does not exist in Ae 3.5.18 Exercise. a) If A ¥Ae and x E A, then
}.\
z
sp(x) =
{O} U { z
l O
}
: z i 0, (z  x) does not eXist .
b) If A = Ae, x E A, and z i 0, then z E sp(x) iff o E sp(x) iff X I does not exist, i.e., iff x is singular. [Hint: If z i 0: a) xy = e iff z I x · zy = e; b)
(�) ° does not exist;
Section 3.5. Banach Algebras
121
Iz
3.5.19 THEOREM. IF x IS AN ELEMENT OF A BANACH ALGEBRA, FOR SOME Z IN Sp(X) , sr (x) :::; i . 3.5.20 Remark. Hence sp(x) i 0.]
[
PROOF. (Rickart [RiD If tJ. sp(x) , then A = Ae and X  I exists. Hence 3.5.16d) implies 1 = sr (e) :::; sr (x) sr (X  I ) . Thus, if sr (x) = then X  I does not exist, i.e., E sp(x) : the result is established if sr (x) = o. If sr (x) �f p > the idea of the argument is to show by contradiction that sp(x) ct D(O, p t . Indeed, if sp(x) C D(O, p t , then
0
0,
0 0,
{ I w l � p} {w tJ. sp(x)} , =?
(w  l x) O exists. From 3.5.15 it follows that f(w ) clef (w  x) is a continuous map from S �f { w I w l � p } to A. Since lim I l w l x ll 0, Iwl+oo i.e.,
W r+
:
the basic inequality
I l yo I :::; 1 ��II�I
'
°
1
=

=
which is valid when
I Y I < 1 by virtue
ii (w l x) O ii
of the calculations in 3.5.15, implies lim = o. Thus f is Iw l +oo uniformly continuous on S. If n in N, the equatiori  1 = has as solutions the n distinct nth . 1 cle=f W l , . . . , Wn cle=f 1 , ei £!!.n , , e i 27r ( nn  l ) roots of unIty: For in C and the general notation Zj �f ZWj , 1 :::; j :::; n, direct cal = culation (via induction) shows that 0 . . . 0
zn
z
The equation
0
•
n
L
k=
l
Wk =
Furthermore, if Z E S, then
•
.
z  n xn (Z� l X) n (Z� l X). l 0 implies that if R �f  L then =l n j
z nxn zj l x =
0
k
k k Zj X ,
Rj . From the identity
aoO =Ooa= a
Chapter 3. Functional Analysis
122
there follow
Since
n
L: R
j =l
j =
0, (3.5.21 )
The uniform continuity of I in S implies that for each positive E there is a a such that a > p and III ( Pj )  I (aj ) 11 < E, 1 :::; j :::; n. However, (3.5.21) implies that for all n in N, (3.5.22) As in earlier calculations, lim II (a)  n xn ll n + =
E
=
0, whence
Because p < a, lim a n xn = 0, whence lim a  n xn ) O = 0. Since n + CXJ n + CXJ ( may be arbitrarily small (and positive), by virtue of (3.5.22) ,
whence
lim IIp n xn li = o. (3.5.23) n +CXJ However, since p = sr (x) , inf II p n xn ll ;; = 1, and (3.5.23) yields a contra n EN diction. D [ 3.5.24 Remark. The derivation above is given in terms the properties of exp as derived in Chapter 1. An alternative proof can be based on Liouville 's Theorem in the theory of holomorphic functions on C, cf. 5.3.29.] 1
3.5.25 LEMMA. IF x IS AN ELEMENT OF A COMMUTATIVE BANACH AL GEBRA A, THEN sr (x) IS sup { I pl : p E sp(x) } �f P(x) . PROOF. a) For some z in sp(x), sr (x) :::; Izl (v. 3.5.19), whence sr (x) :::; P(x) .
Section 3.5. Banach Algebras
123
On the other hand, if w E sp(x) and
Hence, if
y �f w 1 x, for all large
n
I w l > I z l , then
and some t, 00
I l yn l � :::; t < I, and so
n= l converges to (cf. 3.5.6c) ). Hence w 1 x is advertible, whereas, since 1 w E sp(x), w x, is not advertible, a contradiction:
yO
If sr (x)
=?
{{z E sp(x) } 1\ { sr (x) :::; I z l }} { I z l = P (x) } . (1  E)P(X) < P(x), the argument above shows
=
00
converges and its sum is (z  l x) O , a contradiction since
z E sp(x) :
sr (x) = P(x).
[ 3.5.26 Note. Owing to 3.5.25, the notation sr (x) and the term
D
spectral radius of x for P are justified.]
3.5.27 LEMMA. IF X IS A BANACH SPACE AND Y IS A CLOSED SUBSPACE, THE quotient space XjY ENDOWED WITH THE quotient norm
I IQ
:
e
XjY :1 r+ inf {
Il xl
: xj Y =
e}
IS A BANACH SPACE. PROOF. That is a true norm is a consequence of the definitions and the elementary properties of inf. If is a Cauchy sequence in Xj Y, for each n , there is an Xn such that xn jY = and + < Thus and so is a Cauchy + < + sequence. If lim X �f x and �f xj Y, then lim = XjY is a Bana�h space. D Thus [ 3.5.28 Note. By definition, if = xj Y, then the map X :1 x r+ xj Y �f is normdecreasing.]
I IQ
{en}n EN n. x en e n l I l l I l n T Q n {xn }n EN I l xn  xm l l i en  e>n I Q 2 Tm e n 4� en e: n 4� n e I l e l Q :::; I l x l . e
124
Chapter 3. Functional Analysis
3.5.29 DEFINITION. A left (right) ideal IN AN ALGEBRA A IS A PROPER SUBSPACE R SUCH THAT AR C R (RA C R) . A SUBSPACE THAT IS BOTH A LEFT AND RIGHT IDEAL IS AN ideal. THE IDEAL R IS regular OR mod ular IF THE quotient algebra AIR CONTAINS AN IDENTITY. WHEN R IS A REGULAR IDEAL AND Ul R IS AN IDENTITY IN AIR, IS an identity mod ulo R. CORRESPONDING DEFINITIONS APPLY TO LEFT ( RIGHT ) identities
u
modulo right (left) ideals .
Exercise.
u
3.5.30 For a Banach algebra A: a) If is an identity modulo the (regular ) ideal R, then is an identity modulo every ideal that contains R. b) If R is a regular left (right) ideal, then R is contained in a regular left (right) maximal ideal. c) Every maximal ideal is closed. d) An element has a right adverse iff is not a left identity modulo any regular right maximal ideal. e) If A ¥ Ae a proper subset S of A is a regular maximal ideal iff for some maximal ideal Me in Ae and different from A, S = A n Me. and are in R. b) Some [Hint: a) For every version of Zorn's Lemma applies to the poset of ideals containing R. c) The continuity of multiplication applies. d) If has no right adverse, then R �f E A is a right ideal, tJ. R, and is a left identity modulo R, and b) applies.]
x
u
x
x, ux  x
xu  x
x
{ xy  y : y }
x
x
Example.
3.5.31 In the Banach algebra Co(ffi., q , Coo (ffi., q is a dense ideal contained in no maximal ideal: Coo (ffi., C) is not a regular ideal. 3.5.32 THEOREM. (GelfandMazur) IF A IS A COMMUTATIVE BANACH ALGEBRA AND M IS A MAXIMAL REGULAR IDEAL IN A, THEN AIM IS ISOMORPHIC TO C
I I Q,
PROOF. With respect to the quotient norm AIM is a Banach field (with identity and so if e i 0, 0 tJ. sp(e). On the other hand, sp(e) i (/) and if E sp( e), then e  is singular and since AIM is a field, e = The correspondence e r+ is an isomorphism between AlM and C D Owing to the fact that A r+ AIM is an algebrahomomorphism, for a commutative Banach algebra A, each regular maximal ideal M may be re garded as a special element of the dual space A': is a multiplicative lin ear functional. Furthermore, if E A, then whence 1.
z
e)
z
ze
x' x
ze.
x' I l e l :::; I l x l ,
I l x' l :::;
3.5.33 DEFINITION. THE SET OF REGULAR MAXIMAL IDEALS IN A COM MUTATIVE BANACH ALGEBRA A IS Sp (A) , THE spectrum of A. The uses of the word spectrum as in spectrum of [ 3.5.34 (when is an element of a Banach algebra A) and spectrum of A can be misleading. However, the distinction between the two usages is clear:
Note. x
x
•
sp(x) is a set of complex numbers;
Section 3.5. Banach Algebras •
125
Sp (A) is a set of regular maximal ideals.]
3.5.35 THEOREM. IF A IS A COMMUTATIVE BANACH ALGEBRA CONTAIN ING AN IDENTITY, Sp (A) , REGARDED AS A SUBSET OF A', IS A WEAK' COMPACT SUBSET OF THE UNIT BALL B (0', 1) IN A'. PROOF. If A 3 A ft E Sp (A) is a net and (B', B)converges to some x' in B (0', 1 ) , for each pair {x, y} in A,
M>.
M>. (x, M>. ) (y , M>. ) (xy, M>.) .
(J"
=
The left resp. right member of the equation converges to (x, x') (y, x') resp. (xy, x') , whence x' is a continuous multiplicative linear functional: Sp (A) is weak'closed. Owing to 3.4.12, B(O', 1) is weak' compact. D For x in A and in Sp (A) the complex number (x, is denoted and the map ' : A 3 x ft E C(Sp (A), q is the Gelfand map. The preceding development implies that � 3.5.36 If A is a commutative Banach algebra and E Sp (A) , then E sp(x) . [ 3.5.37 If E sp(x), for some in Sp (A), is an If y tJ. identity modulo then  1 ) ° and so Hence 2: sup E sp(x) } �f P(x) .
M
x( M )
x
M)
I l x(M) l oo I l x l .
Exercise. M x(M) Note. Z M l Z� l X M. M, fj(M) (z� x(M) x(M) I l x l oo { I pl : p However, if M E Sp (A) and x(M ) w i 0, then w� l x(M) 1 and W� l X is an identity modulo M. If y �f (W� l X) O exists, then W� l X + w� l xy 0 , =
= z.
=
=
Y 
=
1 + Y  l y = 0,
a contradiction:
W� l X has no adverse, w E sp(x), I w l � P(x) : I l x l oo P(x) . =
In C(Sp (A), q , Sp (A) is to be viewed in its weak' topology. Thus ' may be viewed as a covariant functor from the cate gory of Banach algebras containing an identity (and contin uous ((>homomorphisms ) to the category CF of continuous func tion algebras on compact Hausdorff spaces (and continuous C homomorphisms) [Loo, Ma ] .]
BA.,
c
3.5.38
Exercise.
If (J C ft C is a Cautomorphism, then (J :
=
id .
Chapter 3. Functional Analysis
126
[Hint: If z E
e,
then 8(z) = 8(z · 1 ) .]
3.5.39 Exercise. If A is a commutative Banach algebra, and x E A, then
,
x [Sp (A)] =
{
sp (x ) if A = Ae sp (x ) or sp (x ) \ { O } if A ¥Ae .
Since function algebras of the form C(X, q arise naturally in the study of commutative algebras, the Stone WeierstrafJ Theorem below takes on added importance. The result is phrased in terms of the notion of a sepa rating set of functions in a function algebra. 3.5.40 DEFINITION. A SUBALGEBRA A OF eX IS separating IFF FOR ANY TWO ELEMENTS a, b OF X, SOME f IN A IS SUCH THAT f(a) i f(b) ; A IS strictly separating IFF FOR SOME f, f(a) = = 1  f(b) .
0
3.5.41 Exercise. If A is a commutative Banach algebra,
A, cle=f { X : X E A } is a strictly separating subalgebra of Co (Sp ( A) , q . 3.5.42 Exercise. If A is a strictly separating sub algebra of JR.x , a, b are two elements of X, and { c , d} c JR., then, for some f in A, f (a) = c and f(b) = d. 3.5.43 Exercise. If X is a compact Hausdorff space and A is a II 11 00 closed subalgebra of C(X, JR.) , then A is a vector lattice. [Hint: If f E A, the Weierstrafi Approximation Theorem (3.2.24) implies that I f I is approximable by polynomial functions of f.] 3.5.44 THEOREM. ( StoneWeierstrafi ) IF X IS A COMPACT HAUSDORFF SPACE, ANY CLOSED STRICTLY SEPARATING SUBALGEBRA A OF C(X, JR.) IS C(X, JR.) . ( EACH STRICTLY SEPARATING SUBALGEBRA OF C(X, JR.) IS 11 1i 00 DENSE IN C (X , JR.) . ) PROOF. If f E C(X, JR.) and a , b are two points in X , A contains an fa,b such that fa b ( a ) f(a), fa,b (b) = f(b) . If E then ,
> 0,
=
X
Uab �f { x : fab ( ) < f(x) + E } and Vab �f { x
:
X
fab ( )
> f (x)  E }
are open. If b is fixed, {Uab LE x is an open cover of X. Hence there is a fi nite subcover {Ua1b, . . . , Uapb} and, owing to 3.5.43, fb �f inf fap b E A. l �p �P If x E X, for some p, x E Uap b and fb ( ) < f(x) + E. On the other hand, if X
Section 3.5. Banach Algebras
x E Vb
127
p
�f n Vap b, then Ib(X ) > I (x)
 f.
The open cover {Vb h E X admits p =l a finite subcover {Vb" . . . , VbQ } and 3.5.43 implies ¢ �f sup Ibq E A. If l �q�Q x E X, for some q , x E Vbq and I(x) < ¢(x) < I(x) + Eo D [ 3.5.45 Note. a) If A is merely separating, A can fail to be C(X, JR.) , e.g., if X = [0, 1] ' the set A of all polynomial functions that vanish at zero is separating. However,  f
A=
C(X, JR.) n { I : 1(0)
=
O} .
If JR. is replaced by C in the discussion above, the corresponding conclusions are false, v. Chapter 5. What is true and what follows directly from 3.5.44 is that if A is a closed strictly sepa rating subalgebra of C(X, JR.) and 1 E C(X, q , then 1 = u + iv, {u, v} C C(X, JR.) and by abuse of notation, C(X q = A + iA. There is a corresponding statement if A is merely separating.] .
3.5.46 Exercise. If X is a compact Hausdorff space and a subset A of C(X, JR.) is both Aclosed and vclosed, the I lloo closure of A contains each continuous function approximable on every pair of points by a function in A. 3.5.47 Exercise. a) The set
is strictly separating. b) The smallest algebra A over 1I' and containing
1
,
associated measure T ) . e) For the maps W : [0, 27r] 3 x r+ e ix E 1I', W* : C1I" =
1 r+ l o W E C [O , 2 �l , I I W* ( f ) 11 2 ' f) The set W* (
W* [L2 ( 1I', T )] = L2 ( [0, 27r] , A) and 11111 2 a CON in L2 ( [0, 27r] , A). 3.5.48 Exercise. For a commutative Banach algebra A such that A = A e , if x E A: a) { x(M) : M E Sp (A) } is a compact subset of C; b) for the closure A ( x) of the set of all polynomials n
aoe + L ak xk , ak E C, k= l
n
E N,
Chapter 3. Functional Analysis
128 :
the map h Sp
[A(x) ]
3
M r+ x(M)
E
C is a homeomorphism.
3 . 6 . Hilbert Space
The groundwork for studying Hilbert space S) was laid in the material starting with 3.2.9 and ending with 3.2.18. The function is a positive definite and conjugate bilinear form. The current is devoted to deriving some fundamental results about the character of operators, i.e., for various Banach spaces X, elements of X] c .
Section( , )
[S) ,
x' E S)' , FOR SOME x IN S), (y, x) (x, x' ) .
3.6.1 THEOREM. (F. Riesz) IF
[
==
Remark.
3.6.2 In the preceding sentence there is a possibility for notational confusiOn: the symbol in the left member of the identity denotes the value of the inner product in S); in the right member denotes the value of the functional acting on the vector The content of the THEOREM and the nature of the PROOF should clarify any distinction.]
(y, x)
x' (x, x')
x.
{
x') O} M {m),} x. v v then (v, x' ) j. o. Moreover, �f  E S) \ M and (u, x' ) 1 . For any (v, x' ) m �f ( x') u E M. Thus, if z �f u L (y, a),) a)" then z j. 0, E M � , and if a �f 1 : 1 ' then {{ a} l:.!M } is a CON set in S). Hence, if x �f (a, x' )a, for any y, y (y, a)a + L (y, m),) m), and (y, x') (y, a) (a, x') = (y, (a, x') a) �f (y, x). D 3.6.3 Exercise. The correspondence S) ' x' x E S) is a conjugate linear bijection. If {a, b} C S), then (a, b) (a' , b'). In view of the conjugate character of the bijection, S) and S)' PROOF. The kernel M �f z : (z , = is a closed subspace of S); Zorn's Lemma implies that M contains a maximal orthonormal subset �f If M S), then 0 serves for If M ¥S) and E (S) \ M), ), E A "
w,
Z
w 
=
w,
U
=
=

), E A
), E A
=
sentially the same. 3.6.4 If
=
3
r+
Exercise. { U, V } C [S)] c and {x, y} C S), then {( U (x), x) == [x, V (x)]} {} {( U (x),y) == [x, V (y) ] } .
are es
Section 3.6. Hilbert Space
[Hint:
129
[x ± iy , V (x ± iy ) ] . The replace ( U (x ± xiy ), x ± iy ) is known as polarization.] ±i r+
ment technique
x
y
=
= 3.6.5 DEFINITION. WHEN C SJ, THEN S � �f 3.6.6 a) For any S � is a closed subspace. b) If M is a closed subspace and E SJ there is in ]\,{ x M � a unique pair b such that = a + b: SJ = M ffi M � . c) the projection SJ 3 r+ a E M is in [SJk d) pk and [Hint: A maximal ON subset M �f of ]\,{ is contained in a CO� set H for SJ and a = L
S
Exercise.
x
=
x
S,
:
I PM I l.
PM
=
{ill>. } ),EA ),EA (x, ill), ) ill), .]
3.6.7 DEFINITION. FOR
AND THE image OF
{ x : (S, x) {O} }. PM x {a, }
T IN [SJ]e, THE kernel OF T
IS
T IS im (T) (�f T(SJ ) .
f) id + IS INVERTIBLE. PROOF. The results a)d) follow from direct calculation. e ) On the one hand,
TI T;
I TI T{I I = Ilxsupl/= 1 I TI T{ (x) 1 I TII I I/xsupl/ = I I T{ (x) 1 1 � I TII I · I T{I I = I TII 1 2 , and on the other hand, if I l xl l = 1, then I TI T;(x) 1 I (TI T{ (x),x) 1 I T{ (x) 1 2 . Thus I TI T{II I T{1 1 2 I TlI 1 2 . 2 I l x 1 2 , it follows that f) Since ( [id + TI T{ ] (x), x) = I x l 1 2 + I T{(x) 1 ker (id + TI T;) = {O}. Thus im (id + TI T; ) is dense in SJ. If x E SJ, there is a sequence {Yn } n EN such that lid + TI T; ] (Yn) converges to x. Because id + TI T; is normincreasing, { Yn }" EN is a convergent sequence, and if �
;?
;?
=
=
;?
t Functional Analysis lim Yn �f y, then lid + TI T{ ] (y ) = x: id + TI T{ is bijective and thus is n+ = invertible. D [ 3.6.9 Note. When dim (SJ) < 00, every injective element of [SJ] is invertible. By contrast, 5 in 3.5.8 is injective and is not invertible. An invertible operator T is not only injective but also bijective.] 3.6.10 Exercise. If T E [SJ] c , then ker (T) = lim (T, )] L (cf. 3.4.14) . [Hint: {(T(x),y) O} {} {(x,T'(y)) O}.] 3.6.11 DEFINITION. A T IN [SJ] c IS selfadjoint IFF T' T ; T IS normal IFF TT' = T'T. 3.6.12 Exercise. If T E [SJ] c , then T + T' and i (T  T' ) are selfadjoint; if T is normal, T + T' and T  T' commute. Chap er 3.
130
==
==
=
T
3.6.13 THEOREM. (The Spectral Theorem) IF IS NORMAL, THE (COM MUTATIVE) BANACH ALGEBRA AT GENERATED IN [SJ] c BY {id , IS, VIA THE MAP ' isometrically isomorphic TO C (Sp (AT) , q . PROOF. The map is linear (v. 3.5.28). Owing to 3.5.37, for any m AT , sr ( = Smce = 2 l z;; = U lV, IS. a linear combination of the selfadjoi�t operators U and V. Furthermore, U'  iV' and §t u' {0 5 (cf. 3.4.16 ) . Hence
5. 5'
T, T'}
5) I 5�I I = . . 5 5 + 5' + .5  5' def + . 5 55' UU'  VV' + i (VU' + U'V) , 1 55' 1 = 1 5 1 2 = 1 521 1 (cf. 3.6.9) , 1 5 1 = 11 52 11� . . . = 1 52n I T n sr (5 ) = 1 5 1 = , �
=
=
=
=
t
=
i.e., is an isometry and hence (v. 3.4.2 ) AT is a closed subspace of �
C (Sp (AT) ' q . If M E Sp (AT) , is a selfadjoint element in AT, S(M) iO'id , for L a E JR., and b
�f a + ib, �f 5 + a2 + b2 + 2bO' + 0'2 ::::: I l t l ! : II L I12 = II LL' II = 1 52 + 0'21 1 ::::: 1 5 1 2 + 0'2 . Thus a2 + b2 + 2bO' ::::: 1 5 1 2 , which is false if O'b is sufficiently large. Hence b = O: S(M) E R By definition, if MI and M2 are two elements in Sp (AT)' for some selfadjoint 5 in AT , S (Ml ) j. S (M2 ) and a direct calculation shows that j. 0,
5
:::::
131
Section 3.7. Miscellaneous Exercises
fj �f { S
:
}
S a selfadjoint element of AT is strictly separating. The StoneWeierstrafi Theorem implies fj is dense in C (Sp (AT) , JR.) . Hence fj + ifj ( = AT is a dense (v. 3.5.45) and, as shown above, a closed sub
)
space of C (Sp (AT) , q . [ 3.6. 14 Remark. The isometric isomorphism established yields an interpretation to be given to f (T) when f E C(Sp ( A ), q . This observation is the basis for a functional calculus for normal oper ators. Furthermore, T : Sp (A) 3 M r+ T ( M) E ((: is a continuous injection that permits the identification of Sp (A) with a compact subset sp(T), the spectrum of T, in C. ]
D
3.6. 15 DEFINITION. A LINEAR MAP T : S h r+ Sh (T E [S h , Sh ] ) BETWEEN HILBERT SPACES Sh resp. Sh ENDOWED WITH INNER PRODUCTS ( , ) resp. [ I ] IS: a) unitary IFF T (SJ t ) = SJ 2 AND (x, ) == [Tx I T ] ; b) AN isometry IFF II x ii == II Tx · 3.6. 16 Exercise. a) A unitary operator is an isometry. b) An isometry need not be unitary. c) If T E [SJ] and T is unitary, then T is normal and bijective. Furthermore, T' = T � l . d) If SJ �f ((:2 endowed with the inner product 2 ( , ) : ( ((:2 ) 3 {( X l , X2 ) , ( YI , Y2 ) } r+ XIYI + X2Y2 and T in [SJ] is such that
y
y
l
then T is unitary (hence normal) but T is not selfadjoint. [Hint: For b), 3.5.8 applies.] 3. 7. Miscellaneous Exercises
3.7. 1 Exercise. then
In the context of Young's inequality (3.2.1), if c 2:
lco ¢(s) ds + l<1>0 (C) '1jJ(S) dS = A ([O, C]
x
0,
[O, ¢(c)] ) = c¢(c) .
(The equation above implies Young's inequality without recourse to argu ments based on geometry.) �.7.2 Exercise. If (X, S, f.l ) is a measure space and f E 5, then (3.7.3)
132
Chapter 3. Functional Analysis
iff for some a in JR., sgn (I) . f � e i a . f, i.e . , If I = e i a f. [Hint: If (3.7.3) holds, A �f f dp, j. 0, and sgn (A) = e i a , for
Ix
e i a f �f g �f h + ik, it follows that
Ix g dp, = Ix h dp, + i Ix k dp, = Ix eiaf dp, = I lx f dP, 1 = Ix l fl dp,. Hence Ix g dp, = Ix h dp, � Ix I hl dp, � lx (h2 + k2 ) � dp, = Ix g dp" k 0, Ix g dp, = Ix I gl dp" and a.e., �
° � g = e i a f = l ei a f l = If I = sgn (I)f a.e.]
3.7.4 Exercise. If B and B are Banach spaces and E [B ' B2 ] , then y; in TB;:is continuous iff for Isome S 2in [B;, B�] , each Xl in BIT, and I each
T', T
T'
in which case, S = i.e., is continuous iff exists. [Hint: The Closed Graph Theorem (3.3.18) applies.] 3.7.5 Exercise. a) If 1 � p < 00, the set S of simple functions is dense in LP(X, p,) metrized by I lip · b ) For some (X , 5, p,) , S is not dense in LOO (X, p,). c ) If 1 � p < 00 and LP is derived from a DLS functional I defined on a function lattice L, then L n LP is a dense subset of LP. d ) For some function lattice L and some DT functional I defined on L, L n L 00 is not dense in L00 . 3.7.6 Exercise. If X is a compact Hausdorff space and A is a I 11 00 closed subalgebra of C ( X, JR.) , then A is closed with respect to the lattice operations 1\ and V . [Hint: The relation between the lattice operations and I I and the Weierstrafi Approximation Theorem (3.2.24) apply.] For f in L 2 ([0, 1], A) the sequence of its Fourier coefficients is Q
{ cn �f _1_ Jot f(t)e 2nrrit dt } nEZ . v'2ii
Section 3.7. Misoellaneous Exercises
133
The following relevant functions figure in the Exercises below.
N _1_ e2nrr i t , "" � Cn � n=  N V 271" 1 sin ( 2N + 1) 2 7I"t (Dirichlet's kernel), D N (t) cle=f 71"t 271"
�f
SN (t)
FN (t) �f 
1 271"(N + 1)
3.7.7 Exercise. a) FN = each is of period one. c)
1
[
•
Sln 2 (N sin
� 1)7I"t ) �t sin ( )
(
]
2 (FeJ' er ' s kernel) .
N
1 "" D n . b) D N and FN are periodic and N + 1 n� =O

1
1 1 FN dt = 1, 1 SN (t) = 1 D N ( t  s)f(s) ds �f D N D n dt =
(3.7.8) *
f(t),
1 1 "" S Fn (t  S)f(8) ds cle=f FN * f(t). N (t) r N + 1 n� 0 in = =O N
0
U
0
1 d) If < E < 2 ' then FN I [0,1]\[<,1 <] + as N + 00 e) If f E C([O, 1], q and f(O) = f(l) , then limoo II FN f  f ll oo = (Fejer's Theorem). N
0 [Hint: For e), if E E (0, �), since FN 2': 0, *
+
*
I FN f(t)  f(t ) 1 �
<
.
1 <
1 + 1  + 1� < FN (t  s) l f(s)  f(t) 1 ds
�f I + I! + I!I.
0
Because f is bounded, if N is large, then I + II is small since FN � on [0, E] U [1  E, 1] . If E is near � , since f is continuous, 2 (3.7.8) and the nonnegativity of FN imply II! is small.] 3.7.9 Exercise. Fejer ' s Theorem (3.7.7e)) implies the Weierstrafi Ap proximation Theorem (3.2.24).
134
Chapter 3. Functional Analysis
[Hint: If I E C([O, 1], ffi.) , in C([O, 27r], ffi.) there is an 1 such that 11 [0 1 ] = I while !(27r) 1 (0). A change of scale applies.] , 3.7.10 Exercise. a) If 1 :s; p < 00 I E LP ([O , 1 ] , ), ) , then =
P 0.
= F Nlim += II N I  I II *
[Hint: The results 3.7.7e) and 3.7.9 apply.] For a measure space (X, 5, p,), a sequence {In} nEN of measurable func tions converges in measure to I (In =t f) iff for each positive m
s
nlim += p, ( { x : l in ( x)  I ( x ) I > E })
=
E,
0.
3.7.11 Exercise. If (X, 5, p,) is a measure space, each In increases mono tonely, In =t I, and I is continuous at x (x E Cont (I)) , then m
s
nlim += In (x) = I(x) . [Hint: If h is small, for some J(h), I(x) = I(x + h ) + J( h ) while I J(h) 1 is small. If E > h > and for infinitely many n,
0,
0,
In (x)  I(x) 2': E , then for infinitely many n ,
In (x + h)  I(x + h) 2': In (x)  I(x) + J ( h) ? E + J(h). If h is small, E + J (h) 2':
�. Hence, for infinitely many
n,
{ y : In (Y)  I(Y) } contains a fixed interval of posi tive measure. If I ( x)  In (x) 2': E for infinitely many n, the argument applies for I h l small and h negative.] 3.7.12 Exercise. If (X, 5, p,) is a totally finite measure space, I E S , I is strictly positive a.e. (p,), and for a sequence {En } nEN of measurable sets, then nlim nlim + CXJ }rEn I dp, + CXJ p, (En) =
[Hint: p,
0,
( { x : I(x) > � } )
=
0.
t p,(X).]
135
Section 3.7. Miscellaneous Exercises
Exercise.
3.7.13 (Schwarz) If 15 is a finite set and are two sets of complex numbers, then
{aa} aE 8 and {ba} a E 8
A and B, not both Aaa Bba, 3.7.14 Exercise. (FischerRiesz) If {xa} a EA is a CON set in S) and X E S), then L l ( x, xa ) 1 2 < 00. Conversely, if L l aa l 2 < 00, for the poset aEA aEA b.(A) of all finite subsets of A, the net b.(A) 15 L aaxa converges to some x in S) and for each (x, xa ) aa. ' [Hin" If ' E �(A) , then I � aaxa l � l aal ' ·] 3.7.15 Exercise. If {xa} a E 8 is a finite orthonormal set and X E S), then Equality obtains in the preceding inequality iff for some E 15. zero, = [Hint: 3.2.10 applies.] Cl'
n
Cl' ,
:
3
r+
=
�
[Hint:
3.7.16
Exercise.
(Holder's inequality extended) If
1 < Pi , 1 :::; i :::; I,
I
L P1i = 1, 
i= l
I I and Ii E LPi (X, p,) , then I �f II Ii E L l (X, p,) and 11111 1 :::; II Illi ll pi ·
i= l
[Hint: If PI
�f p, then p'
=
(ti=2 P�)t
1
i= l
and induction applies.]
136
Chapter 3. Functional Analysis
3.7.17 Exercise. If (X, S, Il) is a measure space, f E S, and A �f { p : 0 < p < 00, I lfll p < oo } :
a) A is convex; b) A 3 P r+ I flip is continuous; c)
{ r < p < s}
=?
{ llfll p � max { llfllr , I fils } }
(regardless of whether any of r, p, s is in A) ; d)
{
lim I flip = I l flloo ; { A ¥= 0 } =? p+ oo
}
e) when X E S and Il(X) < 00, {p < q} =? { llfll p � I l fllq } . 3.7.18 Exercise. For each x in [0, 1], the value Bn( f ) (x) of the nth Bernstein polynomial is a convex linear combination of the values of f at the points '5. , 0 � k � n. n
1 00 ,
3.7.19 Exercise. a) If X is a Hausdorff space B is the set of bounded B is a continuous JR.valued functions on X, with respect to the norm I Banach space. b) For x in X, the evaluation map �x : B 3 f r+ f (x) is an element of B' and II �x ll � 1. c) The map I : X 3 x r+ �x E B' is a continuous injection. d) The weak' closure ,6(X) of I(X) is weak'compact. (The space ,6(X) is the Stonetech compactijication of X.) e) If F E C (X, JR.) , for some F in C(,6(X) , JR.) , F l x = F: each continuous function on X has a continuous extension to ,6(X) , cf. 1.7.28. A subset S of a topological vector space V is bounded iff for every neighborhood N of 0 and some real A, S C AN . 3.7.20 Exercise. (Kolmogorov) A Hausdorff topological vector space (V, T) is normable, i.e., T is norminduced by some norm II II , iff there is in V a bounded convex neighborhood of O. [Hint: If V is normable, then N �f { x : I l x ll � I } is a bounded convex neighborhood of O. If N is a bounded convex neighborhood of 0, then N contains a circled neighborhood U for which Conv (U) is bounded. The Minkowski functional pu is a norm that induces T .]
4
More Measure Theory
4.1. Complex Measures
The Riesz Representation Theorem 2.3.2 yields a measure space (X, 5, p,) and an integral representation of a nonnegative element in [Co(X, ffi.)] ' . The more general problem of finding an integral representation of arbitrary el ements in [Co (X, ffi.] ' resp. [Co (X, C] ' deserves a solution. It is given in Section 4.3. The measure in 2.3.2 takes values in [0, 00]; for [Co (X, ffi.)]' resp. [Co(X, q]" one might expect measures taking values in ffi. resp. C. 4.1.1 Example. For the measure space (X, 5, p,), when f L l (X, p,) , the map � 5 3 E r+ f dp, is a Cvalued set function. If {En } nEN is a se
E
Ie
:
quence of pairwise disjoint sets in 5 and E �f U nEN En , the results in Chapter 2 imply:
�
is a countably additive Cvalued set function such that � (0) = o . The situation just described motivates the following development.
4.1.2 DEFINITION. FOR A SET X AND A aRING 5 CONTAINED IN !fj(X) A SET FUNCTION � 5 3 E r+ � ( E ) C IS A complex measure IFF: a) � (0) = 0; b) FOR A SEQUENCE {En} nEN OF PAIRWISE DISJOINT SUBSETS IN 5,
E
:
�
(U nENEn
)
=
00
L � (En )
n= l
(�
IS COUNTABLY ADDITIVE) . IN THE CIR
CUMSTANCES DESCRIBED, (X, 5, 0 IS A complex measure space. 137
Chapter 4. More Measure Theory
138
[ 4.1.3 Remark. Henceforth, according as 1l(5) C [0 , 00]' resp. 1l(5) C [00, (0 ) or 1l (5) C (00, 00] ' resp. 1l (5) C C, (X, 5 , Il) is a measure space, resp. a signed measure space, resp. a complex measure space. For a signed measure space (X, 5, Il), the pos sibility 1l (5) = [00, 00] is excluded. Otherwise, there arises the awkward question of assigning a value to 00  00 or 00 + 00.] 4.1.4 Exercise. For a set X and a aring 5 contained in !fj(X), a set function � is a complex measure iff: a) for every E in 5 and every countable
measurable partition { En }nE ]\j of E, � (U n E ]\j En
)
=
00
L � (En) and b) for
n= l every countable measurable partition { En }nE ]\j, L I � (En) 1 < 00. n= l 4.1.5 Exercise. If (X, 5, 0 is a complex measure space and for E in 5, O'(E) �f �[�(E)] , (3( E ) �f <;S[�(E)] , then a and (3 are complex measures and 00
0'(5 ) U (3(5) C R 4.1.6 Exercise. For (X, 5, Il), if 1l(5) C [0 , (0 ) , then 1]
�f sup { 1l (E)
: E E 5 } < 00 .
[Hint: For some sequence { En }nE ]\j, En C En+ l and
4.1.7 Exercise. For (X, 5, Ili ) , if 1l; (5) C [0, (0 ) , 1 � i � 4 , then is a complex measure. 4.1.8 DEFINITION. FOR A SIGNED MEASURE SPACE (X, 5, Il) , A SET A IS A positive set ( negative set ) IFF FOR EACH E IN 5, (E n A) 5 AND Il (E n A ) � 0 ( Il (E n A ) � 0 ) . For � in 4.1.1, any set where ! is nonnegative is a positive set. If A C B and ! is positive on B (hence also on A) Il [E> (I, 0) ] > 0, then Ed!, O) is a negative set and E> (1, 0) is a positive set. If ! dll < 0, then 4.1.9 Exercise. a) Every measurable subset of a positive set is a positive set. (Hence, if El and E2 are positive measurable sets, then El \ E2 and
E
Ix
Section 4.1. Complex Measures
139
EI n E2 are positive measurable sets.) b) If {En} n EN is a sequence of pairwise disjoint positive sets, U nEN En is a positive set. c) If E is a positive set and A E 5, then E n A is a positive set. d) Similar assertions
obtain for negative sets.
4.1.10 THEOREM. (Hahn) IF (X, 5, p,) IS A SIGNED MEASURE SPACE AND
p,(5) C [00, (0) or p,(5) C (00, 00] , THEN FOR SOME P IN 5 AND Q �f X \ P AND EACH E IN 5,
P n E, Q n E E 5, p,(P n E) 2': 0, p,(Q n E ) :::; 0, p,(E) = p,(P n E) + p,(Q n E) . PROOF. The argument below is given when p,(5) C [  00, (0). A similar argument is valid when p,(5) C (00, 00]. If 1] �f sup { p,(E) : E 5, and E is a positive set }, then 1] < 00 (v. 4.1.6) . If 1] = 0, then (/) serves for P. If 1] > 0 there is a sequence {En} n EN of measurable positive sets such that p, (En ) t 1]. If P �f U En , then
E
P = EI LJ U �=2 (En \ En I ) E 5 and for each M in N, 1] 2': p,(P) = p, (EM ) + whence p,(P)
=
00
L
n=M+ I
1]. If E E 5, then
P, (En \ En d 2': P, (EM ) '
00
p,(P n E) = P, (EI n E) + L p, [(En \ En  d n E] 2': 0 : n= 2 P is a positive set. If Q �f X \ P and Q is not a negative set a contradiction is derived by
the following argument. For some measurable Eo contained in Q, 00 > P, (Eo) > o. But Eo is not a positive set, since otherwise, PLJEo is a positive set and
p, (PLJEo) = p,(P) + p, (Eo) > 1], a contradiction. Hence, for some measurable subset E of Eo, p,(E) < o. In 1
1
1
the sequence 1,  ,   , . . . there is a first, say   , such that for some 2 3 ml 1 measurable subset EI of Eo, p, (Ed <   . Then ml 
p, (Eo \ Ed = p, (Eo)  p, (E d > p, (Eo) > o.
140
Chapter 4. More Measure Theory
The argument applied to Eo now applies to Eo \ EI : for some least positive integer m2 and some measurable subset E2 of Eo \ EI , P, (E2 ) < � and m2 of pairwise disjoint sets and least by induction there is a sequence { E } such that positive integers { } 
n nEN
mn nEN
p, ( E2 ) < 
m12
,
p, [Eo \ (EI U E2 )] = p, (Eo )  p, ( EI )  p, (E2 ) > 0 ,

00
ex:> 1 < and mk L n= l m n implies that A �f Eo \ U nEN En Hence
measurable subset F of
then F U EK c Eo \
+ 00
as k + 00 The earlier argument .
is not a positive set. Hence, for some
1 A and some mK , m K > 2 and p, ( F ) <  m K . But
(u ::11 ) and Ek
mK :
imply a contradiction of the minimality of Q is a negative set and the pair {P, Q} performs as asserted. 0 4.1.11 Remark. The pair (P, Q) is a Hahn decomposition of X. [ Although as constructed, P if X tJ 5, then Q tJ 5. ]
E.S,
4.1.12 Exercise. If (Pi , Qi ) , i = 1, 2, are Hahn decompositions of X for the signed measure space (X, 5, p,) , then for any measurable set
(A n PI ) = p, (A n P2 ) (and hence p, ( A n Q I ) = p, (A n Q 2 ) ) '
A,
p,
[ 4.1.13 Note. If p,(N) = 0 and N \ P j. 0, then a second Hahn decomposition is (P U N, Q \ N). Examples of such N abound. For example, if f �f X [0, l and p, in (JR., 5)" p,) is defined by
1
p, (E) �f Ie f dx,
Section 4.1. Complex Measures
141
then P �f [0, 1] is a positive set and Q �f ffi. \ { [O, I] } is a neg ative set and (P, Q) is a Hahn decomposition. If N �f Q, then IAN ) = 0, N \ P j. 0, and (P U Q, Q \ Q) is a second Hahn de composition. Thus Hahn decompositions are not necessarily u nique but they all produce the same effects.] 4.1.14 DEFINITION. WHEN (X, 5, f.l) IS A SIGNED MEASURE SPACE AND (P, Q) IS A HAHN DECOMPOSITION FOR f.l, THEN FOR E IN 5, f.l +
( E) clef f.l (E n P), f.l ( E) cle=f f.l ( E n Q). =
[ 4.1.15 Remark. In light of 4.1.12, the set functions f.l ± are independent of the choice of (P, Q).] 4.1.16 Exercise. The set functions f.l± are (nonnegative) measures. 4.1.17 THEOREM. IF (X, 5, �) IS A COMPLEX MEASURE SPACE, THE SET FUNCTION
I�I : 5 3 E r+ sup
{�
I� (En)1 : {En} nE N a measurable partition of E
}
IS A MEASURE AND 1�1 (5) C [0, (0 ) PROOF. Since the only measurable partition of 0 is itself, 1�1 (0) = 0. If { En} n EN is a measurable partition of E, then for each n in N and each positive E, En admits a measurable partition {Enk hEN such that 00
'" � I� (Enk ) 1 � I�I (En ) k=1
.
 2: ' Since E
=
U {k, n }CN Enk , it follows that 00
{k, n }CN whence IWE) �
00
L I�I. (En ) :
n =1
I�I is superadditive.
On the other hand, the partition {En} n EN may be chosen so that 00
n= 1
00
n= 1
142
Chapter 4. More Measure Theory
whence I�I is countably additive: I�I is a measure. The set functions a �f � [�] , �f �[�l are signed measures and
(3
0'(5) U (3 ( 5 ) c ffi..
o'±
(3±
Corresponding to the four measures and there is the Jordan decom 0'+  0'_ + i of �. Since the ranges of the measures are contained in [0, (0), it follows that for E in 5 and in the context above,
position �
o'±, (3±
((3+  (3 )
=
00
00
n=1
n=1
The definitions of
M �f sup {
o'±, (3± imply that + ( D ) + _ ( D ) + (3+ (D) + (3 (D) D E 5 } < 00 ,
O'
O'
whence IWE) :::; M. 4.1.18 Exercise. If (X, 5, f.l) is a signed measure space, then
D
f.l f.l+  f.l  and If.l l = f.l+ + f.l  . =
4.1.19 DEFINITION. IF (X, 5, �) IS A COMPLEX MEASURE SPACE,
L l ( X, � ) clef L 1 (X, o'± ) n L =
1
(X, (3± ) ,
4.2. Comparison of Measures
When (X, 5, f.l) and (X, 5, �) are measure spaces or complex measure spaces the relation between f.l(E) and �(E) as E varies in 5 deserves analysis. Ex amples of such a relation are: a) 0 :::; f.l(E) + �(E) < 00; b) I WE) :::; f.l(E) ; c) (more generally) Ll (X, O C L 1 (X, f.l). Although each of the preceding is of some interest, the relations that have emerged as of fundamental im portance are those given next .
Section 4.2. Comparison of Measures
143
(X, 5, Il) AND (X, 5, �), Il IS � (Il « 0 IFF {�(E) = O} {1l(E) = O} ; W HEN (X, 5, Il) AND (X, 5 , 0 ARE COMPLEX OR SIGNED MEASURE SPACES, Il � IFF { I WE) = O} {1l(E) = O}. WHEN AIL E 5 AND Il(E) Il (E n AIL ) , Il lives ON AIL' WHEN Il LIVES ON AIL ' � LIVES ON A� , AND AIL n A� = 0, Il AND � ARE mutually singular (Il 1 �). 4.2.2 Example. If (X, 5, Il) is a measure space, I is nonnegative and 5measurable, and, for each E in 5, �(E) �f Ie I dll , then � « Il. 4.2.3 Example. For the map Ila : 5), E A (E n Ca ) , ([0, 1], 5)" Ila) is a measure space and Ila lives on Ca. If = 0, then A lives on [0 , 1 ] \ Ca and Ilo 1 A. Although the circumstances just described exemplify the relations � « Il and � 1 Il, 4.2.1 DEFINITION. FOR MEASURE SPACES absolutely continuous WITH RESPECT TO =?
�
=?
==
3
a
r+
the following discussion provides a more refined sorting out of the possibil ities for those relations.
(X, 5, Il)
(X, 5,�)
Ila
Ils: Il = Ila Ilsi
4.2.4 THEOREM. IF AND ARE TOTALLY FINITE MEA SURE SPACES, THEN FOR SOME MEASURES AND a) b) + c) 1 d) 1 e) FOR SOME NONNEGATIVE INTEGRABLE AND EACH IN
Ila « Il; Ils Il; Ila Ils ; h, E 5, lla(E) = Ie h d�.
PROOF. (von Neumann) The following argument derives a)e) more or less simultaneously. If �f + and then Owing n to Schwarz's inequality (3.2.11),
p Il �
I E L2 (X, p),
I E L2 (X, Il) L2 (X, �). 1
I d � i I I I dll � i I I I dp � (i 111 2 dP) [p(X) ] � < 00. i ll I l Hence T : L 2 (X, p) I T(f) �f i I dll E C is in [L2 (X, p ) ] ' . Riesz 's result (3.6. 1) implies that L 2 (X, p ) contains a g such that for every E in 5 and every I in L 2 (X, p), (4.2.5) Ie dll = i XE dll = L XEgdp = Ie gdp, L ( l  g ) ldll = i lg dP  i lgdll = i lgd�. (4.2.6) "2
3
r+
•
Chapter 4. More Measure Theory
144
If p(E) > 0, (4.2.5) and the inequality 0 :::; f.l(E) :::; p(E) imply 0 :::;
1)] = 0: 0 :::; g(x) :::; 1 a.e. Modulo a null set l } l:J { x g(x) = 1 } �f Al:JS . Thus, for E in
Thus g ? 0 a.e. and p [E> (g, (p) , X = 0 :::; g(x) < 5,
{x
� · Ie g dp :::; 1.
p( )
:
f.l(E n
:
A) + f.l(E n S) �f f.la (E) + f.l s (E), f.l a « f.l, f.ls 1 f.l.
For n in N, E in 5, and f �f
(� g k ) . X E ,
(4.2.6) reads (4.2.7)
A,
As n + 00, if E C the left member of (4.2.7) converges (by virtue of Lebesgue's Monotone Convergence Theorem) to f.la(E) and the integrand in the right member of (4.2.7) converges monotonely to some h: for any E in 5, f.la(E) h d�. In particular, 0 :::; h E L l (X, � ) . D ==
Ie
[ 4.2.8 Remark. The equation f.l = f.l a + f.ls represents Lebesgue 's decomposition of f.l: f.l a is the absolutely continuous component and f.ls is the singular component of f.l. The equation f.la(E)
==
Ie h d�
is the expression of the RadonNikodym Theorem . The complex of results and assertions is sometimes referred to as the Lebesgue RadonNikodym Theorem or LRN. The function h is the RadonNikodym derivative of f.l with respect to �: h = df.l .
d�
]
4.2.9 Exercise. The RadonNikodym derivative of f.l is unique modulo a
null function (f.l).
4.2.10 Exercise. The validity of LRN persists if f.l is a complex measure and if X is totally afinite (with respect to If.ll). 4.2.11 Exercise. If (X, 5, f.l) and (X, 5 , �) are measure spaces such that X E 5 and f.l(X) + �(X) < 00 ((X, 5, f.l) and (X, 5, O are totally finite) ,
Section 4.2. Comparison of Measures
145
� « p" and � is not identically zero, then for some positive E and some E in 5, E is a positive set for �  Ep,. [Hint: For each n in N, if (Pn , Qn ) is a Hahn decomposition for �  p" then � Qn ° < � U Pn . For some no, n EN n EN 1 p, (Pno ) > 0, E = Pno , and E  .] no
(n )
�
=
(
)
=
4.2.12 Exercise. If X in (X, 5, p,) is totally finite and � « p" then : a)
{
}
r f dP, :::; �(E) I 0; A �f f : O :::; f E L l (X, p,), M(E) �f sup E E S JE
b) for some nonnegative h in S and each E in 5,
l
��).
l h dv
=
M(E); c)
�(E) == h dp, (hence h (The preceding conclusions yield a second proof of LRN.) [Hint: For c), 4.2 . 11 applies.] d� = h, and is in 4.2.13 Exercise. If (X, 5, p,) is totally finite, � « p" g dp, LOO (X, �), then 9 d� 9 h dP,. =
1
=
1
4.2.14 THEOREM. I F (X, 5, p,) IS A MEASURE SPACE AND (X, 5, O IS A COMPLEX MEASURE SPACE, THEN � « p, IFF FOR SOME MAP
J : (0, (0 )
3
THE IMPLICATION {p,(E) < J(E)}
E r+ J(E) E (0, (0 ) ,
::::}
{ I�(E) I < E } OBTAINS. PROOF. If � « p, and no J as described exists, then for some positive E, each n in N, and some En in 5, p, (En) < T n while I� (En ) 1 2': E. Then p, lim En = ° while I�I lim En 2': E, a contradiction. n 4� n 4� If J as described exists, p,(E) 0, and E > 0, then 1p,(E)1 < J(E), and thus I�(E)I = 0. D 1 4.2.15 Exercise. If (X, 5, p,) is a measure space, h E L (X, p,), and for E in 5, �(E) �f h dp" then = h and IWE) l h l dp,. [ 4.2.16 Remark. In the preceding discussion there are references to various special measure spaces, e.g., a (X, 5, p,) that is totally finite.
(
)
(
l
��
=
)
=
l
146
Chapter
4.
More Measure Theory
For a given measure space (X, 5, p,), the following classification of possibilities is useful. A measure space in one class belongs to all the succeeding classes. a) (X, 5, p,) is totally finite, i.e. , X E 5 and p,(X) < 00; b) (X, 5, p,) is finite, i.e. , for each E in 5, p,( E ) < 00; c) (X, 5, p,) is totally afinite, i.e., X is the union of count ably many measurable sets, each of finite measure; d) (X, 5, p,) is afinite, i.e. , each E in 5 is the union of countably many measurable sets, each of finite measure; e) (X, 5, p,) is decomposable, i.e., 5 contains a set F of pairwise disjoint elements F of such that: e1) X = U F; for each
FE F F in F, p,(F) is finite; e3) if p,(E ) is finite, p,(E ) =
L p,(
FE F
E
n F),
e2 )
27
( 4. . 1 )
(whence there are at most count ably many nonzero terms in the right member of e4) if A C X and for each F in F, A n F E 5, then A E 5. A set E in 5 is finite, afinite or decomposable according as, by abuse of notation, (E, E n 5, It ) is totally finite, totally afinite, or decomposable. Discussions of the relations of the hierarchy to the validity of LRN (and hence the validity of the representation theorems in Section 4.3) can be found in [GeO, Halm, HeS, Loo] .]
(4. 2. 1 7));
4.2.18 Exercise. If f E
L 1 (X, p,), then the set K", (f, O) is afinite.
4.2. 19 THEOREM. IF (X, d) IS A METRIC SPACE, (X, aR[K(X)], p,) IS FI NITE, AND EACH x IN X IS THE CENTER OF A acompact OPEN BALL, THEN (X, aR[K(X)] , p,) IS REGULAR.
PROOF. The set R of regular Borel sets is nonempty since (/) E R. The formulre of set algebra imply that R is a ring. If E > 0, R 3 An C An+1 C Un+1 E O(X), and p, (Un \ An ) < 2En ' then 00
A �f U An C U Un �f U E O(X) , nEN n=2 00 00 00 U \ A C U (Un \ An ) , p,(U \ A) � L p, (Un \ An ) < L 2: < E. n=2 n=2 n=2
Section 4.3. LRN and Functional analysis
147
N
U An �f BN , then BN E R. Lebesgue's Monotone Convergence Theon= 1 rem and the finiteness of (X, a R[K (X)], Il) imply that for large N, If
Furthermore, B N contains a compact N such that BN ) + (BN N ) E : Hence N If E > 0, 3 :J and :J
K Il (BN \ KN ) < 2E ' Il (A \ K ) .::;: Il (A \ Il \ K < A E R. R Dn Dn+1 Kn+1 E K (X), Il (Dn \ KNn ) < E, then D �f n Dn :J n Kn �f K E K(X) and Il(D \ K) < E. If n Dn �f EN , n= 1 nEN n=2 for large N, EN E R, (EN \ D) < �, and EN is contained in an open set UN such that Il (UN \ EN ) < 2E ' Hence =
It
D E R.
R
R < < < < .!.
Thus is monotone, whence is a aring (v. 2.5. 11) . Lebesgue's Monotone Convergence Theorem implies that a acompact open ball B ( x , t is regular. If 0 s then B ( x , s t is also regular. If then for each n in N and each k in there is a regular open ball B k t such that 0 and so for some finite set {kih < i
K E K(X), (,
rk
r
r,
rk
K,
tJ K, Un Un ;= 1 then inf { d(x, k) : k E K } �f 15 > 0, and for some n, .!. < 15: tJ Ur" i.e., n K = n Un . Since R is a aring, K is regular: a R[K(X)] C R. D nEN K, K
rk,
x
4.3. LRN and Functional analysis
Among the important consequences of LRN are the characterizations of the dual spaces and when X is a locally compact Hausdorff In particular, the problem raised at the beginning of space, the Chapter can be addressed. It is no exaggeration to state that modern functional analysis owes its current richness to the role played by LRN in establishing the basic relations among what are now regarded as the classical function spaces.
[LP (X, Il)] ' , [Co(X, C)] ' .
Chapter 4. More Measure Theory
148
4.3.1 THEOREM. IF (X, S , p,) IS TOTALLY FINITE, 1 '::;: p < 00, AND L E [U(X, p,)]' ,
Ix
THEN FOR SOME f IN U' (X, p,) , L( g ) == gf dp,. THE FUNCTION f IS UNIQUE (MODULO A NULL FUNCTION) AND Il f ll p ' = I I L I I . [ 4.3.2 Remark. The result above is in one sense a generaliza tion of 3.6.1; the assumption that (X, S, p,) is totally finite limits the generality. On the other hand, extensions to totally afinite measure spaces are available (v. 4.3.4) . 1 PROOF. At most one such f exists since for any set E in S ,
The heart of the argument centers on showing that the ([valued set function � S 3 E r+ L is a complex ,measure, and that � « p,. The dp, serves for f . complex conjugate of RadonNikodym derivative d� The reasons that � is a complex measure and � « p, are: a) :
(X E )
b) L is linear; c) L is continuous (whence for each g in LP (X, p,) ,
Because X is finite, LOO (X, p,) C U(X, p,) . If g E LOO (X, p,) , there is a sequence s n nEN of simple functions such that
{ }
Sn t g
ll ..(. 0
and I lg  s n oo
(v. Section 2.1), whence n+ limoo Ilg  sn ll p
=
O. Thus, if
then (4.3.3) (v. 4.2.13). The next paragraphs show: f E U' (X, p,) and I f l l p'
=
IILII.
Section 4.3. LRN and Functional analysis
149
l ie l
If p = 1 and E E 5 , then I dll .::;: II L II . Il(E) , whence as in the PROOF of LRN, it follows that II (x) 1 .::;: II L II a.e.:
I E L00 (X, Il) [= £P' (X, Il)] .
If 1 < p < 00, then sgn (7) is measurable and sgn (7) 7 = If En �f E:;(II I , n), n E N, and kn �f x En lll pl  1 , then
III (2.4.8).
I kn l P = xEn lllpl and kn E LOO (X, Il ). Hence
1En Illpl dll = Jx
r kn · k� 1 dll
Ix
Ix
= kn lll = kn sgn (7) 7 dll = L [kn sgn (7)] � II L II · llkn sgn (7) li p � II L ll ll kn il p
)
1
)
1
and Il kn ll p P implies pr � l i L l i, n E N. = n Thus Il h ll p .::;: II L II . Holder's inequality and (4.3.3) imply II L II .::;: 11 1 11 pl . Thus II L II =l Il h ll p , 1 � P < 00. D l 4.3.4 Exercise. The conclusion of 4.3.1 holds if (X, 5 , Il) is totally afinite and 1 '::;: p < 00: [£p (X, Il)]' = £P' (X, Il). [Hint: If X = U nEN Xn , Xn E 5 , and 0 < Il (Xn ) < 00, by abuse . of notatlon, there are 5 n cle=f 5 n Xn, Iln cle=f Il I Sn ' ( Xn , 5 n , Iln ) , B anach spaces Bn �f LP (Xn , Iln ), injections
(le
Ill pl dll
(Ix
x En lll pl dll
that identify LP (Xn , Iln ) with closed subspaces of LP(X, Il), and finally Ln �f L I Bn For each n there is in Lpl (Xn , Iln ) an hn such that for g in Bn , Ln ( g ) = r g hn dlln and '
JXn
Each I in LP(X, Il) may be written uniquely in form
00
00
L I IXn �f L ln' n 1 n 1 =
=
150
Chapter 4. More Measure Theory
00 n=1
The result 2.3.2 can be extended to
X Co (X, q f 1 (1 )
4.3.5 THEOREM. (F. Riesz) IF IS A LOCALLY COMPACT HAUSDORFF THERE IS A SPACE AND 1 : 3 r+ E e lS IN REGULAR COMPLEX MEASURE SPACE p, ) SUCH THAT
(X, S,B, 1 (1 ) Ix f(x) dp,.
[Co(X, q]"
=
�[1(1 )] � 0' (1) 'S[1(1)] (3(1), a (3) a
(3
then and are continuous PROOF. If , �f and linear JR.valued maps, by abuse of language, signed functionals. The ar gument reduces to showing that (and similarly is further decomposable into the difference of (nonnegative) DLS functionals to which 2.3.2 applies. There is an echo in what follows of the Hahn decomposition (4.1. 10) of a signed measure. E JR.) �f sup For in and = 2: Thus sup is abbreviated If c 2:
f cto(X, , 0'+ ( 1 ) + { la(g) 1 : g Coo+ (X, q, I g l �f foo.} la (g) l . 0' (1 ) 0' (0) 0 0' (1 ) � I l al l l ll I gl � f
0, then
Igi I fi , i 0'+ (II ) + 0'+ (h ) =
Moreover, if = 1, 2, then the careful application of the identity � sgn (z)z == Izl leads to
10' (gI )l + I g2sup1�h 10' (g2 ) 1 sup [ 10' (g I ) l + 10' (g2 )1 ] 1911Sh 1 92 I S h = sup a {sgn [a (g I )] g l + sgn [a (g 2 )] g2 } 1911Sh 192 1 S h 10' (g l + g2 ) 1 � 0'+ (h + h ) · .::;: 191sup 1Sh sup
Ig l l �h
=
192 I S h
I g l .::;: h 12
0 � h 1\ I g l �f hI .::;: h , 0 .::;: Igl  h I �f h2 � h Fur gI l h I + h2 ,
If + , then thermore, since =
whence
0' + (h + h ) � 0'+ (h ) + 0'+ ( h ) · If f E Coo(X,JR.) and
Section 4.4. Product Measures
151
0'  (1+ ) 2: 0,
0'+ (1 )  0'+ (1)  0' (1 ), 0'  0' .
0'+ (1 ) 2:
then �f then max{O, a(l)} . If Similarly analysis applies to (3. a ± are continuous, and a = Via the DLS procedure, the functionals a±, (3± engender regular measures (± , 1]± .  1]  ) and E Coo(X, and E Coo (X, If f.l �f  C + i
11
then ( )
=
(+
(1] +
r f(x) df.l, and �
ness of Coo (X,
f
1 f.l I (E). 11 1 11 = Esup E�
q in Co (X, q applies.
q
f
q,
Finally, the I ll oo  dense
D
4.4. Pro duct Measures
To measure spaces (Xi, 5i, f.li) , i = 1, 2, there correspond: a) the space X X l X2 and b) the intersection 5 1 8 5 2 of all arings contained in lfj (Xl X2 ) and containing { E l x E2 : Ei E 5i, i = 1 , 2 }. There arises the question of how to define a product measure f.l on 5 1 8 5 2 so that the equation f.l (E l x E2 ) = f.l l (Ed ' f.l 2 (E2 ) is satisfied for all Ei in 5i, i = 1 , 2. The DLS approach to the answer is given below. 4.4.1 Remark. Alternative derivations can be given by prov ing whatever is claimed first for simple 5 1 8 5 2 measurable func tions and then, via the approximation and limit theorems of mea sure/integration, extending the conclusions for wider classes of functions. In such a procedure, the resulta preFubinate mea sure space (X l x X2 , 5 1 8 5 2 , f.l l 8 f.l 2 )can fail to be complete.] Associated with (Xi, 5i, f.li) are the Banach spaces L l (Xi, f.li) , i = 1 , 2, and the nonnegative linear functionals Ii : L l (Xi , f.li ) 3 ft r df.li, i = 1, 2. }xi If Ap , 1 :::; p :::; P, resp. Bq , 1 :::; q :::; Q, in 5 1 resp. 5 2 are sets of finite mea sure (f.ld resp. (f.l 2 ) and apq , 1 :::; p :::; P, 1 :::; q :::; Q, are real numbers, then P,Q �f "'" � apq X A p X Bq p,q=l is in ffi.x 1 X X2 and the set L of all such functions is a function lattice.
�
X
X
[
f
f
f
P, Q 4.4.2 LEMMA. a) IF � "'" apqX Ap X Bq E L, THEN 5 1 resp. 5 2 CONTAINS p,q=l PAIRWISE DISJOINT SETS Eu , 1 :::; u :::; resp. Fv, 1 :::; v :::; V, AND ffi. CONTAINS NUMBERS auv , 1 :::; u :::; 1 :::; v :::; V, SUCH THAT P,Q L apq X Ap XBq = L auv X Eu X Fv ' u , v=l p,q=l
U,
U,
u,v
152
Chapter 4. More Measure Theory
b) FURTHERMORE, P, Q
U, V
p,q=l
u , v=l
L apq h (X AJ 12 (X BJ = L
( J h (X FJ ·
O'uvh X E
PROOF. a) A detailed computational argument establishes the validity of the assertions in the LEMMA. Not only is the argument tedious, it is not really informative and adds little to the understanding of the underlying structure shown in the visible geometry. The basic reasoning is depicted in Figure 4.4.1. The interested reader might wish to provide a formal argument that translates the geometry into the unavoidable prolixity. b) The linearity of the functionals h and h implies the result. D P, Q 4.4.3 Exercise. If L apq X Ap X Bq cle =f f E L, then p,q=l
is welldefined, i.e.,
1( 1 ) is independent of the representation of f.
UI x V3
U2 x V3
U3 x V3
UI x V2
U2 x V2
U3 x V2
UI x VI
U2 x VI
U3 x VI
L�� X I U2 Figure 4.4. 1.
Section 4.4. Product Measures
153 X
4.4.4 DEFINITION. WHEN A C Xl X2 AND X E Xl , THE xsection Ax OF A IS { X2 : (x, X2) E A }. WHEN f E ffi.X 1 X X2 THE xsection fx OF f IS X2 : 3 X2 r+ f (x, X2) . SIMILAR FORMULATIONS APPLY FOR Y IN X2 , Ay , AND fy . 4.4.5 LEMMA. THE MAP I IS A DLS FUNCTIONAL. PROOF. The nonnegativity and linearity of I flow from its definition. Fur
P,Q p,q= l
thermore, if f �f " � apq X Ap X Bq E L, then for each ( fixed ) x in X l ,
fx
=
12 (fx ) =
P,Q L apq X Ap (x)XBq E L I (X2 ' f.l2) , p,q= l P,Q L apq X Ap (x) h (X BJ E L l (Xl , f.l d · , q=l
p
Similar formulre apply for fy . The definitions of the functions imply the fundamental equality: 1(f) = II [12 (fx )] h [h (fy )]. Because (fn ) x ..!. h [(fn U ..!. O. Furthermore,
0,
=
L l (Xl , f.l d 3 h [(fn ) x ] ..!. 0,
whence h {h [(fn U } = I (fn) ..!. o. D Since I in 4.4.5 is a DLS functional, in accordance with the develop ments in Chapter 2, I engenders a complete measure space (X, 5, f.l) , the Fubinate of (X l , 5 1 , f.l d and (X2 ' 52, f.l2) :
When the number of ingredient factor spaces is two or more, the gen eral vocabulary and notation for dealing with product measure spaces are those given in 4.4.6 DEFINITION. a) FOR A SEQUENCE { (Xn , 5 n , f.ln ) } nEN OF MEASURE SPACES, : (Xl x X2, 5 1 52, f.l l x f.l2) IS THE Fubinate OF (Xl , 5 1 , f.l d AND (X2' 52, f.l2) ' b ) FOR GREATER THAN 2,
K
X
l Kl K Kl X 5 , , k k X X X f.lk)
IS THE Fubinate OF
(
k=l
k=l
k= l
154
Chapter 4. More Measure Theory
X k= l k , THE INTERSECTION OF ALL aRINGS CONTAINED IN !fj(Y) AND CONTAINING { X �= Ek : Ek E Sk, 1 ::; k ::; K } IS 6Sk. l k= l ON OSk, Xk f.l k IS A MEASURE DENOTED Of.lk AND THERE EMERGES k=l =l k=l THE preFubinate MEASURE SPACE X �= Xk, g Sk, g f.lk . WHEN ( ) l EACH Si == 5, AND EACH f.l f.l , THE FORMULJE f FOR Y cle = K
Xi
==
K
X
K
K
X,
i
==
f.l) ,S,f.l) 4.4.7 Exercise. a ) If K 2: 2, then Xk Sk :J OSk. b ) If Ek E Sk, then =l k=l Ek) fl f.ldEk) ' (X �From ( X �=l 4.5.16, =l f.lk )4.5.10, and 4.5.18 below it follows that some stances :J in 4.4.7a) is and in others :J is ;;. .
PROVIDE THE NOTATIONS FOR THE KFOLD FUBINATE OF ( X WITH ITSELF AND THE KFOLD PREFuBINATE OF (X, 5, WITH ITSELF. K
K
=
m
=
m
Fubinate measure spaces arise from DLS functionals and are automatically complete. Some preFubinate measure spaces can fail to be complete.
The procedure described in 4.4.1 leads to preFubinate measure spaces. Their completions are the Fubinate measure spaces. One advantage of the DLS approach to product measures is the fact that it immediately produces Fubinate measure spaces.
f.l) )
4.4.8 Exercise. a) If (X, 5 , is a measure space and 5 contains a ring R of sets of finite measure, then a R ( R is afinite. b ) The measure space (X, 5, engendered by I in 4.4.5 is afinite. [Hint: As in 2.2.40, for a) the relevant sets are Eo, the set of all countable unions of elements of R, and for each ordinal number a in (0, Q) , the set En of unions of count ably many set differences drawn from U
f.l)
y
Ey.J
Section 4.4. Product Measures
to
155
The machinery of Chapter 2 applies in the current situation and leads
4.4.9 THEOREM. (Fubini) FOR MEASURE SPACES THE PREFuBINATE MEASURE SPACE
(Xl , S;, IL;), i = 1 , 2, AND
Ix E L I (X2 ' 1L2 ) resp. Iy E L I (Xl , IL l ) , x r+ iXr Ix ( Y) dIL2 (Y ) E 51 resp. Y r+ ixr Iy (x) dILl (x ) E 52 , 2 ! Ix I dlL = Ix (lx Ix d1L2) dILl = 1x (Ix Iy dILl ) d1L2 . !
2
2
!
PROOF. Since L is li ll I dense in. L l (X, IL) (v. 2.2.31), for a sequence contained in L, lim = 0, whence
{fn } nEN
n+ = Illn  11 1 1 r I d L = lim I (fn ) . ix l n+ =
Via, as needed, passage to a subsequence, recourse to the lattice operations 1\ and V, and the replacement of by a g such that g == (g and are indistinguishable in L ) , 0 � t may be assumed. For each in and 0� resp. 0 � resp. each in
Xl
1 (X, IL) IIn I (fn ) y Y X2 , (fn ) x
I
I x
(4.4.10) resp.
(4.4.11)
Owing to the form of each In ,
Ix In dlL = Ix! (lx2 (fn )x d1L2 ) dILl = 1x (Ix (fn ) y dIL l ) d1L 2 . 2 !
Lebesgue's Monotone Convergence Theorem (v. 2.1.9f)) applies. 4.4.13 Exercise. In the context of Fubini's Theorem (4.4.9), if
(4.4.12) D
Chapter 4. More Measure Theory
156
then for x in Xl resp. y in X2 , Ix E L I (X2 , f.l 2 ) resp. Iy E L I (Xl , f.ld and (4.4.12) obtains. [Hint: In L l (X, f.l) there are nonnegative functions 1;, 1 :::; i :::; 4, i ( 13 such that I II =
 14 ) ']
h+
4.4.14 Exercise. For (Xl x X2 , 5 1 8 5 2 , f.l l 8 f.l2 ) and a nonnegative func tion I in L l (X l X2 , f.l l f.l2 ) , for almost every x in X l , X
X
Jx2 Ix d (f.l2 ) E 51 ,
Ix E L I (X2 , f.l2 ) '
Ixl X X2 Ixl (lx2 Ix df.l2) ' =
Similar formul� obtain for almost every y in X2 • Fubini's Theorem and its elaborations describe the circumstances in which an integral of an l over a product space may be calculated by iter ated integration of the sections Ix over the corresponding factor spaces. Al though no conditions, e.g., afiniteness, are imposed on the measure spaces (Xi, 5i, f.li) , i 1 , 2, the PROOF deals only with an integrable function I, and automatically 0) is afinite. The next result describes circumstances in which the existence of an iterated integral of the sections of l over the corresponding factor spaces implies the existence of the integral of l over the product space (in which case the two calculations yield the same result) . =
K",(f,
4.4.15 THEOREM. (Tonelli) FOR A TOTALLY FINITE MEASURE SPACE
(X l
X
X2 , 5 1
X
5 2 , f.l l
IF I IN ffi.X 1 xX2 IS NONNEGATIVE AND 5 1 OF
Ixl (lx2 Ix df.l2) df.ll 1x2 (lxl Iy df.ll ) df.l2
X
X
f.l2 ) '
5 2 MEASURABLE AND EITHER
< 00 ,
(4.4.16)
< 00 ,
(4.4.17)
OBTAINS, SO DOES THE OTHER, THEIR VALUES ARE EQUAL, AND
Ixl xX2 I d (f.ll
X
f.l 2 )
= =
Ixl (lx2 Ix df.l2 ) df.ll 1x2 (lxl Iy df.ll ) df.l2·
(4.4.18)
Section 4.4. Product Measures
157
{In}nEN
PROOF. Since I is measurable, there is a sequence of simple func tions such that t I. Each of the following equalities is validated by Lebesgue's Monotone Convergence Theorem (v. 2.1.19f) ) or by the defi nition of integration of simple functions:
In
}rX1 XX2 i d (f.l 1
x
f.l 2 )
=
nlim t oo }rX1 XX2 In d (f.l 1 f.l2 ) (fn )y df.l1 ] df.l2 }�+� 1x2 [lx l 1x2 [J�� Ixl (fn )y df.l1 ] df.l2 l (fn ) df.l 1 df.l 2 = 1x2 [/XI n �� y ] Iy df.l 1 ] df.l2 . = 1x2 [/XI X
=
=
Hence, if the last (iterated) integral is finite so is the (double) integral and Fubini's Theorem implies the conclusions of the theorem. D 4.4.19 Exercise. Tonelli's Theorem is valid if: a) each measure space (Xi, 5i, f.li) , i 1, 2, is afinite; b) I is 5 1 x 5 2 measurable.
=
4.4.20 Exercise. a) If E E 5
n ).
� n , there is a Borel set F such that
(E \ F) U (F \ E) ( = Ef:l.F)
�n
is a null set ( A x b) If I is 5  measurable, there is a Borel measurable g such that 1 == g. c) If h : ffi.2 r+ ffi. is Borel measurable and I : ffi. r+ ffi. is Borel measurable, then I 0 h is Borel measurable. [Hint: For c) use b).] 4.4.21 Exercise. If
{J, g} C L 1 (ffi., A ) , then :
h ffi.2
3
(x, y)
r+
I(x  y)g(y)
l I(x  y)g(y) dy �f 1 * g(x),
the convolution of I and g, exists a.e., and
Chapter 4. More Measure Theory
158
[Hint: The functions f and g may be assumed to be Borel mea surable; JR.2 3 (x, y) ft x  Y E JR. is continuous. Fubini 's Theorem applies to r f(x  y)(g(y) d).. X 2 (x , y).] JK{2
[ 4.4.22 Note. If # (A) 2: No, an indexed family ((X)"
S)" 1l), )} ),EA
of measure spaces, X X), has a standard meaning. In constructand on some aring ing a measure Il related to the set awkwardness is avoided if the conditions: a) related to for each >.. , X), E and b) Il)' (X),) = 1 are imposed.
{S)' hEA S ),
),E),
{ 1l), hEA
The customary approach is to form, for each finite subset
the rectangle R �f E)' l . . . x E), x X X)" then the algebra ), $ a F, and the aalgebra A generated by all such rectangles. For each X
n
n
R, Il( R) �f II Il), k (E)' k )' The set function Il is extensible to a k= l
measure on A and this measure behaves properly with respect to of given measures. the set
{ 1l), hEA
Details of this procedure and of the associated Fubinoid theorems can be found in [HeS] .]
4 . 5 . Nonmeasurable Sets
The existence of nonmeasurable sets in the context of a measure space (X, Il) can be established trivially in some instances and in others, only appeal to sophisticated aspects of set theory permits a satisfactory resolu tion.
S,
R �f for Il : R 3 E ft 0, (X, R , Il) is a complete measure space and every nonempty subset of X is nonmea surable. b) The set C consisting of and all finite or countable subsets of JR. is a aring and C 3 E ft #(E) is counting measure. If X �f JR., the measure space (X, C, ) is complete, and every noncountable subset of X, e.g., JR., is nonmeasurable. (Note that if E E R resp. E E C and M e X, then M n E E R resp. M n E E C.) By contrast, the following discussion demonstrates that for the com ).. ) , there is a set M such that if )"(E) > 0, then plete measure space (JR.,
4.5.1 Example. a) If X i (/) and v :
{(/)},
(/)
v
S)"
Section 4.5. Nonmeasurable Sets
159
M n E tJ 5), . The result is applied to the study of the completeness of product measure spaces that arise from complete measure spaces. 4.5.2 Example. The complete aalgebra 5 generated by the set of all arcs A:> ,(3 �f { : 0 ':::; a < (J < (3 .:::; 27r } in 'lI' may be endowed with the mea sure 7 (v. 3.5.47) such that 7 ( A ,(3 ) �f (3  a. If (J E JR., then Aa,(3 E 5 and 7 ( Aa,(3 ) = (3  a: ('lI', 5, 7) is In the group 'lI' there is the subgroup �f { : (J E Q } . The Ax iom of Choice implies the existence in 'lI' of a set 5 meeting each coset of in precisely one element. Thus { g5 : 9 E consists of count ably many pairwise disjoint sets such that U E g5 = 'lI'. If 5 E 5 and 7(5) = then g G for 9 in 7(g5) = for each in N, 7('lI') 2: and so = O. Since 'lI' = U E g5, it follows that 7('lI') 0, a contradiction. Thus 5 C!fj('lI') . g G 'f=The map exp : JR. 3 r+ E 'lI' (v. 2.3.13) permits a transfer of the dis cussion above from ('lI', 5, 7) to (JR., 5)" A). The conclusions reached are paraphrased loosely by the statement: 'lI' and JR. contain nonmeasurable sets. 4.5.3 Exercise. a) For disjoint sets E l and E2,
e iO e'O G,
a translationinvariant. e iO G e iO G G} a, n na, a
a:
=
x e ix
(4.5.4 )
b) for each F in 5 and E in H (5), p, * (F n E) + p,* (F \ E) = p,(F) ; c) if E E H (5) and p,* ( E ) < 00, then E E 5 iff p,* ( E ) p,* ( E ) . a) For the first inequality in (4.5.4) there are sequences {An} nE N resp. {Bn} n E N contained in 5 and such that =
[Hint:
An C El and P, (An) t p,* (E d , Bn :J E2 and p, (Bn ) ..(. p,* (E2) ;
for the second inequality, there is an 5sequence {en} n E N such en :J en+ l and p, (en) ..(. p,* (El l:!E2 ) . Then p, (An n en) t p, * (E d and P, (Bn n en) ..(. p,* (E  2). For b) , a) applies.] 4.5.5 Example. If a is an irrational real number, � �f A
�f { C : n E Z } ,
and B �f { en
e 27ria ,
: n E Z} ,
then: a) B is a subgroup of 2 in A; b) B n �B (/), and A = Bl:!�B; c) because a is irrational, both A and B are (countably) infinite dense subgroups of the compact group 'lI'. Zorn's Lemma implies there is a set 5 consisting of exactly one element of each 'lI'coset of A. For M 5B , if MM l n �B i (/), i.e. , if
index
=
�
160
Chapter 4. More Measure Theory
since '][' is abelian, S 1 S; 1 E �B C A. Hence, owing to the nature of 5, S 1 = 82 . Thus X 1 X; 1 = b 1 b; 1 E B, i.e., X 1 X; 1 E �B n B = 0 , a contradic tion: MM� 1 n �B = 0 . If L is a measurable subset of M and T(L) > 0, then M M� 1 :J LL � 1 , which contains a ,][,neighborhood of 1 (v. 2.3.9) and thus an element of the dense set �B, a contradiction. It follows that the inner measure of M, i.e., the supremum of the measures of all measurable subsets of M, is zero: T* (M ) = O. Because T is translationinvariant, T* (� M) = O. For x in '][' there is in 5 an s such that XS� 1 a E A. If x tJ. M, then a tJ. B: for some b in B, x = s �b E 5�B = �M. Thus '][' \ M c � M, and so T* ('][' \ M) = O. For each measurable set P ,
�
T * (P n M) + T* (P \ M)
(v. 4.5.3), whence T * (P n M)
=
T * (M)
•
T(P ) ,
T(P), in particular,
=
1 > 0 = T* (M) .
(}� 1 (M ) � M in JR. has properties analogous to those of M . The set M is nonmeasurable, ), * (M ) 0, and ), * ( M )
The set •
=
= 00 .
=
The set M is thick and for every measurable subset P of JR., ), * (P n M)
=
0 while ), * (P n M)
=
),(P)
(whence if )'(E) > 0, then M n E is nonmeasurable).
[
4.5.6 Note. The use of Zorn's Lemma or one of its equivalents is unavoidable in the proof that !fj(JR.) \ 5), i 0 . Solovay [Sol] shows that adjoining the axiom: Every subset of JR. is Lebesgue measurable. to ZF, the ZermeloFraenkel system of axioms of set theory, yields a system of axioms no less consistent than ZF itself.] 4.5.7 Exercise. For (X, 5, f..l ) , if f..l* resp. f..l* are the associated �uter and inner measures induced by f..l on H(5), then a f..l * afinite E is in 5 iff
4.5.8 Exercise. a) If f : JR. r+ JR. is (Lebesgue) measurable and p is a polynomial function, then p 0 f is measurable. b) If g E C(JR., JR.) , then g 0 f is measurable. c) For the Cantor set Co, the corresponding Cantor function
Section 4.5. Nonmeasurable Sets
161
'1jJo (v. cPo and the function 5
( �)
2.2.40 and 2 . 2 .41 ) are continuous and: c1) �f J contains a set that is Lebesgue nonmeasurable;
'1jJ (10 , 20 ) = 2 ' 1 1 l c2) '1jJ (5) �f g(5) is Lebesgue measurable; c3) h �f X ,p  l ( S ) is Lebesgue
measurable; c4) h o g is not Lebesgue measurable. In sum:
continuous function composed with a measurable func tion is measurable; a measurable function composed with a continuous function can fail to be measurable. [ 4.5.9 Note. For measure spaces (Xi , 5 i , f.li ) , i = 1 , 2, the Fu A
binate (Xl x X2 , 5 1 52 , f.l l f.l2 ) derived in Fubini's Theorem is automatically complete (v. 2 .1.4 2 d ) ) and 5 1 x 5 2 :J 5 1 8 5 2 . There follow instances (4.5.10) where :J is = and others (4.5.18) where :J is �.l X
X
4.5.10 Example. If 5 i �f R or 5 i �f C, i = 1, 2, in 4.5.1, then
4.5.11 THEOREM. FOR FINITE MEASURE SPACES ( Xi , 5 i , f.li ) , i = 1, 2, THE FUBlNATE (X l X2 , 5 1 5 2 , f.l l x f.l2 ) IN 4.4.9 IS THE COMPLETION OF THE PREFuBINATE (X l X2 , 5 1 8 5 2 , f.l l 8 f.l2 ) ' PROOF. In the context and notation of the general DLS construction of Chapter 2, L is li ll I dense in L l ( X, f.l). Hence, if E E D and f.l (E) < 00, then X E is the II Il l limit of a sequence {fn} nEN in L. In the current con text (cf. 4.4.2 and 4.4.5), each In is a simple function with respect to 5 1 8 5 2 . Via passage to a subsequence as needed, it may be assumed that lim In �f I exists a.e., and with respect to f.l l x f.l 2 , I � X E . If each In n+ = is replaced by gn cl=ef 1 if In i ' o otherwise X
X
X
{
0
lim= gn �f g exists a.e., and with respect to f.l l then {gn } nEN C L, n+
If Fn �f { (x, y) · l Fn n+In = 1
=
clef F
x
f.l2 ,
Chapter 4. More Measure Theory
162
exists and is in 5 1
x
5 2 , and f..l 1
x
(
f..l2 Ft:J.E
)
=
O.
D 4.5.12 Exercise. The conclusion to 4.5.11 remains valid if the measure spaces are afinite. 4.5.13 COROLLARY. THE FUBINATE OF (JR.k , s T k , ),0k ) AND (JR.1 , S T1 , ),0 1 ) IS THE COMPLETION OF THE PREFuBINATE (JR.k+ l , S� (k+ l ) , ),0 ( k+ l ) ) . 4.5.14 Exercise. If L �f Coo (JR.n , JR.) and I is nfold iterated Riemann in tegral: I : Coo (JR.n , JR.) 3 f r+ f dXn . . . dX 2 dX 1 , then I is a DLS functional. The result of applying the DLS procedure in the con text of L and I is a measure space that is the n fold Fubinate of (JR., 5)" ), ) with itself. 4.5.15 Note. The nfold Fubinate of (JR., 5), , ),) with itself is denoted (JR.n , S),n ' ),n ) ; ),n is ndimensional Lebesgue measure .
l (l C·· (l ) ) )
[
]
4.5.16 Example. For the set M of 4.5.5, 5
�f {O}
x
M C {O}
x
JR. E S� 2
and ), X 2 ({O} x JR.) = 0: 5 is a subset of a null set (),X 2) . If 5 E S�2, then = M E 5)" a contradiction: (JR.2 , ST2, ), 0 2) is not complete; (JR.2 , S�2 ' ), ) 2 is complete. 50
4.5. 17 THEOREM. IF {k, l} e N, THE COMPLETION OF
\
IS (TTll k+ l , 5 ), k + l ' A k+ l : S 0), ( k+l) 5 ),k + l ' PROOF. Since JR.n is totally afinite with respect to both ),n and ),xn, the results in 2.5.13 and 2 .5.14 apply. D m.
)

4.5.18 COROLLARY. S� (k+ l ) � S� ( k+ l) . 4.5.19 Note. When n > 1 and confusion is unlikely, the notation (JR.n , 5),, )') replaces the unwieldy (JR.n , S)'n ' ),n ) and the associated usages are simplified.
[
]
4.5.20 Exercise. The measure space (JR.n , 5),, )') is translationinvariant and rotationinvariant, i.e., for T a translation or a rotation of JR.n , if E E 5)" then T E E 5), and )' T E = ), E .
( )
[ ( )] ( )
163
Section 4.5. Nonmeasurable Sets
[Hint: The aring generated either by the set of all open balls or the set of all open rectangles
n
contains 5 � .J
(lR.nn, x
4.5.21 THEOREM. IF 5), n ' f.l ) IS A MEASURE SPACE SUCH THAT EACH HALFOPEN INTERVAL k l [a k, bk) IS OF FINITE f.lMEASURE AND f.l IS TRANSLATIONINVARIANT, =FOR SOME NONNEGATIVE CONSTANT p,
l) n )
[o, l) n
n
PROOF. If f.l ( [0, is the union of 2 k pairwise �f p, then, since disjoint halfopen subintervals Ij , each the translate of any other,
Thus f.l ( E) = p for every halfopen interval E and hence the equality ) obtains for every E in the aalgebra generated by the set of halfopen ��. D As remarked earlier, 4.4.9 imposes no restriction on the underlying measure spaces. Since f E L l (X, f.l ) , E,,,, (f, 0) is afinite, i.e., the integration with respect to f.l is performed over a afinite set. However, if only 5measurability for f is assumed, K", (f, O) can fail to be afinite and the conclusion of Fubini's Theorem can fail.
An ( E
4.5.22 Example. If X l = X2 = 5 1 = 5 )', 5 2 = f.l l = ).., and f.l2 is counting measure then E �f { (x, y) : x y } E 5 1 5 2 �f 5, the char acteristic function f �f X E of the diagonal E �f { (x, y) : x y } is 5measurable, and v,
!fj(JR.) ,
JR.,
=
X
=
In this instance, K", (f, O) is not afinite. Indeed, f is not integrable with respect to f.l l x f.l2 .
164
Chapter 4. More Measure Theory
4 . 6 . Differentiation
The symbol
dy  used .
dx �
m
: S), 3 E
�: that appears in LRN is reminiscent of the classical symbol elementary calculus. If f E e ([0, 1 ] , q , y clef lX f(t) dt, and Ie f(t) dx � �(E), then � « >. and =
r+
dy dx
lim >.([x ,x+h] ) +O h>,o
>.([x,x+h])+O h>,o
lim
=
d� d>'
0
J:+ h f(t) dt
(4 . 6 . 1)
h �([x, x + h] ) >.([x ' x + h]
(4.6.2)
 = f(x).
(4.6.3)
The display above suggests that for the measure space (JR., a complex measure space (JR., , p, ) such that p, « >.,
S),
>' ( E ) +O >' ( E)>'O
lim
p,(E) .
S)" >.) and (4.6.4)
>.(E)
��
However, the left member of (4.6.4) is a point function, i.e., E ClR , while there is no reference to any point of JR. in the right member. The resolution of this difficulty is addressed in the following paragraphs. A particular con sequence of the development is a useful form of the Fundamental Theorem of Calculus (FTC) (v. 4.6.15, 4.6.16, and 4.6.33) : If f E L l (JR., >.) and F(x) �f
and F' � f.
lx f d>' , then F' exists
a.e. (>.)
In the wider context of Lebesgue measure and integration, FTC is a corollary to several more general results that will emerge as the discussion proceeds. [ 4.6.5 Note. Most of the next conclusions are true almost every where ( a.e.) , not necessarily everywhere. The calculations involve measures of unions of sets. Because measures are count ably addi tive, the arguments are eased if the constituents of the unions are pairwise disjoint. The context is a complex measure space (JR.n , p, ). The Jordan decomposition of p, and LRN permit the assumption that p, is nonnegative and totally finite, whence regular (v. 4.2. 19) . Thus attention can be focussed on open sets, since the measure of an
S)"
Section 4.6. Differentiation
165
arbitrary measurable set E can be approximated by the measures of open sets containing E. The following constructions allow much of the argument to be con fined to special collections of halfopen cubes Q and associated open cubes C. . . . These are used well in a different context in Section 7.1 . ]
U
'"
as
�f (kl , . . . , kn ) in Zn , cl=ef { x : cl=ef (X l , . . . , xn ) , k; Xi k is the halfopen cube vertexed at (k, m) cl=ef ( l m, >. [8 = 0, and thus >.(N) �f >. [ U For m in N and k
Qk,m
X
2m :::;
ki + 1 . , 1 :::; l :::; n <� kn
2m ' . . . , 2m
( Qk,m)]
Since (lR.n ,
>.(
S) = O.
k Ez n,m EN
}
) . For each k and
8 (Qk,m)
]
= O.
S)" >.) is complete, every subset S of N is in S), and All null sets S emerge as ignorable in the following
presentation. Unless the contrary is stated, null set, a.e., mea sure zero, =, etc., are to be interpreted as referring to Lebesgue measure >.. The open cube Qk, m [= Qk, m \ 8 ( Qk,m )] is denoted Ck ,m , while the closed cube Qk, m is denoted Rk , m ' For the remainder of this section, ffi.n will be regarded as endowed with its customary (product) topology. If A and B are two cubes each of which may be halfopen (a Q . . . ) or open (a C. . . ) and A n B i 0, then one is a subset of the other: A and B are disjoint or nestle. Hence any union of such cubes is the union of pairwise disjoint cubes. •
•
If m E N, then ffi.n = k EZ n Qk ' m ' If X E ffi.n , then x is in some unique Qk J , (.x ) , in some unique Qk2 , (x), . . . , i.e., there is a decreasing sequence
U
2
I
(4.6.6) of uniquely determined halfopen cubes such that x = •
If x E ffi.n \ N, there is a unique decreasing sequence
n
m EN
Qk = , m (X) .
(4.6.7) such that x =
n
m EN
Ck= ,m (X ) .
Chapter 4. More Measure Theory
166
4.6.8 LEMMA. IF U IS AN OPEN SUBSET OF JR.n , THEN: a) U IS THE UNION
Q ...
OF COUNTABLY MANY PAIRWISE DISJOINT HALFOPEN CUBES ; b) FOR SOME NULL SUBSET S OF N, U \ S IS THE UNION OF COUNTABLY MANY PAIRWISE DISJOINT OPEN CUBES PROOF. a) Since U is open, if U, then some rt is contained in is contained in U. For large m, some rt. Since the set of all halfopen cubes is countable, U is the union of a countable set of halfopen cubes. Since any two such are disjoint or nestle, if they nestle, the smaller of the two may be discarded. The open set U remains the union of those not discarded. b) If, as in the argument for a) , U = U , then
xE
Qk, m
C.... B(x, B(x, Q ...
and U \ S = U
C....
D
4.6.9 If U is an open subset of JR., then U = (/) or U is the union of a unique finite or countable set of pairwise disjoint open intervals.
Exercise.
4.6.10 DEFINITION. FOR A COMPLEX MEASURE SPACE (JR.n , SA , IN JR.n , WHEN IS THE SEQUENCE IN (4.6.7) ,
x
{Ck= ,m (x)} mEN '
f.l) , AND
xE
IF N OTHERWISE '
. f.l [Ck= , m (X)] EXISTS, IT IS THE derivative D f.l(x) x tJ N AND m+hm = [Ck= , m (X)] at x of f.l with respect to A. 4.6.11 LEMMA. a) IF r E JR. AND U �f { x : M(x) > r }, THEN U IS OPEN, ) i.e., M IS lsc. b) IF r > 0, THEN A(U) :::; If.ll (JR.n . r PROOF. a) If r :::; 0, then U JR.n . If r > 0 and x E U, then for· some m, If.ll [Ck= , m (X)] > rA [Ck= ,m (X)]. If Y E Ck= ,m (x), Ck= ,m (Y) = Ck= ,m (X) , whence M (y) > r, i.e., y E U: Ck= , m (x) C U. Since Ckm , m (x) is open, U is open . b) Each x in U is in some open cube C.. .(x) contained in U and such ] that I �I �:: (�� > r. If {Cp} EN is an enumeration of the cubes C... (x) P [ x varies over U, there is a subsequence {Cpq } qEN defined inductively as follows: WHEN
\ /\
=
as
167
Section 4.6. Differentiation .' CP1 = C 1 ; .' if CP1 , , •
•
•
Cpq have been defined, are pairwise disjoint, and u
=
:
u jq= 1 Cp
J
the induction stops. If U i U = 1 Cpj , Pq + 1 is the least P greater than q Pq and such that C is disjoint from U = C J 1
.
p
p
J
. •
Since two cubes C . are disjoint or nestle, U = U
Cpq and for each q, I IL I/Cpq/ > r. Thus )'(U) "' ), (Cpq ) '" IILI (CpJ ::::: IILI (l�n ) . D �q r �q r ), Cpq 4.6.12 Exercise. If f E L 1 (l�n , IL), IL : 5 3 E r+ IL(E) �f Ie f d )" and =
<
q
), [Ck,m1 (X) ] 1Ck.m (x) I f(y )  f(x)1 d)' (y ) then D IL (X) f(x) (v. 4.6. 15 ). 4.6.13 Exercise. If g E C(l�n , q, then for each x, lim 1 (X)] 1Ck,m (X) I g(y)  g(x) 1 d),(y ) m+ = ), [Ck,m hm
.
m+ =
=
0,
=
4.6.14
=
O.
Exercise. If f E L 1 (l�n , ), ) and > 0, then ), ({ x : I f(x)l > r }) ::::: ll1.lh r , sup 1 1 I f(y )1 d)' (y) > r } ) ::::: ll1.lhr . 1 :$m<= ), [Ckm ,m (X) ] Ckm ,m(X) r
f E L 1 (l�n , IL), THEN lim 1 (X)] 1Ck,m (X) I f(y)  f(x) 1 d)' (y) � O. m+ = ), [Ck,m
4.6.15 THEOREM. IF
[
Remark. The set Lf of x where lim 1 ( If(y)  f(x)1 d)' (y ) m+ = ), [Ck , m X ) ] 1Ck.m(x)
4.6.16
=
0
Chapter 4. More Measure Theory
168
f.
is the Lebesgue set of In view of 4.6.12, 4.6.15 implies that for the measure f.l 3 E r+ , the RadonNikodym
. df.l . f d)" IS
' denvatlve
PROOF. The map f.lx
:
:
S),
a.e.
Ie f(y) d).. (y)
]
S), 3 E
r+
le l f(Y)  f(x) 1 d)" (y ) is a measure. The
assertion to be proved is that Df.lx (x) exists a.e. and that the result is zero a.e. The conclusion is equivalent to the statement that if 0 the set of f.lx .IS a nu11 set . . . 1' pomts x not m N and wh. ere 1m Since (lR,n , q is a dense subset of L 1 (lR,n , if E 0 there is in
> [Ck= ' m (X)] > m+ = ).. [Ck= , m (X)] o C o ).. ) > n Coo (lR, , q a g such that I l f  g il l �f I l h l l < E. Furthermore, ).. [Ck:,m (X)] ik=. = (X) I f(Y)  f(x) 1 d)" (y ) ' = (X) I h(y ) 1 d)" (y ) + Ih(x)1 � ).. [Ck:, m (X)] ik= + ).. [Ck:,m (X)] ik= . = (X) I g (y )  g(x)1 d)" (y ) clef Im + IIm + IIIm' r
r
=
However, 
r
m+ = IIm > 3 } lim
Owing to 4.6.13, the last of the three summands above is empty; owing to 4.6.11 and 4.6. 14, the measure of each of the first two summands does 3E not exceed  . D r
4.6.17 Exercise. a) For some xfree constant Kn , if x E lR,n \ N and there is a
Ck,m (X) containing B(x,
r
r
> 0,
t and such that
Section 4.6. Differentiation
169
b) For some xfree constant L n ,and each Ck, m (X), there is a B(x, rt con taining Ck , m (x) and such that
> L n >. [Ck1,m (X)] '
1
>. [B ( x, r)O]
4.6.18 Exercise. If f E L l (lR.n , >.) and r ..(. 0, for all x off a null set,
1 1 I f(y)  f(x) 1 d>.(y) = 0, f(x) = m . >. [B(x,1 r)0] 1 f(y) d>.(y).
hr+O p
.
>. [B( x, r )0]
B (x,r) O
hrto
B (x,r) O
[Hint: For Ck ,m (X) and B(x, rt as in 4.6.17a)
K" >. (B(x,1 r) ) }r 1 (x)] 1 >. [Ck ,m 0
B (x , r) o
�
If( y )  f(x) 1 d>.( y )
Ck,m (X)
If( y )  f(x) 1 d>.( y ).]
[ 4.6.19 Remark. In [Rud] nicely shrinking sequences and in [Sak] sequences with parameters of regularity are introduced to define alternative versions of Df.l. Since two sets A and B under lying those treatments can fail to be disjoint or to nestle, covering theorems related to Vitali 's Covering Theorem are used to cope with this situation. The cubes C... require no appeal to such de VIces. The burden of 4.6.17b) is that for each x in not in N there is a nicely shrinking sequence {Ck ,m (X)} m EN ' Similarly, 4.6.17a) says that for each x not in N there is a sequence { B (x, rm ) ° } mEN that is nicely shrinking relative to some sequence in Q.]
There remains a collection of results dealing with three classes of func tions specified in the terms and symbols of 4.6.20 DEFINITION. WHEN P IS A RIEMANN PARTITION (cf. 2.2.50) OF [a, b] AND h E ClR : n a) THE Pvariation OF h IS varh (P) �f L I h h THE total
variation OF h ON [a, b] IS
k=2 (Xk) (Xk dl ;
P a Riemann partition of [a, b] };
Chapter 4. More Measure Theory
170
b) WHEN var h([a, b]) < 00 , h IS OF bounded variation on [a, b] :
h E BV([a, b]) ; WHEN
sup
oo< a < b
var h([a, b] )
< 00 ,
h I S O F bounded variation:
h E BV; c
) h IN
ffi.[a , b] IS monotonely increasing ON [a, b] (h E MON([a, b] ) ) IFF
{a � x < y � b} ::::} {h(x) � h(y) } ; d) WHEN 9 E [0, ) [a , b] AND h E MON([a, b]) , THE upper RiemannStielt jes sum AND resp. lower RiemannStieltjes sum ASSOCIATED WITH THE TRIPLE { g , h, P } IS (0
n
Sp(g, h) �f L sup g(x) . [h (Xk)  h (xk  d ] k= 2 X k  l :SX:SXk resp.
n
sp (g, h) �f L k linf g(x) . [h (Xk)  h (xk  d ] . < < k X _ X X k=2  b THE RiemannStieltjes integral 9 dh OF 9 WITH RESPECT TO h
l
EXISTS WHEN
inf { Sp(g, h) : P a Riemann partition of [a, b] } = sup { sp(g, h) : P a Riemann partition of [a, b] }
lb g dh IS THE COMMON VALUE OF THE TWO MEMBERS ABOVE; b b b dl) WHEN E ffi.[a , b] , l dh EXISTS IFF l g+ dh AND l g  dh EX b b b IST AND l g dh l g+ dh  l g  dh; b d2) WHEN E ([[a , b] , THE RIEMANNSTIELTJES INTEGRAL l dh AND
9
9
=
9
EXISTS IFF THE RIEMANNSTIELTJES INTEGRALS
lb 'S(g) dh EXIST; FURTHERMORE
9
1 �(g) dh AND b
Section 4.6. Differentiation
171
e) h IS absolutely continuous on [a, b] (h E AC([a, b]) ) IFF FOR EACH POS ITIVE E THERE IS A POSITIVE r5 SUCH THAT WHENEVER
THEN
WHEN h E ClR AND h IS IN AC([a, b] ) FOR EVERY [a, b] , THEN h IS absolutely continuous (h E AC) .
4.6.21 Exercise.
(2.2.41)
a) The Cantor function cPo: is in MON([O, 1]) and in BV([O, 1]) but not in AC([O , 1] ). b) If f E L l ([a, b] , >.) and, for x in x [a, b], F(x) �f f(y) d>.(y) , then F is in AC([a, b]) . c) The map
l
g : [0, 1]
3
x r+
{ XC o
sin
�
if x E (0, 1] if x = 0
is in BV([O, 1] ) if c > 1 but is not in BV([O, 1]) if c = l.
4.6.22 Exercise. 4.6.23 Exercise.
MON([a, b] ) U AC([a, b]) C BV([a, b]) . a) If h E BV([a, b]), the map x r+ varh ( [a, x]) is mono tonely increasing. b) If h E BV, then a lim var h ([a, x]) �f Th(x) exists, t oo Th E MON, and lim Th(X) < 00 x oo .
t
The next sequence of results leads to Lebesgue's Theorem
{I E BV} ::::} {If exists a.e. } . The approach is due to F. Riesz. An alternative proof can be based on
Vitali 's Covering Theorem .
4.6.24 LEMMA. IF f E BV([a, b]) , FOR SOME p AND q IN MON([a, b]) , f (p  q). PROOF. If a :::; c < d :::; b, then (v. 4.6.20) =
.
varf ( [a, d])  f(d)  [varf ( [a, c])  f(c)] = var f ( [a, dj)  var f ( [a, c])  [J(c)  f(d)]
and since 0 :::; If(c)  f(d) 1 :::; varf ( [a, dj)  varf ([a, c]) , both varf �f p and varf  f �f are in MON([a, b] ) and f = varf  (varf  f) = p D
q
q.
Chapter 4. More Measure Theory
172
4.6.25 Exercise. If I E BV, then; a) I I I is bounded: 11111 00 :s; K < 00 ; b) I is the difference of two bounded functions in MON.) The name Running Water Lemmasuggested by Figure 4.6.1is often given to 4.6.26 LEMMA. (F. Riesz) IF F E ]R.(O, l ) , THE SET D, CONSISTING OF x FOR WHICH THERE IS SOME x' IN (x, 1) WHERE lim F(y) < F (x'), IS OPEN.
IF D iTHEN
(/) AND D
=
y= x
U l <:: n < M <:: 00 (an , bn ) (cf. 4.6.9) AND x E (an , bn ) , F(x) :S; lim F(y). y= bn

PROOF. If x' > x E D and F (x') > lim F(y) y=x
x' tJ N(x), and for some N1 (x) , sup F(y)
clef =
<
Lx ,
then for some N(x) ,
F (x') .
If y E N(x) n N1 (x) �f N2 (x) , then y < x', and YENJ (x)
lim F(z) :S; sup F(y) < F (x') ,
z =y
yEN2 (x)
i.e., N2 (x) c D: D is open. If x E (an , bn) and F(x) > L bn , then Lx 2: F(x) > L bn ' A contradic tion emerges as follows. Since bn tJ D,
(4.6.27) By definition, for some x' greater than x, F (x') > Lx : F (x') > Lbn • But (4.6.27) implies x' E (x, bn l . If c �f sup { x' : x < x' :s; bn , F (x') > L x }, then an < x < c :s; bn . If c < bn , then for some c', c' > c, and As in the previous argument, c' E (c, bn l. However, by definition of sup, c' :s; c, a contradiction. Hence c = bn and L bn 2: F (bn ) 2: Lx > L bn , a final contradiction. Thus F( x) :s; L bn ' D
Figure 4.6.1.
xaxis
Section 4.6. Differentiation
173
[ 4.6.28 Remark. Riesz's result is applied below to the proof that monotone functions are differentiable a.e. Its analog for se quences leads to a perspicuous proof of G. D. Birkhoff's Pointwise Ergodic Theorem, [Ge3] .]
If f E ffi.lR and x i y, the ratIO gy (x) question of the existence of f'. •
clef f(y)  f (x ) is central to the =
yx
4.6.29 THEOREM. I F f E MaN, THEN f ' EXISTS OFF A NULL SET. PROOF. If
�f limg y (x), Dl ( X ) �f limgy (x), ytx ytx DT(x) �f limgy (x) , DT (x) �f limgy (x), ytx ytx D1 (x)
then 0 :::; Dl (X) :::; D1 (x) :::; 00 and 0 :::; DT (X) :::; DT(x) :::; 00. If a) DT(x) < 00 a.e. and b ) DT(x) :::; Dl ( X ) a.e., then a ) and b ) apply to x H f( x), which is also in MON. Hence 0 :::; D T (x) = Dl (X) < 00 a.e. and the argument is finished once a ) and b ) are proved. a ) . If Eoo { x : DT(x) = oo }, then for each x in E, any positive A, and some y such that y > x, gy (x) > A. Hence, for the map 
�f
F : x H f (x)  A x and its associated set D in 4.6.26 , there is a sequence { (an , bn )}l :s; n < M :S; oo depending on A and such that D = U < n<M < oo (an , bn ) . Furthermore, lE c D and 
F ( an ) :::; lim F ( x ) , A (bn  an ) :::; lim f ( x)  f ( an ) , x =bn
x=bn lim f ( x )  f (an ) :::; f( l )  f(O) . (bn  an ) :::; A l :S;n<M:S;oo l :S;n<M:S;oo x =bn
L
L
' /\' (D) < f( l )  f(O) , and SInce ' cont alne S·Ince E IS ' ' d In ' the open set D , sInce A A may be arbitrarily large, >.(E) = O. b ) . If _
E<
�f { x
r
: Dl (X) < DT(x) } , Q+ '3 < R E Q+ , ET R = { x : Dl (x) < < R < DT (x) } , r
Chapter 4. More Measure Theory
174
then E< C
U
{ r,R }CQ
ErR. The inequality Dl (X) F1 : x H
lead to a set D 1
<
r and the function
f( x) + rx
�f U 1 :5n < M:5 oo (an , bn ) and the inequality ( 4.6.30)
f
Inside each (an , bn ) there is defined the function Fn : x H (x)  Rx and the possibly empty open set Dn �f U < k< K < oo (an k, bn k). There is the 1corresponding inequality
(4.6.31 ) The combination of (4.6.30) and (4.6.31) yields
L (bnk  ank) ::::: i L (bn  an ) . n
k,n
(4.6.32 )
Induction yields a sequence {SN} NEN of sequences of open intervals. Each SN is a refinement of SN  1 . Consequently
Since ErR C U , (3 E ( ;3), ).. (ErR) = 0: ).. (Ed = o. ( a ) S2N The Fundamental Theorem of Calculus asserts the validity of 0,
f ( b)  f(a) =
lb f'(t) dt
f. In the context of Lebesgue integration there is 4.6.33 THEOREM. IF f E MON([O, l] ) , THEN: a) f' E L 1 ([O, l], )"); b ) fs : [0, 1] '3 x f (x)  f(O) fox f'(t) dt IS IN BV([O, 1] ) ; c ) f.� 0 ; d ) fs 0 IFF f E AC ([O, 1] ) ; e ) f (x)  f(O) 2': fox f'(t) dt AND EQUALITY OBTAINS IFF f E AC([O, 1] ) .
for suitably restricted
H
�

=
D
Section 4.6. Differentiation
175
1
PROOF. a) If 0 < c < 1, then for large n and x in [0, c] , x +  E [0, 1] and if n f'(x) exists, then f'(x) = nl�� n f x +  f(X) . Hence f' E SA , and c c 2.1.14a) implies f'(x) dx � l�� n f x  f(X) dx. Moren
[ ( �) ] l l [ ( + �) ] over lC n [f (x + �)  f(X)] dx = n [l c+ � f(t) dt  l � f(t) dt] � c+ � f(c) dt  1 f(O) dt] � n [l � f(c)  f(O)
�
f' E L 1 ([0, 1] , >.).
f(l)  f(O) :
lx
b) Both 9 [0, 1] '3 x H f(x)  f(O) and h : [0, 1] '3 x H f'(t) dt are in MON([O, 1] ) . c ) According to 4.6.18, g' � f' � h'. d) If fs = 0, then f(x) = f(O) + f'(t) dt, whence f E AC([O, 1]). If f E AC([O, 1]), then 4.6.15 and 4.6.18 imply fs = o. e) The argument based in a) on Fatou's Lemma may be repeated to f'(t) dt. Furthermore fs 0 iff f E AC ([O, 1] ) . imply f(x)  f(O) 2': D 4.6.34 Example. a ) If :
lx
lx
f(x) �f
=
{�n(x)
if O < x � 1 if x = 0
then f'(x) � .! and f' tJ L 1 ( [O, 1] , >.). b) If f = x
x E [0, 1] ,
lx f'(t) dt
=
cPo , then f' � 0, whence if
0 and f(x) = fs (x) .
4.6.35 Exercise. If  00 < a � b < 00 , assertions like those in 4.6.33 are valid for functions in BV ( [ a, b] ). 4.6.36 Example. If f(x) == x, then f E MON n AC and f' == 1. Hence f' tJ L 1 (JR., >' ) , and the formula f(x) = fs (x) + f'(X) dx is invalid.
[Xoo
4.6.37 Exercise. If f' exists everywhere on [ a, b] and f' (a ) < C < f' (b) , then for some � in (a, b) , f'(�) = C. A derivative enjoys the intermediate value property.
Chapter 4. More Measure Theory
176
[Hint If g(x) �f f(x)  ex, then g'(a ) < 0 and g' (b) > O. Hence, for some � in ( a , b), g(�) = min g(x ) .] :
as;xS;b
4.6.38 THEOREM. IF: a) f E ]R.[O, 1 l ; b) I' EXISTS THROUGHOUT [0, 1] ; AND c) I' E L 1 ([0, 1] , ),) , FOR EACH x IN [0, 1 ] , f(x) 
f( O ) =
1x f'(t) dt.
[ 4.6.39 Note. The result is in sharp contrast with 4.6.34e) . Thus the assumption b) provides the crucial ingredient for the validity of the conclusion.]
PROOF. The line of argument is to show that f(l)  f( O ) �
11 I' (t) dt. 1x t(t) dt.
That conclusion is applicable to  f and leads to f(l)  f( O ) = The same kind of reasoning can be used when 1 is replaced by any x in [0, 1] .
Although, as a derivative, I' enjoys the intermediate value property, that alone does not suffice for the present purposes. In [ GeO] there is an example of a null function that assumes every real value in every subinterval. The technique is rather to exploit the lower semicontinuity of an aux iliary lsc function 9 that approximates I' from above. The DLS construction implies that if E > 0, there is in lsc([O, 1] ) a 9 such that 9 2': I' and
11 °
g dt <
1
1 I' dt + 2E. Hence
9
+ E �f P E lsc([O, 1] ),
1 1 p dt < 1 1 I' dt + E. 1 If, for every positive E, 1 p(t) dt  f( l ) + f(O) + E 2': 0, then
p > 1', and
f( l )  f(O )
:::;
1 1 p(t) dt + E < 1 1 f'(t) dt + 2E,
11
whence f( l )  f(O) :::; I'(t) dt. The desired conclusion is reached via the following argument showing that if
G, (x) �f 1x p(t) dt  f(x) + f(O ) + EX, then for every positive E, G, (l) 2': O.
Section 4.7. Derivatives i.e.,
Since G, (O)
=
177
0, if G,(l)
<
0, for some largest x in [0, I), G, (x) = 0,
(4.6.40) {x < x' :::; I} ::::} { G, (x') < O} . The hypothesis b) implies that f' (x) exists: for some positive h, x + h < 1, and if x :::; y :::; x + h, then f(y)  f(x) < f'(X) (Y  x) + E. Since p is lsc, for some positive k, x + k < 1 and if x :::; t :::; x + k, then p(t) > J'(x) . Hence, if J �f min{h, k} « I), then G, (x + J) = G, (x + J)  G, (x)
=
>
lx+8 p(t) dt  f(x J) f(x) EJ + + + x
J'(x) J  J'(x) + E J + EJ = 0,
[
]
in denial of (4.6.40). 4.6.41 Remark. Many more theorems about derivatives can be found in the literature, e.g., (DenjoyYoungSaks) If f E for some set A, )'(A) = 0, and if x tJ A, then
[
D
ffi.lR ,
•
f ( x +_ )h' f (x + h)  f (x) = = lim _ _f'...(x') lim ' h h "tO h "tO h or f' (X) exists, 00
,_
_
v. [SzN] ; In the metric space ( O , the set of nowhere differen tiable functions is of the second category, cf. 3.62 in [Ge3] .]
C [ 1], ffi.) ,
•
4.6.42 Exercise. If f E BV and for some g , f = g' , then f E
C(ffi., C).
4 . 7 . Derivatives
The developments in Section 4.6 lead to a rigorous proof of the change of variables formula of multidimensional calculus. . cle For x cle =f (X I , . . . , Xn ) and y =f ( Y I , . . . , Yn ) In ( x, y ) cle=f '" � XkYk mm m. ,
n
n
k= 1
and I lxll � �f L x� .
k= 1
ffi.n m'3 x n [ffi. , ffi. ],
4.7.1 DEFINITION. A FUNCTION f entiable at a IFF FOR SOME Ta IN
:
H
Y �f f(x) E
f(a + h )  f(a)  Ta ( h ) 11 2 lim Il II h l 1 2
h + O h >' O
=
0;
ffi.m IS differ
eo
Chapter 4. More Measure Th ry
178
I AT a: T.a
=
=
[Hint: a) For any C... , if E �f H n C... , Fubini's Theorem im plies dA n = b) Each M is the intersection of finitely many hyperplanes like H . 4.7.6 a) If [!R.n , !R.m ] and E SAn ' then E SA= ' b) If m = n, the map SAn '3 E A [/(E)] is a possibly trivial translation invariant measure. c) For some nonnegative constant p(f ) ,
le I
Exercise.
O.
f..l :
f..l ( E )
]
IE
==
E
ft
I( ) E
p(f) A ( E ) and D f..l (x) == p(f ).
I
[ Hint: For an open cube C . , (C, .) is open or a null set.] The result in 4.7.6 motivates
x E U, I E C (U, !R.n ) ,
4.7.7 THEOREM. IF U IS AN OPEN SUBSET OF !R.n , AND S,(3 !R.n AND EXISTS, THEN
I (B(x,rn E ( ) A [I (B (x, r ) ° )] p [I lim (4.7.8) ' (x)] . T+O A [B(x,r)o] PROO F. Since I (B(x, r )O) U I {B [x, (1  T n ) r] } and each summand n EN in the right member above is compact, I (B(x, r t ) is acompact, and hence in S,(3 (!R.n ) . Since A is translationinvariant, the conclusion when x = 0 1(0) I'(x)
=
=
=
implies the general case. The line of the argument depends on whether the linear map 1'(0) �f T is: a) nonsingular; or b) singular. For a) , if 9 �f T 1 then g' (0) = T 1 (0) = id , and the desired equality is A{ = lim g (4.7.9) T+O A 0
I,
I' [B(x, r)O]} [B(x, r)o] 1 ,
Section 4.7. Derivatives
179
v. a') after the PROOF of 4.7. 10. For b) the range im (T ) is a subspace of dimension less than n. Thus (4.7.5) im (T) is a null set (A), whence p(T) = ° and the desired result is that the left member of (4.7.9) is zero, v. b') after the proof of a') . When = ° = 1(0) and T = id the numerator of the basic difference quotient is which motivates
x 11/(x)  x1 1 2 ' 4.7.10 LEMMA. IF g E e (B(O, 1), ]R.n ), E E (0, 1 ) , AND ( 4 . 7. 1 1 ) {llxl 1 2 = I} {l l g(x)  xl 1 2 < E} , THEN (B (0, 1 n ::) B (0, 1  E t . PROOF. If the conclusion is false, for some y , I Y l1 2 < 1  E and y tJ g (B(O, ln , a contradiction. Since I II x l 1 2  Il g (x) 11 2 1 � Il g (x)  x1 1 2 , (4 . 7 . 1 1 ) implies that if I x l 1 2 = 1, then I l g (x)1 1 2 > 1  E > I l y 1 2 , i.e., not only is y not in (B(O, I n but y is not in g[B(O, 1)] . Thus G : x II yy  g/x)X)II 2 IS a welldefined continuous map of B(O, 1) into itself. Brouwer's Fixed Point Theorem (1.4.27 and 1.4.36) implies that for some xo , G (xo) = Xo . If IIxol1 2 = 1 , the equation G (xo) = Xo and Schwarz's inequality [(3.2. 11)] imply I l y  (xo) 1 2 = (xo, Y  (xo)) = (xo, y ) + (xo , x  (xo))  1 < Ilyll + E  1 < 0, a contradiction. If I xol1 2 < 1, since II G (x) 11 2 == 1 , Xo i G (xo), a final con tradiction. D a') (T is nonsingular) By definition, if E > 0, for a positive J, '*
9
9
H

9
9
9
9
'
As a consequence,
1 /(x)1 1 2 < (1 + E) l xI 1 2 , i.e., 1 [B(O, rt] C B[O, (1 + E)r] o . By virtue of 4.7.10, if E E (0, 1), then B(O, ( 1  E)rt C 1 (B(O, rn. A { [B(x, r )O] } Hence, if E E (0, 1) , then (1  E) n � J (1 Et and (4.7.9) A [B(x, r) O ] � + obtains. b') (T is singular) If E > ° and U, �f { x : inf I l x  Y l 1 2 < E } ' yElm (T) then U, is the open set consisting of all x within a distance of E from
some point in im (T) . If K is compact, K n U, is covered by finitely many
Chapter 4. More Measure Theory
180
CK , A (K :::; C ro , r :::; ro CK, :::; CK2 . Ar I l f(x)  J'(O)x I 1 2 :::; Ar l x l 1 2 , I x l 1 2 < r, Kr rtO Ar [B(O,r)O]} < ArCKrn rn ArCKr . f[B(O , r)] , A {AJ[B(O, D r r)o] [ 4.7.12 Note. In (4.7.9) each B(x, rt may be replaced by an open cube Cr(x) with edges parallel to the coordinate axes, edge length 2r , and containing x. Corresponding to (4.7.9) is A {g [Cr(x)]} [f'(x)] . r� A [Cr(x)] There is a constant Kn such that for all x in ffi.n , the norm open balls of radius E. Hence, for a constant KEn . n U,) C Furthermore, if then For a positive if and there is an such that and lim = O. The facts just stated imply that when is the compact set
Kl K2 ,
=
1·
=
P
I l x l ' :::; IIxl 1 2 :::; Knl l x l ' , i.e., I I ' and I 1 2 are equivalent . x and of radius r for the norm I I ' is B'(x, rt �f { y : I l x  yll' < r } ,
satisfies The open ball centered at
i.e., the open cube with edges parallel to the coordinate axes, edgelength and centered at Corresponding to 4.7.10 is the statement:
2r,
If 9 E then 9
x.
C (B'(O, 1t,ffi.n ), E E (0, 1), and {l l x l ' = I} {llg(x)  xi i ' < E} , (B' (0, 1 n B' (0, 1 Et .] ::)
'*

4.7.13 Exercise. The assertions in 4.7.12 are valid. Together with 4.7.6, 4.7.14 below forms the basis for the change of variables formula of multidimensional calculus (4.7.23 below).
N IS A NULL SET (A) IN ffi.n , f E (ffi.n ) N , AND FOR ) f ) IN N , inf sup I l f (�  fr 11 2 < 00 , THEN f(N) IS A NULL N(y) N(y)3x#y Ix 2
4.7.14 LEMMA. IF EACH SET
Y
(A).
Y
Section 4.7. Derivatives
181
N is the union of the count ably many sets Ekp �f { X : x E N, I l f (x)  f( y ) 11 2 � kllx  yI1 2 , E B (x, � ) n N } , k ,p E N, if each f (Ekp ) is shown to be a null set ( ), ) , the result follows. For the rest of the argument, the subscript kp is dropped. If E, 1] > 0, E is contained in an open set U such that ), (U) < 1]. As in the PROOF of 4.6.8, U is the union of pairwise disjoint halfopen cubes Q ... and L... ), (Q .. . ) < 1]. Since each halfopen cube Q ... is the union of PROOF. Since
Y
all the smaller halfopen cubes it contains, E ( = Ekp ) can be covered by pairwise disjoint halfopen cubes of diameter less than � . If ... E E , then ... there ob ... , diam . . . ) t , whence for some constant tains for the sum of the measures of all the open balls [x.. . , diam ... W corresponding to all the ... in all the Q ...
Q C B [x
p x n Q ... K, (Q B
(Q Q
L), {B [x... , diam (Q . . . ) t } K L ), (Q ... ) < K1]. =
Hence, if
1] �, then L ), {B [x. . . , diam (Q . . . ) t } < E . =
According to the definition of E ( = Ekp ) ,
f(E) C U B [f (x... ) , k · diam (Q . )] , whence
L ), {(B [f (x... ) , k · diam (Q ... )]} � kn L ), { B [x.. . , diam (Q . . . ) t } D
V f' V, f () () f(V): {{,>, (N) o} 1\ {N C V}} {,>, [f(V)] o} ;
4.7.15 COROLLARY. IF IS AN OPEN SUBSET OF ffi.n AND EXISTS AT EACH POINT OF THEN: a) MAPS NULL SETS ), CONTAINED IN INTO NULL SETS ), CONTAINED IN
f
=
'*
V
=
b) MAPS LEBESGUE MEASURABLE SETS INTO LEBESGUE MEASURABLE '* {J(E) E SETS: {E E PROOF. a) 4.7.14 applies.
S ), }
S ), }.
Chapter 4. More Measure Th ry eo
182
b) Since ), is regular and JR.n is a countable union of compact subsets n (JR. is acompact) , if E E S)" then for some acompact set 5 and a null set N, E 5'0N. Hence f(E) f(5) U f(N). Moreover, f(5) is acompact; owing to a), [f ( N ) ] O. D 4.7.16 E xercise. There is no conflict between 4.7.15 and the following facts: a) the Cantor function cPo is differentiable a.e. ( ),); b) the Cantor set Co is a null set ( ),); c) ), [cPo (Co )] l. = 4.7.17 Exercise. a) If f E (JR.n ) lR f (x) �f (/1 (x), . . . , fn (x)) and J' ex8fi (x) eXIsts, . . then . 1 < l. < n , 1 < . < m, and t he matnx ' representmg Ists, 8Xj the linear transformation J' with respect to the standard basis e l , . . . , en is ( 8f8xj(X) ) j �f J(I) , J' (x ) I S i S n, I S Sm the Jacobian matrix of f. b) The entries in the Jacobian matrix J( I ) are = Lebesgue measurable functions. c) If f E [lRm , JR.n ] (c (JR.n ) lR ) and n f (ei ) �f jL aijej , 1 � i � m, "" l 8fi (X) 1 � l. � n, 1 � . � m . then aij ' 8X · 4.7.18 THEOREM. IF: a) f E (JR.n ) B ( O , 1) o; b) J' EXISTS THROUGHOUT B(O, It; c) f IS INJECTIVE AND sup { llf(x)1 1 2 : x E B(O, It } < 00; d) g : JR."' ft [0, (0) IS LEBESGUE MEASURABLE; e) p IS THE FUNCTION IN 4.7 .6, THEN BOTH (I' ) 1 B ( O, I) O AND g o f . p ( I' ) 1 B ( O, I) O ARE LEBESGUE MEASURABLE AND J f ( B (O, I) O ) d)' irB ( O, 1) 0 (g f) . p ( I' ) d)'. PROOF. The measurability of p ( l' ) I B ( o , 1) 0 is a consequence of the results in 4.7.17 . If E E S)" 4.7.15 implies: a) p,(E) �f ),[f(E)] exists; b) (JR.n , S)" p,) is a complex measure space (direct calculation); c) p, « ),. Owing to 4.6.12, if E E S ), n B(O, It, then p, (E) Ie Dp, d)'. Hence, for x in B(O, It and small r, p, [B(x, r t] ), [f (B(O, r)O)] (4.7.19) ), [B(x, r)o] ), (B(x, r)O ) Owing t o 4.7.6 and (4.7.19), Dp,(x) � p [J' (x) ] (v. 4.7.12). =
),
=
=
=
_
=
=
_
_
J
_
i
J
J
P
=
9
=
0
Section 4.7. Derivatives
183
If A E S,B and E �f f 1 ( A ) , then X E = X A 0 f: E E S,B ( C SA ) . Thus 
J
X
f(B(0, 1) 0)
A
d)" = r X 0 f . p (I ' ) d)" . lB(0, 1)0 A
( 4.7.20)
If N is a null set ().. ) , for some A in S,B, A ::) N and )"(A) o. Thus (4.7.20) is valid with A replaced by any Lebesgue measurable subset E of B(O , It. The standard approximative methods extend the validity of (4.7.20) to the case where X A is replaced by an arbitrary nonnegative Lebesgue measurable g. D [ 4.7.21 Note. The PROOF of 4.7.18 shows that g o f . p (I' ) is measurable; g o f need not be measurable (cf. 4.5.8).] =
4.7.22 Exercise. If: a) V is open in ffi.n ; b) X is a Lebesgue measurable subset of V; c) f V '3 x ffi.n is continuous, f is injective throughout X, and f ' exists throughout X; d) )"(V \ X) = 0; and e) 9 ffi.n [0, (0) is Lebesgue measurable, then
ft
:
J
f (V)
9
:
d)" = r l
v
90
ft
f . p ( I' ) d)".
[Hint: The open set V is the countable union of open balls; 4.7.18 applies.]
4.7.23 THEOREM. IF f E [ffi.n ] AND f ' EXISTS, THEN p [!, (x)]
=
Idet {J[f(x)]} I .
PROOF. If T �f !' (x), e is a constant, and
if k = i otherwise '
then I det { J [f (x)]} I = l ei . On the other hand, if the edgelengths of a cube C .. are S 1 , , S n and the corresponding edgelengths of T ( C. ) are s�, . . . , s � , then sj = sj if j i i, and s ; = l ei · Si, whence p(T) = l e i : Idet {J[f(x)]} 1 = p(T) . If if k sl {i, j} if k = i if k = j ..
.
•
•
•
Chapter 4. More Measure Theory
184 T(ABCD) A(ABCD)
=
=
ACED
A.(ACED),  "
E
, ,
,
D tC'+.,.,, C ,
A
Figure 4.7. 1.
B
then l[f(x)] is the result of interchanging rows i and j of the identity matrix: Idet {l[f(x)]} 1 = 1. Because the measure of any cube C... of edge length £ is regardless of the labeling of the coordinate axes, p(T) = 1: Idet {J [ f (x) ] } I = p(T) . If T (ek ) = ei + ej if k = i otherwise ' ek
£n ,
{
then Idet {J [f (x)] } I 1. If n = 2, by direct calculation (integration or elementary geometry), (C. . . ) = [T (C . . . )]. The situation is illustrated in Figure 4.7.l. If n > 2, via Fubini's Theorem and induction, (C. .. ) = [T (C... )]: Idet {J [f (x)]} I = 1 = p(T) . Since every linear transformation is the composition of a finite number of transformations, each like one of the three just described, the product rule for determinants applies [Ge2] . D 4.7.24 Exercise. a) If + > 0, the map =
),2
), 2
),n
),n
a 2 b2
is bijective and !, exists. b) What is p ( I' ) ? When (X, S, p, ) and (Y, T, � ) are measure spaces an F in yX is defined to be measurable iff { E E T} ::::} {J  l (E) E S } [Halm] . In the current context, since [ffi. ] may be regarded as , !' as a function of x may be ]Rn regarded as an element of
n
(ffi.n2 )
ffi.n2
4.7.25 Exercise. Both f ' and p ( I ' ) are Lebesque measurable.
Section 4.8. Curves
185
4 . 8 . Curves
4.8.1 DEFINITION. A curve IN A TOPOLOGICAL SPACE X IS AN ELEMENT "/ OF C([O, 1] , X ) . THE SET im h) �f "/* �f { "/ ( t ) : t E [0, 1] } IS THE image of"/. WHEN X IS A METRIC SPACE (X, d) , THE ( POSSIBLY INFINITE) length fih) OF "/ IS
=
WHEN fih) IS FINITE, "/ IS rectifiable. WHEN ,,/ ( 0) ,,/ (1 ) , "/ IS closed. WHEN "/ �f hI , . . . , "/n ) E C ( [O, 1] , ffi.n ) , f E C h* , c) , AND "/ IS RECTIFI ABLE, THE RiemannStieltjes integral of f with respect to "/ IS
4.8.2 Example. For the curves
"/ 1 : [0, 1] 3 t
ft e 27rit E C, "/2 : [0, 1] 3 t ft e 37rit , "/3
"/ � = "/� = ,,/; , while fi h d is not a closed curve.
=
271", fi (2 )
=
371", fi (3)
=
:
[0, 1] 3 t
ft e47rit ,
4 71" . Furthermore, "/2 =
+
4.8.3 Exercise. For the Cantor function
"/0: : [0, 1] 3 t ft t + i <Pa (t) , what is fi ho: ) ? 4.8.4 Exercise. If if ° < t ::; 1 if t = °
= 00 .
then fih) 4.8.5 Exercise. For a rectifiable curve "/ : [0, 1] 3 t ,,/(t) E ffi.n and a Riemann partition P �f {td l :S: k :S: n of [0, 1] : a) the formulre
"/P [Otk
ft
+ (1  O)tk+ d
=
0,,/ (tk)
+ ( 1  0)"( (tk+ d
° ::; 0 ::; 1 and 1 ::; k ::; n  1 ,
Chapter 4.
186
More Measure Theory
define a rectifiable curve, "/p and £ ("!p) ::; £ (,,/); b ) if E > 0, then for some P, £ ("!p » £("!)  E; c) {£ (,,/p » £ ("! )  E} ::::} {l hp  "/ l oo < E}. d) if n = 2, f E C ( [0, 1], C) , and E > 0, for some P, I I I f d,,/p  1 1 f d"/ I < E. [ 4.8.6 Note. The curve ,,/p is piecewise differentiable; on each interval (tk , t k+ d, "/� is constant. The set b (t k )} l < k < n of its vertices is a subset of ,,/*. If f E C([O, 1], C) , then
f d,,/ is eased if "/ is well 1 behaved or, like ,,/P , only piecewise wellbehaved. For example, if
More generally, the calculation of
{tk } l :$k :$n is a Riemann partition of [0, 1] and
then
1 f dz = 11 f
0
"/ . "/' dt .]
,,/ �f a + ib, {a, b} c C ([O , 1], ffi.) , THEN "/ IS RECTIFI {a b} BV([O, 1]).
4.8.7 THEOREM. IF ABLE IFF , C
PROOF. The triangle inequality (v. 3.2.12) implies that
{ I a (tk+ d  a (tk ) 1 , I b (tk+ d  b (tk ) l } ::; b (tk+ d  "/ (tk )1 ::; l a (tk+ d  a (tk )1 + I b (tk+ d  b (tk )l · Thus "/ is rectifiable iff {a , b} c BV ([0 , 1]). [ 4.8.8 Note. If "/ is rectifiable, then £ ("! ) is not only n l sup L h (t k+ d  "/ (tk ) 1 0 = to < t l < ' " < tn 1, n E N , k=O max
{
=
:
}
i.e., the supremum of the lengths of the associated polygons with vertices on
"/* but, by abuse of notation, £ ("! )
=
11 1 · I d"/ I .]
D
Section 4.9. Appendix: Haar Measure
187
4 . 9 . Appendix: Haar Measure
A
topological group G is a Hausdorff space and a group such that G G '3 (x, y) xy  l E G X
H
is continuous. For a locally compact group G, there is a measure defined on aR[K(G)] �f S(G) . For there obtains:
 Haar
measurep, p, {{ E E S(G)} !\ {x E G}} {{ xE E G} !\ {p,(xE) p,(E)}} , i.e., p, is translationinvariant. Complete proofs of the existence of Haar measure p, and derivations =
'*
of its most important properties are given in [Halm, Loo, Nai, We2] . Below is an outline of the fundamental idea behind the existence proof.
a) The motivation lies in the next alternative definition of the Riemann and Nf E N there is a nonempty family C integral. If f E Coo of nonnegative constants such that of sets
(ffi., ffi.+ ) { cn}  Nt ::;n ::;Nt (2Nt ) 2_ 1 f(x) � L k=O
Furthermore,
1 f(x) dx lR
=
Nt i�f L cn · n=  Nt
The integral stems from the
majorization of f by linear combinations of translates of x . b) Haar's idea was to imitate the procedure described in a), namely to majorize one function by linear combinations of translates of another and thereby approximate some kind of DLS functional. For a locally compact group G and a pair f, in Coo (G, there is a (possibly empty) family C of sets of nonnegative of G constants such that for some subset S �f
ffi.+ ) { g} { cn }  Nt 
Nt f(x) � L cng (Yn x) . n=  Nt Nt If g =t= 0, then C i (/) and (f : g) �f i�f L Cn E ffi.+ . The functional n=  Nt (f : g) enjoys the following properties. (All functions below are in
Chapter 4. More Measure Theory
188 9
f[y] (x) �f f (yx).) (4.9.1 ) (f[y] : g) (f : g) , (f + k : g) (f : g) + ( k : g ) , {t o} {(tf : g ) t(f : g ) } , { J ::=; k } {(f : g) ::=; ( k : g ) } , (f : h) (f : g) . (g : h), (f ' ) > IIll fg ll oooo ' : For a fixed nonzero fo in Coo (G, ffi.+ ) , if Ig(f) �f ? . g\ , then fo . 1 ::=; lg (f ) ( f : fo ) . (fo : g) The choice of fo is arbitrary. Once fixed, fo sets the scale of the Coo (G, ffi.+ ) ; neither
nor h is identically zero;
2:
'*
::=;
=
=
'*
::=;
. 9

9
::=;
•
•
•
measure. It is to be expected that if 9 ::=; h, then Ig approximates the de sired functional more satisfactorily than h . The set (G, ffi.+ ) is a poset with respect to ), as defined by: h} {} ::=; h} (smaller functions succeed larger functions) . If E is compact, for the (G, ffi.+ ) , since
{g )
Coo
{g
[ (fo 1: g ) ' (f : fo)] 1 , (f : fol there is a net n : Coo (G, ffi.+ ) 3 g rt 1g(f) E [ (fo : g) convergent cofinal net. Its limit is defined to be 1 ( f). Owing to (4.9.1), I is a translationinvariant linear functional: I (f[y] ) 1(f). f Coo
=
c) Arguments using partitions of unity lead to the conclusion that I can be extended , say to J , a DLS functional having as domain ( G, ffi.) . The machinery of Chapter 2 applies and produces the (left) translationinvariant Haar measure p, defined on S aR[K(G)]: if E E S and x E G, then p,(xE) = p,(E) . The following are the basic facts about Haar measure: if V is a nonempty open set in S, then p,(V) > 0 ; if K is compact, p,(K) < 00; if � is a (left translationinvariant) Haar measure, then for some positive constant c, � = cp, ; c depends upon the choice of in the construction above.
Coo
�f
•
•
•
fo
Section 4.9. Appendix: Haar
Measure
189
For the locally compact groups ffi. resp. ffi.n resp. 'lI' resp. Z, A resp. An resp. T resp. are (the essentially unique) translationinvariant measures. The convolution of two functions and g in is defined by V
L l (G, p,)
f
With respect to pointwise addition and multiplication defined as convo lution, is a Banach algebra the The group algebra: a) is commutative iff is abelian; b) has an identity iff is discrete. , If is abelian, for the measure : 5 '3 H P,
L l (G, p,)
G
�(xE) p, ((XE)  l ) = If E E  \ then �(E) p,(E). multiplicative constant, � = p, . =
=
=
P,
A(G), group algebra of G. G G � E (E  l ) (x  l E  l ) p, (E  l ) = �(E) . =
Since Haar measure is unique up to a
The following observations are valid when 1) If
M E Sp
G,
G is abelian.
' i 0, [A(G)] ' f E A(G), a E f[a] ( x) clef f ( ax ) , and f(M) =
then
[a] ( M ) (4.9.2) G '3 a {f(M) is independent of the choice of f and is a continuous open homomor phism of G into 'lI'. Furthermore, G denoting the character group of G, i.e., aM :
,'H ';;: aM
the set of all continuous open homomorphisms of G into 'lI',
g
:
Sp (A) '3 M H a M E G
is a bijection. As in the case of Banach spaces and their duals, if "( E G the notation (x, "( ) is used for ,,((x). 2) Since G and Sp are identified, the latter inherits a topology via a (A', A) for A' . Since the unit ball of is a (A', A)compact, G is a loc�lly compact abelian group. If x E then hx G '3 "( H (x, ,,() E 'lI' is , in G. Moreover (Pontrjagin) '3 x H hx E G is an isomorphism (in the category of topological groups and continuous open homomorphisms) . 3 ) If E A(G), the j, also called the may be viewed as a function on G . For j there ob'3 H f ( x ) ( x, "( ) dp, ( x ) �f jb) . Furthertains the formula: ' :
[A( G)]
G,
A'
G f Fourier transform Fourier transform, A(G) f i
:
Gelfand
C) , and since the quotient map is a normdecreasing homomorphism, 1 1 1 00 � I l f l l l
more, j E Co (G ,
'
Chapter 4. More Measure Theory
190
4) If {f, g , h} c Coo(G, q , Holder's inequality and the translation invariance of p, imply (4.9.3)
On the other hand, f E £P(G, p,) and h E £P' (G, p,). Hence, if Il h llp � 1, ' Fubini's Theorem implies
I i [i f (y 1 X) 9(Y) dP,(Y)] h(x) dp,(x) I Ii (J[y] , h) g(y) dp,(y) I =
� II f[y] l i p . Ilgll l '
Thus 3.3.6 implies
(4.9.4 )
5) Equation (4.9.3) resp. (4.9.4) may be extended and interpreted as follows: If 9 E Coo(G, JR.) , the map *
Tg : Coo(G, JR.) '3 f H f 9 E Coo (G, JR.) is defined on the I li pdense subset Coo ( G, JR.) of
£P(G, p,), 1 � p <
00.
According as Coo (G, JR.) is viewed as a subspace of the Banach space Co(G, JR.) resp. LP(G, p,), 1 � p < 00, Tg may be normed:
II Tg (f) ll oo  M  I 9 II 1 �f  sup li T II �f p o I f lip Ilfll p,t 9
resp.
'0
00
II Tg (f) ll p I 9 II p' �f  Mp, p, . Ilfllp ,tO I f li p
,>up li T II p �f 9
4.9.5 Exercise. Under Tg , II II p Cauchy sequences map into II Cauchy sequences. [ 4.9.6 Note. By continuity, Tg may be defined uniquely on all Co(G, q resp. LP(G, p,) , 1 � p < 00. Furthermore, Tg is definable for 9 in L 1 (G, p,) resp. LP' (G, p,) . In all these elaborations, the numbers M.1 o resp. M.1 .1 remain unchanged. ] P
P P
The following extension of (4.9.4) is due to W. H. Young.
1 00 
Section 4.9. Appendix: Haar Measure . { 4.9.7 LEMMA. IF 1 :::; mm
191 clef
p, q} , P1 + q1  1 = 1 > 0, AND
(I, g ) E U (G, /l ) *
X
r
L q (G, /l ) ,
THEN Il f gil T :::; Il f ll p . Ilgll q · 1 1 [ 4.9.8 Remark. If = then  + 1 and, by abuse of no tation, = 00. Then the conclusion of 4.9.7 reduces to (4.9.4).]
P q=
q p' ,
r

=
. 1ent to t he · 1 + 1 1 clef 1 > 0 IS. 1ogica11y eqmva P ROOF. The cond ·Itlon assumption that there are positive numbers (3, such that
P q 
1
1 (3
"I = I, P 1
++o
1

1
r
0,
1 1 (3 '
q
0
1
0
"I
1
"I
,
0
Thus the factorization
=
r.
(4.9.9)
I f (y  l X) 1 ' lg (y) 1 = I f (y  l X) I ;; Ig (Y ) I � ' I f (y l x) I P ( �  ± ) . lg (y) l q ( i  ± )
(
)
(three factors ) and 3.7.16 when *
I f g (x) 1 :::;
[L
0
(
P
= I , (3 =
)
P2 , "I = P3 imply
I f (y  l X) I P .* Ig (y) l q d/l (y)
. [L
(
]
)
1
;;
I f (y  l x) I Pi3 ( � � ± ) d/l(y)
]
1
i3
(4.9.10)
Of the last two factors in the right member of (4.9. 10) , the former is inde pendent of x ( because /l is translationinvariant ) and is I f li p ; the latter is Ilgll q . The translation invariance of /l implies also that
Direct calculations using the relations (4.9. 10) lead to the result. [ 4.9. 11 Note. The derivation above is based on nothing but careful use of Holder's inequality and its extension. In 1 1.2.13 the same result is seen to be one of several consequences of the M. Riesz Convexity Theorem, 11.2.6, of fundamental importance
D
192
Chapter 4. More Measure Theory
in the study of Fourier series and Fourier transforms, the basic ingredients of harmonic analysis on locally compact abelian groups [Loo, We2] .]
�f
{O})
4.9.12 Exercise. a) The set C* (C \ is a locally compact abelian group when its topology is that inherited from the customary topology of C and multiplication (of complex numbers) is the group binary operation. 1 b) The map f.l : S), (C* ) 3 E H Jr 2 dx dy is a Haar measure for C* . E x + y2 [Hint: The discussion in 4.7.24 applies.] 4 .10. Miscellaneous Exercises )'
n
4.10.1 Exercise. (The Metric Density Lemma) If E E S ' then 1. ). [E n B (x, r )] 1m rtO ). [B(x, r )] ':;:=:'..,.,''
exists a.e. The limit is 1 for almost every x in E and is ° for almost every x in ffi.n \ E. [Hint: The results 4.6.11, 4.6. 144.6.17 apply when f X E .] 4.10.2 Exercise. If n = 1, 4.7.18 can be proved without recourse to Brouwer ' s Fixed Point Theorem.
�f
4.10.3 Exercise. If
{an } nEN C ffi., {dn} nEN C (0, 00), and if t < a if t 2: a '
00
�f L dnjan is in MON and Discont ( I ) {an } nEN ' n= l For in (0, 00) and a Lipschitz function (v. 3.2.31)
then f
=
a
L : ffi.2
3
{x, y}
H
L(x, y ) E (0, 00),
an f in ffi.lR is in Lip (L, O' ) iff If(x)  f(y) 1 :::; L(x, y) l x  y in . 4.10.4 Exercise. a) If L is a constant and f E Lip (L, 1 ), then f E AC. The converse is false. b) If f E C l ([O, 1] , ffi.) , for some constant Lipschitz function L, f E Lip (L, 1) . The converse is false. 4.10.5 Exercise. If {J, g } c AC, then { J ± g , fg } C AC. For some f in AC and some g in Coo (ffi., C) , fg tJ AC.
Section 4. 10. Miscellaneous Exercises
193
4.10.6 Exercise. a) If 1 E AC ([a, b] ) and 9 E C([a, b] , q , then
ib
b) If
9
dl
=
{
ib
9
.
f' d).. .
1 (x) �f O �f 00 < x :::; 0 , 1 1f O < x < 00 1, then 1 E BV and 9 E C(JR., q , and g ( O) =
1=
l
9
dl >
l
9
. f' d)..
=
O.
c) For the (continuous) Cantor function cPo :
4.10.7 Exercise. If 9 E C([a, b] , q and h E BV([a, b] ) , then: a) exists; b) the following formula for integration by parts is valid:
ib g dh
[Hint: a) Only the situation 9 E C([a, b] , JR.) , h E MON([a, b] ) need be addressed. b) The formula (Abel summation) n 1
L1 � ) [ X ) k= 1 ( k g ( k+I =
9
Xk
( )]
)  9 (x I ) 1 (6 ) 9 (x n ) 1 (� n 
I
n 1
L g X ) [I �k )  1 (�k I ) ] k=2 ( l,; (
applies to the Riemann sums approximating 4.10.8 Exercise. If 1 E C([a, b] , JR.) ,
F(t) then: a) F E BV([a, b] ) ; b)
=
9
ib 1 dg . ]
E BV([a, b] ) , and
it 1 dg , t E [a, b] , =
F(t)  F(t) I(t) [g (t)  g (t)], F( H )  F (t ) = l (t) [g (H)  g (t)];
Chapter 4 . More Measure Theory
194
c) F' � f · g' .
4.10.9 Exercise. If U E 0 (JR.n ) , for a sequence { Kn} nEN of compact subsets: a) Kn = U; b) Kn C K�+l c) if K (JR.n ) 3 K c U, for some N, nEN
U
Kc
N
U Kn .
n= l
[Hint: For each n and m in N, the union of the finitely many Rk,m (v. Section 4.6) contained in B(O, nt n U is a compact subset of U. If K is a compact subset of U, and d is the Euclidean metric for JR.n , then inf { d (x, u) x E K, u E U } > 0.] :
4.10.10 Exercise. If then flh) = 11"1' 11 2 dt.
11
"Ii E AC ([O, 1] ) , 1 � i � n, and "I
clef =
( "1 1 , . . .
, "In ) ,
4.10.11 Exercise. If G is a topological group and H is an open subgroup of G, then H is closed (whence, if H ¥ G, G is not connected) . [Hint: Each coset of H i s open.]
4.10.12 Exercise. The value of a M in (4 . 9.2) is independent of the choice
of f.
4.10.13 Exercise. If G is a locally compact group with Haar measure p, and for each neighborhood V of the identity of G: a) Uv is a nonnegative uv (x) dp,(x) = 1 , the net function in A(G); b) Uv = ° off V; and c)
i
V H Uv f converges [in A(G)] to f and ( uv ) a ( a ) as a function of a , converges uniformly to ( a , a ) . [ 4.10.14 Note. The Banach spaces L l ([O, 1], >.) and L l ([O, 1), >.) are essentially indistinguishable since >. ({I}) 0. The map *
=
is a continuous bijection between [0, 1) and 'lI'. The topology T f { E : E C 'lI', \11  1 (E) E O{ [0, I)} } is that inherited by 'lI' from JR.2 and with respect to T, 'lI' is a topological group. The measure spaces ([0, 1] ' 5,6([0, 1]), >.) and ('lI', 5,6 ('lI') , T) are isomor phic via the bijection \II . Thus L l ([O, 1] , >') and L l ('lI', T) are iso morphic in the category of Banach spaces and continuous homo morphisms.]
�
195
Section 4. 10. Miscellaneous Exercises
4.10.15 Exercise. With respect to convolution as multiplication, i.e., with respect to the binary operation
the Banach space L 1 (1I', T ) is a Banach algebra A(1I') . 4.10.16 Exercise. If G is a locally compact group, f..l is Haar measure, 2: 1, E > 0, and U(G, f..l ) , in N(e) there is a V such that
IE
p
is the set of Fejer's kernels (
4.10.17 Exercise. If {FN}
v.
3.7.6) and
I E U([O, 1 ] , ), ) , then FN INEN exists for each N and I I O. Nlim t oo II FN  l p [Hint: The result is true if I E C( [O , 1] , q .] n 4.10.18 Exercise. If {J, g } c (ffi,m ) lR and both I ' and g' exist at some n a in ffi,n , then: a) h �f (f, g) E ffi,lR ; b) h ' exists at a; c) *
h ' (a)
=
*
(f, g) ' (a)
=
=
(f' (a), g) + (f, g' (a) ) .
[Hint: (f(a + x) , g(a + x) )  (f(a) , g(a) ) ( f (a + x) , g(a + x) )  (f(a + x), g(a)) + (fa + x) , g(a) )  (f(a) , g(a)) .] =
4.10.19 Exercise. If E c ffi, and )'(E) = 0, then ffi, \ E is dense in ffi.. 4.10.20 Exercise. For 1 m2 '3 (X1 , X2 : m.
) {
how do the iterated integrals
H
(x i  x� ) ( x i + x� ) O
if X21 + x 22 > 0 otherwise
Chapter 4. More Measure Theory
196
compare? 4.10.21 Exercise. For the measure space (X, 5, f..l ) that is the Fubinate of ( X l , 5 1 , f..l d and ( X2 5 2 , f..l2 ) , 5 contains the aring 5 consisting of all empty, gnite, or countable unions of sets of the form E l X E2 , Ei E 5 i , i = 1 , 2. Is 5 necessarily 5? [Hint: The case (Xi , 5 i , f..li ) = ( [0, 1] , SA, >') , i = 1 , 2, is relevant.] '
4.10.22 Exercise. If
X I  X 2 E Q } and {A l , A 2 } C (5A2 n E) ,
E = ffi? \ { (X l , X 2 )
then >. ( A I X A 2 ) = 0. 4.10.23 Exercise. Is there a signed measure space ([0, 1] , 5, f..l ) such that f..l =t= 0, f..l « >., and for all a in [0, 1] , f..l ( [0, aD = O? 4.10.24 Exercise. If f E MaN, then f* : JR. '3 x
H
limx f(y ) yt
�f f(x)
exists and is in MaN and f* is leftcontinuous, i .e., f* (x) = lim f* ( y ) ; fur ytx thermore, f* � f. Similar results obtain for * f (x)
�f lim f(y ) �f f(x + ) . ytx
4.10.25 Exercise. If f E BV, then: a) U* , f * } c BV; b) there is a countable set 5 such that off 5, f* = f * ; c) for the jump function j(x)
�f L I f* (y)  f* (y) 1 y <x
associated with f, f  j E C(JR., q . [Hint: If f E MaN and f(x+)  f(x)
>
0, then
Q n (f(x ) , J(x+ ) ) :;to 0.]
4.10.26 Exercise. If 5
then f : JR. '3 x H
L
an '::; x
00
�f {an }N EN C JR., {jn }nEN C C, and L Un l < 00,
Un l i s in f E MaN and
* I f* (x)  f (x) 1
= { I jn l if x E � . otherwIse °
n= l
197
Section 4. 10. Miscellaneous Exercises
4.10.27 Exercise. If (JR., S)" p,) is a complex measure space and for x in JR., f(x) p,[(oo, x)] , then: a) f E BV; x lim f(x) = 0; c) 
�f
t oo
f(x)
=
f(Y) ] . [�f lim ytx
f(x)
d) The function f is continuous at x iff p,({x}) = O. (Properties a)c) characterize functions of normalized bounded variation. The set of all such functions is NBV) .
4.10.28 Exercise. a) If f E JR.lR n NBV, then : Co(JR., JR.) '3 9 ft is a DLS functional. The associated measure p, is totally finite and
I
Tf (x)
clef p, [(oo, x) ] .
l
9
df
=
b) If f E ClR n NBV, there is a corresponding complex measure p, and
Tf (x)
�f 1p,1 [(oo, x)].
4.10.29 Exercise. If f E BV( [a, b] ) , then varf is continuous at c in (a, b) iff f is continuous at c. 4.10.30 Exercise. a) If 9 E BV([O, 1] ) , for some a and b, 
g ([O, 1]) C [a, b] . b) If, for [a, b] as in a), f' E C([a, b] , JR.) , then f o g E BV([O, 1] ). *
*
*
Littlewood's Three Principles
In closing this discussion of real analysis there is an opportunity to mention some general guidelines [Lit ] that lie at the root of many of the arguments and ideas that have been presented. Real analysis began with Newton in 1665. In his time, a function was usually given by a formula and most formulre represented functions that were (at worst) piecewise differentiable. As real analysis grew and developed over the succeeding 300 years, there appeared functions defined by expressions of the form
f(x)
{A i f xES = � �;
B if x E T
xEU'
Chapter 4. More Measure Theory
198
The study of trigonometric series gave rise to highly discontinuous functions and led Cantor to discuss the sets of convergence and sets of divergence of the representing series. He turned his attention to set the ory itself and started an investigation that climaxed in 1963 with P. J. Cohen ' s resolution of the Continuum Hypothesis [Coh] . Other significant outgrowths of Cantor's work were general topology, Lebesgue's theory of measure, DLS functionals, abstract measure theory, probability theory, er godic theory, etc. The subject of functional analysis arose in an attempt to unify the methods of ordinary and partial differential equations and integral equa tions. The techniques were approximations that permitted modeling the equations by systems of finitely many linear equations in finitely many un knowns. In passing from the solutions of the approximating systems to what were intended to be solutions of the original equations, limiting processes were employed. At this point there appeared the need to conclude that the functions found in the limit were within the region of acceptable solutions. Therein lies the virtue of the completeness of the function spaces LP and the condition that a Banach space be normcomplete. (H. Weyl remarked that the completeness of L 2 is equivalent to the FischerRiesz Theorem. More generally, the normcompleteness of L 1 (hence of LP ) is essentially equivalent to the three basic theoremsLebesgue's Monotone Convergence Theorem, Fatou's Lemma, and the Dominated Convergence Theorem of integration.) Modern methods of TVSs relax the completeness con dition somewhattopological completeness replaces normcompleteness but some effort is made to insure that limiting processes do not lead out of the spaces in which the solutions are sought. Topology gave rather general expression to the notions of nearness (neighborhoods) and continuity. Measure theory elaborated the notion of area since integration was for long viewed as the process of finding the area of a subset { (x, y) 0 :::; y :::; f(x) } of ffi? At the start, f was a continuous function and area was approximated by the areas of enclosed and enclosing rectilinear figures. The idea for this goes back first to Riemann, then Euler and Newton and ultimately Archimedes, who, using exhaustion, determined the areas and volumes of some nonrectilinear figures. With the advent of Lebesgue's view of integration, rectangle took on a new meaning, in the first instance, the Cartesian product E X [a, b] of a measurable subset E of ffi. and it closed interval [a, b] . Nevertheless, behind all the generalities lay the intuitive notion of the graph of a wellbehaved function. When topology and measure theory were alloyed, the evidence became clear that many of the results were derivable by appeal to approximation of the general situations by others where the comfort of continuity and simplicity were available. :
Paraphrased, Littlewood's Three Principles read as follows.
Section 4. 10. Miscellaneous Exercises
199
1. Every measurable subset of ffi. is nearly the union of finitely many intervals. 2. Every measurable function is nearly continuous. 3. Every convergent sequence of measurable functions is nearly uniformly convergent. In the context of locally compact topological spaces and Borel measures the statement corresponding to 1. is: 1'. Every Borel set is nearly the finite union of compact sets.
4.10.31 Exercise. In what contexts is l' valid? 4.10.32 Exercise. Which of the results in Chapters 1 4 exemplify Littlewood's Three Principles?
C O M PLEX ANALYSIS
5
Locally Holomorphic Functions
5 . 1 . Intro duction
In ClR the sets C k (JR., C), k = 0, 1, . . " satisfy the relations
Indeed, if fo
0,00) and for k in N,
= X[
/k (x) �f
{ oLXoo
/k l (Y) dy if x 2': ° if x < °
then /k E C k (JR., JR.) \ C k+1 (JR., JR.) . Furthermore, for no x in JR. \ { a} is /k(x) representable by a power series of the form
00 cn (x  a) n (�f P(a, x). L n =O
On the other hand, as the developments in this Chapter show, if
f E (C'c , r > 0, and
z
z �f I' ( z )
f ( + h)  f ( ) lim h > o h>'O
h
open
o
z in D(O, rt �f { z : z00E c, I z l < r }, the disc f dius r, then there is a power series L an zn �f P(a, z ) such that for all 00 z in D(O, rt, f ( z ) = L cn zn , i.e., the series in the right member conn=O verges if I z l < r and the sum is f ( z ) . Moreover, by abuse of notation, oo exists for all
fEC
ra 
(D(O, rt, q .
The striking contrast between the situations described in the two para graphs above is one of many motivations for the study of CIC . The principal 203
204
Chapter 5. Locally Holomorphic Functions
tool used in the investigation is complex integrationa special form of inte gration, for which Chapters 2 and 4 provide a helpful basis. The principal results from those Chapters are the THEOREMs about functions "( in BV
11
and associated RiemannStieltjes integrals of the form J d"(. In Section 1 . 2, fl represents a special ordinal number. Henceforth, unless the contrary is stated, the symbol fl, with or without affixes, is reserved for a region, i.e., a nonempty connected open subset of C regarded ffi.2 endowed with the Euclidean metric: as
d : ffi.2 3 [(a, b) , (c, d)] H J(a  c) 2 + (b  d) 2 E [0, (0).
Thus an element of C is regarded either as a pair (x, y) of real numbers or as a complex number z �f x + iy. When S e C, J E CS , and (x, y) E S, the notations J(x, y) and J(z) refer to the same complex number. Since polar coordinates r, fJ are frequently useful, when z = x + iy = re iO = a + re it , the notations J(r, fJ) and J (a + re it ) are also used for J(x, y) and J(z). For a complex number z �f a + ib, the complex conjugate of z is a  ib and is denoted z . This use of  conflicts with  used for closure in topology. Henceforth, for a subset S of C, S denotes the set of complex conjugates of the elements of S, whereas SC denotes the closure of S re garded as a subset of C endowed with its Euclidean topology. The term curve is reserved for a continuous map
"( : [0, 1] 3 t H "((t) E C;
the image of "( is "(* �f "(([0, 1]). When ,,( E BV([O, 1] ) , i.e., when "( is rectifi able, the integral t Jb(t)] d� (t) is sometimes' written J dz, or J dz, or
1
io
"I
1 1 J (a + re27l"it ) 27rire 27l"it dt.
5.1.1 ExerCise. a) For n E Z+) J(z)
b) If J(z) = !, then
z
1
"I
=
n
z , and "((t)
J dz = 27ri.
=
e 21rit :
1
"I '
1 J dz
=
0.
Complex integration is used below to show that a function J differentiable in a region fl is locally representable by a power series: such that for some If a E fl, then C contains a sequence { } positive r(a) and all z for which I z  a l < r(a),
cn nEN
J(z) =
00
L cn (z  a) n . n=O
Section 5.2. Power Series
205
Thus the study of differentiable functions in CIC is reduced, in part, to the study, pursued with particular vigor and success by Weierstrafi, of power series convergent in some nonempty (open) discs of the form:
D(a, r �f { z : z E C, I z  a l < } . r
r
Later developments (v. Chapter 10) are concerned with methods of ex tending when possible, such functions to domains properly containing the disc within which the representing series converges.
(analytically continuing),
5 . 2 . Power Series
5.2. 1 THEOREM. (CauchyHadamard) THE ( FORMAL ) POWER SERIES 00
Cn Zn � n=O
5 cle=f ,",
1 cle z = 0 OR I Z I < ==. 1 . =f Rs; 5 FAILS TO CONVERGE lim I cnl :;;: n ( 5 diverges) IF I z l > Rs.
CONVERGES IF
..... oo
1 1 [ 5.2.2 Note. The conventions = 00 and = 0 are observed. 00 o Hence the of 5 is in [0, 00] . For example, if = n!, n E N, = 00. It is occasionally = 0; if = n. convenient to denote by to emphasize the dependence of of the radius of convergence upon the sequence �f coefficients in 5.] 

radius of convergence Rs Rs Cn \' Rs Rs Rc
Cn
c
{cn } nEZ+
Rs :::; 00 and I z l < Rs, for some in (0, 1 ) , I z l nlim I Cnin < whence, for some s in ( 1 ) , all large N , and all k in N, r
PROOF. If 0 <
_
r,
..... oo
1
r,
M
Cn zn , M E n , are a C auchy � n=O
· 1 sums 5M cle=f ,", · 0 < < 1 , the part la S mce sequence.
s
1').T
206
Chapter 5. Locally Holomorphic Functions
If Rs < 00 and
I zl
=
Rs
+ r5 > Rs , for some { nk h E N' nk t 00 and
(v. 1.7.32) .
o
5.2.3 Exercise. a) If Rs > 0 and S (k ) is the power series derived from S by kfold termbyterm differentiation, then RS ( k ) = Rs . b) For each S (k ) , convergence is uniform on every compact subset of D (0, Rs t . c) If
rectifiable curve, "1* C D (0, Rs t , and f =
00
L cn zn , then n=O
"I is a
[Hint:
The CauchyHadamard formula and induction apply for a) and b); b) implies c) .]
i zn dz
"I
5.2.4 Exercise. If is a rectifiable then 0
closed curve ["((0)
=
"1(1)], and n E N,
=
5.2.5 THEOREM. SOME n IN Z,
IF "I IS A RECTIFIABLE CLOSED CURVE AND C tJ. "1* , FOR 1 dz n. 1 (5.2.6) Ind ,), (c) . 1 27rl z C clef = 
c
')'
=

PROOF. Since tJ. "1* , the integral in the right member of (5.2.6) is well defined. Furthermore, since = 1 iff (by abuse of notation) E 27ri . Z (v. Section 2.3) , the result is valid iff for some n in N, exp [Ind ,), (c)] = exp(n) . If E E (0, 1), there is a linear h;' is a polygon) such that
ea piecewise
a
"IP
"I�
(v. 4.8.5 and 4.8.6) . Thus, since is piecewise continuous (indeed, piecewise constant, whence piecewise absolutely continuous) ,
O(s)
=
Of
[18 "IP t1)  d"lp (t)] exp [1 8 "IP t� "I� (t) dt]
clef =
"IP
(
exp
o
(
(s) � O( s) "Ip"I(s� (s) )c o
C

C
Section 5.2. Power Series
207 ==
)
' 0, i.e., (_0_ == o. How "IP  C o 0 is constant on ever, E AC ([t k ' t k + d ) (v. 4.8.5) , whence "IP  c "IP  c each interval [tk' tk + d . Thus, since 0 is continuous on [0, 1] ' for some "IP  c o constant K, = K. "IP  c Consequently, Hence ( yp(s)  c)O'(s)  (yp (s)  c) ' O(s)


1 "Ip(O)  c

__
0 (0) = K = 0 (1) "Ip(O)  c "Ip(l)  c
0 (1) = KO(I), "IP ( 0 )  C
i.e., 0 (1) = 1 { = exp [27riInd ,),p (c)] } and so for some mp in Z,
For some sequence ("(Pk
hEI\P each like "IP , klim + oo Ihpk "11 1 00
=
0 and
Hence, for some n in Z, Ind ')' (c) = n. 5.2.7 Exercise. The last sentence above is valid.
[Hint:
For all appropriate P, the integrand in the first term of the right member above is small and f! ("IP ) is bounded. The absolute value of difference of the approximating RiemannStieltjes sums for the second term is small.] [ 5.2.8 Remark. The discussion above suggests that for many purposes the hypothesis that "I E BV([O, 1] ) may be replaced by either of the (weaker) hypotheses a) "I is piecewise linear; b) off a finite set, "I' exists and "I' is piecewise continuous on [0, 1] .
o
208
Chapter 5. Locally Holomorphic Functions
Furthermore, each
11
11 f
0
'Y(t) d'Y(t) , i.e., each
f f dz, may be re
placed by f 0 'Y(t) . 'Y' (t) dt. The last integral lends itself to the methods of elementary calculus and avoids the use of Riemann Stieltjes integration.]
[ 5.2.9 Note. If l' is a rectifiable closed curve, the function ind 'Y in 2.4.20 and Ind are the same, v. 5.4.28 and 5.4.37. The common value of ind 'Y (c) and Ind 'Y (c) is the win ding number of l' with respect to c, v. 5. 2.10. If l' is not rectifiable, Ind 'Y is not defined.] I'
5.2.10 Exercise. If 'Y(t)
=
e 2mrit , n E Z, then ind 'Y (O) = Ind 'Y (O) = n.
5.2.11 Exercise. a) If {'Yk } �= 1 is a finite set of curves, there is one and K
only one unbounded component C of C \
U 'YZ . b) If c E C,
k= 1
K
2:)nd 'Yk (c) = o.
k=1
K
c) On each of the components of C \ 2.4.20c) .
K
U 'YZ , Ind 'Yk (c) k= 1 k= 1
L
IS constant, v.
K
[Hint: a) Since
U 'YZ is compact, for some positive r,
k= 1
K
U 'YZ ¥D (O, r )
k= 1
K
and C \ D(O, r) is connected. b)
( Ql 'YZ ) . c) 1 dz hm 1 . 'Yk z
L Ind 'Yk (c) is constant in each
k= 1
component of C \
I c l too
C
= 0, 1 � k � K.]
Section 5.3. Basic Holomorphy
209
The developments above set the stage for a general discussion of (lo cally) holomorphic Junctions, i.e. , functions that behave like many of those
appearing above. They all enjoy the local propertydifferentiability which reveals itself as a crucial characterization of local holomorphy. 5 . 3 . Basic Holomorphy
5.3.1 DEFINITION. WHEN U E O(C) , AN J IN CU IS holomorphic IN U (f E H (U) ) IFF I'(z) EXISTS FOR EACH z IN U. MORE GENERALLY, WHEN S e C, THE NOTATION H (S) SIGNIFIES THE SET OF FUNCTIONS HOLOMORPHIC IN SOME OPEN SET CONTAINING S:
{ J E H (S) } {} {3 U {{ S C U E O (C) } 1\ { J E H (U) }}} . The next two results show that despite the restrictiveness of the condi tions defining holomorphy, if E C and R > 0, H [D(a, Rr] is substantial.
a
5.3.2 Exercise. a) If the radius of convergence of
in D(O, Rr, J(z)
7ri
00
�f L
00
L cn zn is R and, for z
n=O n Cn z , then J E H [D(O, Rr]. b) If 0 < r < R and
n=O 2 t l'(t) = re , 0 ::; t ::; 1 , then
i J(z) dz
=
O.
[Hint: a) Induction yields the identity w n  zn n  l = (w  z) (w n  2 + 2zw n  3 + · · · + (n  1)z n  2 ) . w  z  nz Thus when max { I z l , I z + h i } ::; r < R and I h l is positive,

I
J(Z + h)  J(Z) h = Ihl
::;
l�

� ncn z n  1 Lt
n= l
1
Cn [ (z + h t  2 + 2z(z + ht  3 + . . . + (n
� f n(n  1 ) l en I rn  2 .
I I
 1)zn2] I
n= 2
Then 5.2.3 applies. b) 5.2.4 applies.]
5.3.3 Exercise. If l' is a rectifiable closed curve, g E C ( 1' * , C) , and, for w) z not in 1' * , J (z) �f g ( dw, each component C of C \ 1' * is a region
and J E H (C).
1
y w  z
Chapter 5. Locally Holomorphic Functions
210
5.3.4 THEOREM. IF 1 E H (Q) ,
I(z) �f I(x + iy) �f u(x , y) + iv(x, y) [�f �(f) + i CS (f)], AND z E Q, THEN THE CauchyRiemann equations
ux (x, y) = V y (x, y) and uy (x, y) = vx (x, y) OBTAIN. PROOF. If z = x + iy, then !' ( z) = lim h�O h ";O
1 ( z + h)
h

1 (z)
= Ux + ivx I(z) = lim I(z + ih) ih =

h�O h ";O

o
iuy + v y .
[ 5.3.5 Remark. In terms of the operators
(�
a �f � �f � az 2 ax
_
)
i� ' ay
a �f
� �f � az
(�
+ i� 2 ax ay
a the CauchyRiemann equations reduce to a1 = azl
=
)
o. m
In this book the same symbol a appears in different contexts: au analysis,  , the partial derivative of u with respect to x; in topolax ogy, a(U) , the (topological) boundary of U ; in complex analysis, al as introduced above. Nevertheless, the intended meaning of a whenever it occurs, is clear.
a al . . 1 denvatIve · . · · fy t he partIa The" notatlOn resp. l does not slgm az az of 1 with respect z resp. z. The alternative notation a1 resp. a1 is less likely to be misinterpreted.] =
5.3.6 Exercise. If: a) Q is a region in C; b) u and v are in ffi.r1 ; c) the derivative of the map T Q 3 (x, y) r+ [u(x, y), v (x, y)] E ffi.2 exists; d) in Q, the CauchyRiemann equations obtain, i.e., Ux = Vy and uy = Vx , then for some 1 in H(Q), I(x + iy) = u(x, y) + iv(x, y). (If partial differentiability of u and v only off a countable set and the CauchyRiemann equations only :

Section 5.3. Basic Holomorphy
211
a.e. in Q are assumed, a result of Looman and Menchoff implies nevertheless that J E H(Q) , v. [Sak] , (5.11), p.199.) The next development leads to the connection between differentiability, i.e., holomorphy, of a function J in en and, for each D(a, rt contained in Q, the representability of J (z) as a power series P( a, z) ( converging at each point z of D(a, rt). The fundamental tools are Cauchy 's Integral Theorems and Cauchy 's Jormul(£;.
5.3.7 Exercise. a) If {A, B, C } c e and a(ABC) is the 2simplex deter mined by A, B, and C,
1
8[a(ABC)] b ) If
/,(t) =
=
[A, B] U [B, C] U [C, A] . 1
. 0
then /' E BV([O, 1]) and /'*
=
1 13
8[a(ABC)] . c ) If J E C b * , C) ,
1 J dz d f l[A,B] + 1[B,C] + 1[C,A] J dz 'Y
�
(5.3.8)
r �f Ja[a(ABC)] J dz =
1
2
0
13 J /'(t) (3B  3A) dt + 13 J + 1 1 J /'(t) (3A  3C) dt. 3
3
0
5.3.10 Exercise. If n E Z+ and J(z)
r
Ja[a(ABC)]
0
/,(t)(3C  3B)
�f z" ,
J dz
=
o.
[Hint: Since /" exists and is continuous on the intervals
the three integrals in (5.3.8), (5.3.9) can be calculated directly. ]
(5.3.9)
Chapter 5. Locally Holomorphic Functions
212
5.3.11 THEOREM. (Cauchy's Theorem, basic version) IF
AND
f E C(Q, q
[a(ABCW C Q n H (Q \
{P}), FOR "I AS IN 5.3.7,
if dz
=
o.
[ 5.3.12 Remark. The point P is not specified. The hypothesis
f
f E H (Q \ {P})
(weaker than E H (Q)) adds strength to the conclusion at the cost of complicating the proof. However, in the treatment of the elab orations of Cauchy's Theorem and in the treatment of Cauchy's formula, the strong(er) conclusion plays an important role.] PROOF. For simplicity, the heart of the argument is carried out under the additional hypothesis that P tJ. [a(ABCW. In that circumstance, if dz �f K i 0, then as in Figure 5.3. 1 , the barycenters (31 , (32 , (33 of
Jf 'Y
[A, BJ , [B, CJ , [C, A] engender four simplices
Direct calculation reveals
c
A
Figure 5.3.1.
Section 5.3. Basic Holomorphy
213
and thus the absolute value of at least one of the summands in the right For the corresponding simplex, the pro member above is at least cedure just applied may be repeated and induction produces a sequence a( ABC), a l , . . . , an , of subsimplices such that
I�I .
...
and
Ir
a(ABC) ::) a l ::) . . . ::) an ::)
�(an l
I I�I .
1 dZ 2':
4
...
Since
8[a(ABC)] } · ( an ) < diam [a(ABC)] and fl [8 ( an )] _ dlam , Q n n 1:
2
2
n (an t = Q. Since P tJ. [a(ABCW, P i Q nEN and f'(Q) exists. Hence, if Q i z E ( an t, for some Q in [a(ABCW,
I(z)  I(Q)  f'(Q)(z  Q) �f 8(z), zQ and E > 0, then for some no , 8(z) < E if n > no . Hence, by virtue of 5.3.10, if n > no ,
I4�I
�
Ir
Ja(an l
l Ir Ir
1 dz = =
11(1
Ja(an l
[J(Q) + f'(Q)(z  Q) + 8(z)(z  Q)] dz
l
l
8(z) ( z  Q) dz Ja(a n l � diam [ BC)] . f! E [8 (an )] f! {8 [a(ABC)] } < E · diam [a(ABC)] n 2 2n � E · diam [a(ABC)] . f! {8[a(ABC)] } .
��A
. ����
1(
Hence = o. More generally, when P E [a(ABCW, there remain the following pos sibilities: a) P E {A, B, C}; b) P E 8[a(ABC)] \ {A, B, C}; c) P E a(ABC). For a), if, e.g., P = A, on (A, B] there is a sequence {An } nEN such that lim n An = P ( = A) . The original argument applies for a (An BC) :
+=
r
Ja[a(A n BC l ]
lim r 1 dz 1 dz = 0, n+ = Ja[a(A n BC l ]
=
r
Ja[a(ABC l ]
1 dz.
Chapter 5. Locally Holomorphic Functions
214
For b ) , if, e.g., P E (A, B), the argument for a ) applies to cr(APC) and cr( PBC) . Furthermore,
+ r r 1 dz 0 + o. 1 dz = r Ja[a( APC)] Ja[a(PBC)] Ja[a(ABC)] For c) the argument for a) applies to cr(ABP) , cr(PBC), and cr(PCA), + r + r while r 1 dz r 1 dz. 0 Ja[a(ABC)] Ja[a(ABP)] Ja[a(PBC)] Ja[a(PCA)] 5.3.13 Exercise. The PROOF above can be conducted via barycentric subdivisions of cr( ABC). is a closed polygon, there are finitely many 5.3.14 Exercise. a) If K K C simplices {crk h� k�K such that (crk / c Q and 8 (Sk) ' b ) If k= 1 k= 1 I(z) dz O. 1 E H ( Q ) , then =
=
'Yp
1
'YP U
U
=
"iP
5.3.15 THEOREM. IF Q IS convex AND 1 E C(Q, q n H ( Q \ {P}), FOR SOME 1 IN H ( Q ) , 1 ==
(.1) ' .
PROOF. If {Q, z} c Q, the convexity of Q implies [Q, z] c Q. The formula 1(z) �f r I(w) dw uniquely defines T Since Q is convex, if R E Q, then J[Q,z] [cr(QzRW C Q and 5.3.11 implies 1(R) = r + r I(w) dw. Thus, J[Q,z] J[z,R]
 
1
[J(w)  I(z)] dw I(R)  I(z) [z,R] If E > 0 and R is if R i z, I(z) Rz Rz near but different from z, owing to the continuity of I, =
Ir
J[Z, R]
l
[J(w)  I(z)] dw � E I R  z I .
J
'Y
o
5.3.16 COROLLARY. IF Q IS convex, IS A RECTIFIABLE CLOSED CURVE, AND C Q, FOR 1 AS IN 5.3.15, 1 dz = O.
'Y*
'Y
"i
PROOF. If is piecewise linear, since
(.1)' = I, FTC implies
i 1 dz = 1 1 (.1) ' h(t)] 'Y'(t) dt =
1 1 [10 'Y(t) ] , dt = 1 0 'Y I � .
215
Section 5.3. Basic Holomorphy
Since ,,/ is a closed curve J 0 "/ I � = O. If "/ is a rectifiable closed curve, as in the last part of the PROOF of 5.2.5, there is a sequence {"/Pn } nEN such that lim 1 dz = 1 dz. n+= 'Y n Each integral in the left member above is zero. 0 5.3.17 Exercise. If Q is convex, {P, Q } c Q, and "/1 , "/2 are two rectifiable curves such that:
J
a) b)
J
"I
P
"/� U "/� C Q, "/1 ( 0) = "/2 (0) = P, "/1 (1) = "/2 (1) = Q,
J"11 1 dz = J"12 1 dz: the integral is independent of the path. If, in 5.3.15, the hypothesis that Q is convex is dropped, J 1 dz can then for any 1 in H (Q),
"I
be different from o.
5.3.18 Example. If Q cle=f C \ { 0 } and 1 ( z) =  , then 1 E H (Q), and Q IS. Z not convex. If "/ ( t) = e 27rit , 0 :::; t :::; 1, then "/ * C Q and
1
On the other hand, since Ind 'Y (O) = 27ri, the presence of 27ri in the right member above suggests the possibility of a formula that relates Ind (0) and some integral involving I. In the derivation of the formula, the importance of the hypothesis 1 E C(Q, q n H (Q \ P } ) in 5.3.11 becomes clear. "I
{
5.3.19 THEOREM. (Cauchy's integral formula, basic version) IF Q IS A CONVEX REGION, "/ IS A RECTIFIABLE CLOSED CURVE SUCH THAT "/ * C Q AND 1 E H(Q) , FOR EACH a IN Q \ ,,/* ,
1
I(a) · Ind 'Y (a) = . 27rl
J I(z)a dz. "I

Z

(5.3.20)
[ 5.3.21 Note. If Ind 'Y (a) = 1 and g(z) �f I(z)(z  a) , (5.3.20) yields I(z) dz = g(a) = 0, i.e., a significant generalization of 5.3.11. The larger message of (5.3.20) is that the value of 1
i
at any point a in Q is a complex weighted average of the values of 1 on "/* . Mere continuity of 1 is insufficient for such a conclusion since, absent the assumption of the differentiabilty of I, 1.2.41 implies that for some continuous I, f(a)=l while 1 1 "1 ' = 0.]
216
Chapter 5. Locally Holomorphic Functions
PROOF. Since J(a)Ind ,),(a) = ity of
dz = o. 1')' J(z)z  J(a) a
� 27rl
1')' zJ(a) a dz it suffices to prove the valid
To this end 5.3.16 is applicable because, by definition,
{
= Fa : (""\ 3 z clef H
J(z)  J(a) za f' (a)
if z ¥= a if z = a
(5.3.22)
is in C(Q, C) n H (Q \ {a}) and thus 5.3.16 implies (5.3.20).
o
5.3.23 COROLLARY. IF a E Q, D(a, r ) c Q, AND J E H (Q), FOR SOME SEQ U ENC E { cn (a) } nEZ + IN C AND ALL z IN D(a, r t ,
J(z) =
00
L Cn (a)(z  a) n .
( 5.3.24)
n=O
PROOF. For "( : [0 , 1 ] 3 t ft a + re 27rit , Cauchy's formula and 5.2.11 imply that if z E D(a, r t , then
(1 Fz (w) dw 1 J(z) d ) = �27rl (0 1')' J(z) d ) J (w) = _1 dw . 27ri ')' (w _ a) ( _ z  a )
J(z) =
Since
� 27rl 1
I I za wa
+
')'
1
< 1
')' W
wa
for all w on "(* ,
Z
+
W
1
f (� � : )
= 1 _ z  a n=O wa
counting measure, Fubini's Theorem applied to
W
n
Z
W
v
. When is
justifies the subsequent interchange of integration and summation:
Thus cn (a) =
� 27rl
1(
j +1
J (w
dw . an 5.3.25 Exercise. If ( X , S, f.l ) is a complex measure space, ')'
W
g E 5, U E O(C), and g(X) n U = 0,
o
Section 5.3. Basic Holomorphy
then
f
:
U3
z
r+
z
217
rx g�xp,)(x) z is in H (U).
i
a
[Hint: If E U, for some in U and some positive
z E D(a,rt c U. z  a :::; I z  al < 1 and For each x in X , I r g (x)  z I
r,
(z  a) nn . g(x)  a (g(x)  a) ( 1  z  a ) n=O (g(x)  a) + l g(x)  a 1
1
Again, when
v
(Xl
=L
is counting measure, Fubini ' s Theorem applies to
. J g (dp,)(x) = �� Jx (g(x(z)  a)a)nn+ 1 '] n=O m
and valIdates
X
X
Z
f( ) exists and f(rn) (z) = n=L n(n  1) · · · (n  m + l)cn (a)(z  a) n rn rn (whence f ( rn ) E H(Q) ) . In particular, 1 f(w)n + dw. (5.3.27) Cn (a ) = f(nn!) (a) = 27ri J'Y (w  a) l Furthermore the radii of convergence of the series representing f and f ( rn ) are the same. 5.3.28 Exercise. If M(a, r) �f max I f (w)1 �f max I f (a + re 2 7l"it ) I , O�t � l I w  a l =r 1 then M(a, r) 2 z 1 I f (a + re27l"it ) 1 2 dt and 5.3.26 Exercise. In the context of 5.3.23, if m E N, 00
00
2 r 2n . c i (a)1 n n=O [Hint: Fubini's Theorem applies; {e 27l" nit } nEZ is ON on [0, 1] .] =L
Chapter 5. Locally Holomorphic Functions
218
The inequality
00
L I cn (a)1 2 r2n � M(a, r ) 2
n=O
Gutzmer's coefficient estimate: which implies the (weaker) Cauchy estin! M (a, r) mate.. I f (a ) 1  r A function in H (C) is an entire function. The set of entire functions is
is
<
( ) n
denoted
.
n
E.
5.3.29 Exercise. (Liouville) If f E E and sup � K < 00 , then for zEIC some constant == a bounded entire function is a constant function. Gutzmer ' s estimate applies for all positive Hence, if > 0, then = 0.]
I f(z)1
c, f c: [Hint: n cn (O)
r.
I f (�) 1 z1
5.3.30 Exercise. If E E , a < 1, and sup zEIC 1 + constant. Gutzmer's estimate applies.]
f
[Hint:
Q
<
00 ,
then
f IS a
n E N, p(z) L ak zk and an i 0, then for some �, k= p(�) = 0 [the Fundamental Theorem of Algebra (FTA)] . [Hint: If the assertion is false, g �f 7 E and Liouville's Theorem ==
5.3.31 Exercise. If
n
O
E
applies.]
5.3.32 COROLLARY. I F A IS A BANACH ALGEBRA, x E A, THEN sp(x) i 0 (v. 3.5.20 ) .
PROOF. If sp(x) = 0 , then for all
z, (x  ze)
calculation reveals that the function

1 exists, and if x' E A�, direct
f : C z x' [(x  ze)  1 ] E C is entire. As I zl + 00 , I f (z)1 + o. Liouville's Theorem implies f I I 3
r+
== 0: for every x', (X , x' ) = O. Since X i 0, the HahnBanach Theorem is con tradicted. 0 �f If 0 ix + for some unique (J in [0, 271") , = If 00
z I z l eio reiO. �f a n +ibn , {an , bn } C JR., 0 � R and z = Re iiJ,
z � iy,
f(z) �f L
n=O
n Cn z , Cn
<
He ,
Section 5.3. Basic Holomorphy
then
219
00
00
n=O
n=O
�f � [J(z)] + iCS[J(z)] �f u(x, y) + iv(x, y) �f U(R, 0) + iV(R, 0). In these notations, the following items reveal some of the power of the basic version of Cauchy's formula. 5.3.33 Exercise.
� (C ) = 217r 1 271" U(R,O) dO, CS(co) = 217r 1 271" V(R,O) dO, 1 1 271" n Cn = 7rRn U(R' 0) e  i O dO, n o. [Hint: If n E \ {O}, 1 271" einO dO = 0.] 
o
0

0
0
r .../..
;Z
5.3.34 LEMMA. IF
00
U(R, O) :::; M < AND n E N, THEN I cn l :::; 2 [M ��c (co)] .
PROOF. If n ::J 0,
Cn = 7rR1 n 1 271" [_ U(R,O)e inO] dO = 7rR1 n 1271" [M  U(R, O)einO] dO. Because M  U(R, 0) � 0, 5.3.33 implies l en I :::; 7r�n 1 271" [M  U(R,O)] dO 2[M  � (co)] . and as R t Rc the right member above approaches R� 5.3 .35 Exercise. If {In } nEN C H(Q) and for each compact subset K of Q, In I K� 1 1 K ' then I E H ( Q) and for each k in N, I�k ) I K � I ( k ) I K " [Hint: The formula in (5.3. 27) applies.] 0
0
0
The Maximum Modulus Theorem and the Open Mapping Theorem discussed below are shown to be logically equivalent. The first is proved by appeal to the following fundamental principle:
Chapter 5. Locally Holomorphic Functions
220
S
The average of a finite set of real numbers lies between max and min Thereupon the validity of both assertions is established.
{})
{S}.
(
) I
a
5.3.36 THEOREM. Maximum Modulus Theorem I F E H (Q) AND E Q, THEN 2': sup IFF IS A CONSTANT FUNCTION: THE ABSOLUTE zEn VALUE OF A NONCONSTANT FUNCTION, HOLOMORPHIC IN Q, CANNOT ACHIEVE A MAXIMUM VALUE IN Q. PROOF. If l'(t) �f 0 t 1, Cauchy 's formula and the averag
I I (a)1
I I (z)1 I
re27rit , :::; :::; a + ing principle cited above imply
II(a)1 :::; 1 1 I I (a + re27rit ) I dt :::; max I I ( a + re 27rit ) 1 = max I I (z)l. O�t �1 zE8[D(a,r)O] If max II(z)1 I I (a)l, the argument for the Gutzmer estimate zE8[D(a,r)o] yields L I cn ( a ) 1 2 r 2 n :::; I I (aW I (a)1 2 , whence cn ( a ) 0 , n E N, i.e., I n=O is a constant function. Hence, if I is not a constant function, at every interior point a of Q, I I(a)1 is not a local maximum value of I I I : II(a)1 < sup I I (z)1 zE n 5.3.37 Exercise. ( Minimum Modulus Theorem ) If I H (Q), 0 tJ. I(Q) and for some a in Q, II(a)1 2': inf { 1 1 (z)1 : z E Q }, then I is a constant. [Hint: The function g �f 7 is in H (Q).] 5.3.38 COROLLARY. I F Q IS BOUNDED, I E H (Q ) , AND FOR EVERY a IN inf sup I I (z)1 M, THEN sup II(z)1 :::; M. 8( Q ) , N (a)EN(a) zE n zEN(a)nn PROOF. Only if M < and I is not a constant is an argument required. For those circumstances, there is a sequence { Km} m EN of compact sub sets such that Km C K� +1 ' m E N, and each compact subset K in Q is contained in some Km , v. 4.10.9) . Moreover 5.3.36 implies that =
00
=
co
=
0
Eo
00
=
(
II(z)1 �f Mm is achieved at some point Zm on 8 (Km). For all m, Mm :::; Mm + 1 . Since Q is bounded, {Zm } m EN contains a convergent subsequence, again denoted { zm}mEN" If mlimtoo Zm �f Zoo, since Km C K� +1 ' m E N, Zoo E 8(Q) The max
zEK=
.
221
Section 5.3. Basic Holomorphy
E > 0 and m is large, Mrn < M + E. Hence, for z Q, If(z)1 :::; + E. 5.3.39 THEOREM. ( Open Mapping Theorem ) I F f E H(Q), f(Q) IS EITHER A REGION OR A SINGLE POINT , i.e., f(Q) IS A NONEMPTY REGION OR f IS A CONSTANT FUNCTION. PROOF. If !' =j=. 0, f' i 0 in some nonempty open subset V of Q. If a E V, then for some positive r, D( a, r t c V, and if 0 :::; fJ :::; 27r, then f (a + reiO)°0:f(a) = f (a) + Er' e· ':r& For small positive r, I ET, o l < � 1 !, (a)1 and I !, (a) + ET, o l > � 1 !, (a)1 �f 15. Hence, if r is small, min I f (a + reiO)  f(a)1 > r r5 �f 2d. The geome °SOS 2". try of the situation reveals that if w E D(f (a), dt, hypothesis implies that if any in M
0
I
If
g �f w  f and 0 tJ. g(D(a,rn, 5.3.37 implies d > l w  f(a)l 2': min { l w  f (a + reiIJ) 1 : 0 :::; fJ :::; 27r } > d, a con b D( a, rt, w = f(b), D(f(a) , dt c f (D(a, rn [ 5.3.40 Note. The PROOF above of the Open Mapping Theorem resorted to the Minimum Modulus Theorem, a consequence of the
i.e., tradiction. Hence, for some in
0
Maximum Modulus Theorem:
{ Maximum Modulus Theorem } ::::} { Open Mapping Theorem } . Conversely, the Open Mapping Theorem implies that a non constant function in maps onto a region: if E for some positive the open set is contained in and for some positive C If = each ion satisfies = O. If the halfline starting at and passing through goes on to meet in a point such that
f H(Q) Q a Q, r, f (D(a, rt) f(Q) s, D(f(a), s) f(D(a,rn. f(a) f(a)0, 0, f(b) 8[D (f(a), stJ 0 If(b)1 > I f (a)1 f ( a) 8[D (f(a), srJ f(b) I f (b)1 > I f (a)l: { Open Mapping Theorem } ::::} { Maximum Modulus Theorem } . J
Chapter 5. Locally Holomorphic Functions
222
J
a E Q, AND J  1 1 J[D(a,r) O]
5.3.41 THEOREM . (Inverse Function Theorem) IF E H (Q) , IS INJECTIVE AND i 0, FOR SOME POSITIVE r,
J'(a)
JI D(a,r) O
IS HOLOMORPHIC. PROOF. Since i 0, for all in some
J'(a)
z
D(a, rt, J'(z) i O. J(w)  J(z) if z i w G : Q x Q (z, w) r+ { I'(z) wz if z = w which is in C (Q2 \ { (z, w) : z = w }) is shown next to be in C ( Q x Q, q . Since I' E H (Q) , I' is continuous. If z is near but different from w , and l'(t) �f (1  t)z + tw, then 1'(0) = z, 1'(1) = w, 1" (t) = w  z, G(z, w)  G(z, z) = w 1 z 10 1 {Jh(t)]}' dt  G(z , z), 1 = 1 [I' h (t)]  I'(z)] dt, whence I G (z, w)  G(z, z)1 is small: G E C ( Q x Q, q . Thus, for some positive < r and if max{ l z  ai , I w  al } < then I G(z,w)1 � IJ'(a)1 > 0, 1 i.e. , if {P, Q } C D(a, ) then IJ(P)  J(Q)I 2 1 1'(a)I ' I P  Q I . Hence J I D(a.s) O is injective: for some g defined on J (D(a, t ) , g o J(z) z. If b E Q 1 and Q 1 w i b, for some P,Q in D(a, t , PQ g(w)  g( ? ) (5.3.42) wb J(P)  J(Q) " Since Q E D(a t , J'(Q) i o. As w + b, P + Q and the right member of 1 (5.3.42) converges to I' (Q) ' [ 5.3.43 Note. If 0 = a = J(a), w = J(z) = L an Zn , 0 :::; Izl < r, a 1 i 0, n=1 and when I w l is small, z g(w) = m=1 (5.3.44) L bm wm . The function
3
_
s,
s, s
�
s ,
�
3
,s
s
s
==
0
00
=
00
Section 5.3. Basic Holomorphy
223
Thus
Comparison of coefficients leads to a sequence
(recursively)
cal of formulre from which the sequence {bn } n EN is culable in terms of the sequence {an } n EN . (Similar but more com plicated formulre obtain when a and I(a) are more general.) The Cauchy estimates imply for M �f M(a, r),
If lzl < r, the series l a 1 I z 
oo
M n M 1 z = l a 1 I z  2 z 2 Lz reprer rn 1
n =2 r sents a function F, a of I, i.e., a function F for which the power series coefficients in absolute value the power series coefficients for I. The equation W = F(z) is quadratic in z, and if
majorization majorize
then G o F(z) = z. At and near 0,
whence, for some positive p depending only on 11'(0) 1 r and M, G' exists in D(O, p t . There is a sequence {cn } n EN such that for W in D(O, pt , G(W) = Cn Wn ( = z). The recursive formulre n=1 for the sequence {cn }n EN show Ibn l :::; Icn l , n E N. Thus the series (5.3.44) converges if Iwl < p: 00
L
1 (D(O, r n ::J D(O, pt · J 5.3.45 Example. If I(z) = then 1 E though 1 is it is not
E,
= I, and f' is never zero. Al locally injective eZ, globallyI'injective: for n in Z, e 2mri = 1.
224
0 lal, 0
Chapter 5. Locally Holomorphic Functions
5.3.46 Exercise. If < < for some L in H ( D (a , rr ) , eL ( z ) = z. [Hint: The function L in 2.4.21 serves.] The condition !, (a) i plays a central role in the PROOF of the major part of the Open Mapping Theorem, 5.3.39. There is a refinement that deals with the circumstance: 1 is not a constant function, but !, (a) = O. r
5.3.47 THEOREM. I F 1 E H(Q) , a IS IN Q, AND 1 IS NOT A CONSTANT FUNCTION, FOR SOME m IN N, SOME NEIGHBORHOOD N(a) , A g IN H [N(a)] ' AND ALL z IN N(a) I(z) = I(a) + [g(z)] Tn . FURTHERMORE, FOR SOME b AND A POSITIVE g[N(a)] = D ( b, r �f V, g I ( a ) ' IS INJE CTIVE, N AND FOR SOME h IN H(V) , h 0 g I ( a ) (z) == z. r,
r
N PROOF. For some m in N and some N(a), if z E N(a), then I(z) = I(a) + (z  a) Tn
[� a)n1 �f cn (z 
I(a) + (z  a) Tn k(z)
k' k' and k I ( a ) i O. Hence k E H [N(a)] ' and for some h, h' = k ' If z E N(a) , N then {k(z) . exp[h(z)] } ' = exp[h] [k'  kh'] = O. Hence, for some con'0 , k exp (h) = stant de=f on N(a). Since Tn h k= exp ,
M I M l e"
[I M I ,!;
M
(
:iO )]
0
if g (z ) �f (z  a)k(z) , then I(z) = I(a) + [g(z)]Tn. Moreover, g ( a ) = and g' ( a ) = k(a) i o. 0
0,
5.3.48 THEOREM. (Morera) IF U E O(q , a E U, 1 E C(U, q , AND FOR EACH 2SIMPLEX CONTAINED IN U \ {a}, r 1 dz = THEN 1 E H (U). a
Ja(a) [ 5.3.49 Note. The open set U need not be connected, e.g., the conclusion is valid if U �f D(O, lrl.,JD (3, lr �f l!Jl.,JD(3, l r .] r
0
r
PROOF. If > and w E D(b, r c U, the hypothesis implies (even when r 1 dz unambiguously defines an F in b = a) that the formula F( w) J [b , wl cD( b , r)o . Moreover, FTC implies F' exists throughout D(b, rr and F' = I. Hence F E H (D(b, n and thus F' (= f) E H (D(a, r ) 0 5.3.50 Example. The hypothesis 1 E C(U, q in 5.3.48 cannot be omit
�
r
,
r
ted. Indeed, if
1 ( z) =
{ 0:2
0
if z iotherwise
·
Section 5.3. Basic Holomorphy
and
a
225
is 2simplex contained in C \
r I dz = o. Nevertheless {O}, then lara)
there is no entire function g such that g l lC\ {o} = I. Another aspect of holomorphy is highlighted by contrast in
1 ( ( ))k
5.3.51 Example. For each k in N, the function
j" ll!. " X h
Ik ( � )
27r SIn ' 2
;
0
if x
i 0
otherwise
i
is continuous on JR., 0 , k, n E N, and if k l, h and II are different functions: Two (different!) functions in C(JR., JR.) can assume the same values on an infinite set, e.g., �f , such that 0.
S
=
{.!.} nEN
S· i
n
The following result, which is of general importance in the context of holomorphic functions, provides the contrast to 5.3.51. 5.3.52 THEOREM. (Identity Theorem) IF I E H(Q) ,
S �f {adk EN C Q,
and Q
::)
S· i 0,
THE VALUE OF I(z) IS DETERMINED FOR ALL z IN Q BY THE VALUES I PROOF. By hypothesis, for some in and some subsequence =I contained in Q, lim = If g E H (Q) and g n E N, for
{ (ak )}kEN ·
a S· {an }nEN ) a a. (a (a ) , n n n n+= some positive I  g �f h is representable by a power series in D( a, t and h(an ) = 0, n E N. If h i 0, for some M in N and all z in D (a , t , r,
r
an are in D( a, ) 0 , for all large n, 0 = h (an ) = (an  a)M (� Crn (an  a) rn ) . Since a n  a i 0, o = for all large n, Co + L Crn (an  a) rn = o. The second term in the left rn= and Co
i o.
r
But since all but finitely many
1
r
Chapter 5. Locally Holomorphic Functions
226
member above converges to zero as a n
I
h D(a,r o = 0 ) If Q = D( a, r t
>
a,
whence
Co
=
0, a contradiction:
the argument is complete. If b E Q \ D( a, r t there is a set {[ zn , zn + dh �n� N of complex intervals such that Z 1 = a, ZN = b, and P �f
N
U [zn, zn+ d c Q (v.
n= 1
1.7.11). If w E P, for some positive r(w),
h
is representable for all Z in D[w, r(w)]O by a power series. For each r, the open cover {D[w, r(w)]O} wEP of the compact set P , admits a corresponding finite subcover: { D (wi , ri t L < i
The argument of the preceding paragraph implies h l D( . . ) 0 = 0, 1 � i � I. Wz , Tz
[ 5.3.53 Remark. Loosely paraphrased, the Identity Theorem says that the behavior of a function f holomorphic in region Q is determined by the behavior of f on any infinite subset S such that S· c Q.]
o
The next result is an application of 5.3.52.
5.3.54 THEOREM. IF Q �f C \ {O}, FOR NO f IN H(Q) IS THE EQUATION exp[f(z)] = Z VALID (THROUGHOUT Q) . ·0
PROOF. If f def = u + iv and Z = e' , 0 � () � 271", then u ( z ) = 0 and for some k '][' Z, V (e i ) = () + 2k (e i ) 71". Since v and () are continuous, k is a con stant. As () r 271" , v (e i ) v(l) while () + 2k71" (}(1) + (2k + 1)71" =I v ( l ) , a contradiction. Thus f is constant on '][' and 5.3.52 implies f is constant on Q. Since u(2) =I 0 = u ( l ) a contradiction emerges. 0 :
f>
O
O
>
O
>
A region Q is starsh aped if for some a in Q, Q is the union of halfopen complex intervals [ a , b). 5.3.55 Exercise. A convex region Q is starshaped; the cOnverse is false. 5.3.56 Exercise. If Q is starshaped and 0 tJ. Q then H(Q) contains an f such that exp[J(z)] = z. [Hint: For some a in Q , f Q 3 Z r dww serves.] J[a,zl :
f>
5.3.57 Exercise. If f �f u + iv E H (Q) , then dcl l1u dcl = Uxx + U yy = l1v = Vxx + Vyy = o.
227
Section 5.3. Basic Holomorphy
[Hint:
a + ib Q,(a,f
Near each point of is representable by a con vergent power series. Hence near b), both and are representable by convergent power series of the form
u(x, y)
v(x, y)
00
Tn ,n=O
u
v
Thus and are infinitely differentiable. The CauchyRiemann equations apply.]
[ 5.3.58 Note. The expression t:m Uxx U yy is the In terms of the operators and t:m = As intro duced, is applied to functions in C 2 JR.) ; nevertheless, is applicable to functions in en so long the partial derivatives en and in = 0, then is Uxx and U yy exist. If and the set of functions harmonic in is, in analogy with Owing to the linearity of the operator
f + � of u. a a 4aau. Laplacian l1 u (Q, l1 u uE Q, Ql1u u harmonic H(Q), Ha(Q). l1, { f �f u + iv E Ha(Q) } {} { {u,v} C Ha(Q) n JR.n �f HaJH:(Q) } , H(Q) C Ha(Q). &'i
Hence the study of harmonic functions can be and is confined to Rvalued functions. On the other hand, because the real and imaginary parts of holomorphic functions are harmonic, there is a strong connection between the theory of functions in and the theory of func tions in The relationship of the two theories is discussed in Chapter 6.]
H(Q) HaJH: (Q). The set HaJH:(Q) consists of the JR.valued functions that are harmonic in Q. When u E HaJH:(Q) and for some v in HaJH:(Q), f : Q 3 (x, y) u(x, y) + iv(x, y) �f f(x + iy) is in H(Q), then v is a harmonic conjugate (in Q) of u. If v is a harmonic conjugate of u and c is a constant, v + c is also a harmonic conjugate of u. ft
1 5.3.59 Example. The equation In I z l =  In I z l 2 and direct calculation re2 veal that e\ ft In I z l is in However, if 3
u : {o} �f Q (x , y) HaJH:(Q). f �f u + iv E H(Q), i.e., if a global harmonic conjugate v of u exists, for z in Q, 1 exp[J ( z ) ] 1 = 1 4
Chapter 5. Locally Holomorphic Functions
228
z l = 1, for some k(z) in Z, I (z) = 2k(z)7ri. As in the PROOF of 5.3.54, I isI constant, whereas �[J (z)l (= In I zl ) is not.
If
Global harmonic conjugates need not exist.
5.3.60 THEOREM. (Vitali) IF: E H(Q) , n E N AND sup a)
In nEN I l ln l oo M < 00 ; ak �f ao E Q \ { adkEN ; b) { ak hEN C Q AND lim + oo k limoo In ( ak ) EXISTS FOR ALL k IN N; c) n+ FOR SOME I IN H(Q) AND EVERY COMPACT SUBSET K IN Q, :::;
PROOF. If K is a compact subset of Q, for some positive r,
aUE K D(a, 3rt C Q.
a, b} c K and 1 b  al < r, Cauchy's formula implies I ln (a)  In (b)1 :::; M27r 1 1I za l= 2r Ib r� al dzl = 2M l br  al equicontinuous. The ArzelaAscoli Theorem (1 .6.9) shows that {In }nEN iscontains {Iis na}normal nEN family.a subsequence converging uniformly on K, i.e., {In }nEN There are compact sets Krn such that Krn C K;;' + I , m E N, and each compact subset K in Q is contained in some Krn , (v. 4.10.9). Hence there are subsequences {Inrn } n , Tn E N such that: a) If {
1"l
{Inrn }
Krn . Inn l K � 11 K . diagonalization method
converges uniformly on Hence there is a function b) as n + 00, In particular, such that if K is a compact subset of Q, then for each n, E H (K� ) , i.e., I E H(Q) . In sum: The sequence derived from the just applied converges uniformly on every compact subset of Q. fails to converge, owing to the hy If, for some in Q, pothesis a) , contains two subsequences {I�i ) } ' i = 1, 2 , con
I
I
{Inn } nEN a {In (a)}nEN {In }nEN nEN vergent at a and such that A l �f lim 1�1 ) ( a ) i lim 1�2) ( a ) �f A2 . The n+CXJ n+CXJ
229
Section 5.3. Basic Holomorphy
diagonalization technique applied to each of these subsequences yields sub subsequences { } nEN resp. { } nEN that converge uniformly on each compact subset of Q to functions resp. in H (Q). For k in N, = whence = Furthermore, 5.3.52 im(I) } plies u { nn } and { Inn = C { n } nEN . Hence
f�� f�� (ak ) f�� (ak ), f( l ) f( 2)
f�� f( l ) f(2) f( l ) (ak ) /2) (ak ). 1( 2) nEN f nEN f( l l(a) = n+= lim f��(a) ( = A d lim f��(a) ( = A2 ) . f( 2)(a) = n+= The leftmost members above are equal, whereas A l j. A2 , a contradiction. [ 5.3.61 Note. The terminology locally uniformly convergent is
o
used to describe a net of functions converging on each compact subset of an open set U.
Thus 5.3.60 may be paraphrased as follows. A sequence of functions holomorphic, uniformly
bounded in a region Q, and convergent on a sub set having a limit point in Q is locally uniformly convergent in Q.
Similarly, a
locally uniform Cauchy net resp. a locally uniformly
bounded net is a net that is uniformly a Cauchy net on every com
pact subset of U resp. a uniformly bounded net on every compact subset of U. These notions and their specializations are of particular utility in complex analysis, e.g., in the application of the ArzelaAscoli Principle in conformal mapping, the Great Picard Theorem, etc. The precise ways in which the applications occur are described in the course of the remainder of the book.]
f
1R(f)
5.3.62 Exercise. If E H (Q) and == 0 resp. is an imaginary resp. real constant on Q.
[Hint: The Open Mapping Theorem applies.]
CS(f) == 0 on Q, then f
Chapter
230
5. Locally Holomorphic Functions
5.4. Singularities
5.4.1 Example. If
if z ¥= ° f( z) = { 6Z otherwise ' then as defined, f E H(C \ {O}), f is not continuous at 0; f tJ. H (C) . How ever, since f(z) = z on C \ { a}, f(O) can be redefined to be 0, and then the newly defined f is in H (C) . If if z ¥= ° otherwise Since is large when is small and positive 0 ) , there is no way to define at ° so that the
g E H (C \ {O}). I g (z)1 unbounded near (Iresulting g( z) 1 is function is in H (C) .
then
{
If
h( Z) = � ± then:
z z z iffi.
I zl g
z ¥=
if ° otherwise '
0, I h(z)1 is large; 0, h(z)1 is small; z ¥= 0, 1 z) 1 =I 1 , whence there is no entire function E
a) if is positive and near b) if is negative and near c) if E and h( such that E IIC\ {o} = h .
The phenomena illustrated above motivate the following
s E fl, AND f E H(fl \ {s}), s isolated singularity f. 5.4.3 THEOREM. IF s IS AN ISOLATED SINGULARITY OF f, EXACTLY ONE OF THE FOLLOWING OBTAINS: a) FOR SOME POSITIVE R, IF 0 < r < R, THEN I f (z)1 IS UNBOUNDED IN D(s,rt \ {s} : s IS A pole OF f; b) I f (z)1 IS BOUNDED IN SOME NEIGHBORHOOD N(s) : s IS A removable singularity OF f, i.e., f IS DEFINABLE AT s SO THAT THE RESULTING FUNCTION IS HOLOMORPHIC IN N(s); c) (WeierstraBCasorati) IF N(s) C fl, THEN f[N(s) \ {s}] IS DENSE IN C: s IS AN isolated essential singularity OF f. PROOF. If c) fails, for some N(s) contained in fl, some a in C, and some pos 1 itive r, f(N(s) \ { s}) fails to meet some D(a,rt, F � __ is bounded f a near s, and is holomorphic in D( s, rt \ { s}. 5.4.2 DEFINITION. WHEN fl IS A REGION, IS AN OF
Section
5.4. Singularities
231
G(z) �f ( Z S) 2 F(z), then G E H [N(s) \ { s }]. Direct calculation G'(s) exists and G(s) = G'(s) = 0. Thus near s, and for some N in N \ {I}, G(z) (z  s)N L cn (z  s) n and Co i 0. If n=O If shows

=
00
n=O ¢ E H[N(s)]. For all z in some N1 (s), ¢(z) i ° and ¢1 E H [N1 (s)], 1 i.e., for z near s, ¢( z) is representable by a power series: 1 ¢(z) = nL=O dn (z  s) n , do i 0.
then
ex:>
ex:>
s, f(z) = a + (z  s1) N 2 n=O L dn (z  s) n . If N  2 = 0, then f is definable at s so that f E H [N (s)] : a ) obtains. If N  2 > 0, for I z  s I positive and small, If (z) I is large: b ) obtains. Further discussion of holomorphy and singularities is simplified by the following notations and terminology, when U is an open subset of C. When f E H(U), Z(f) �f { a : a E U, f(a) = o }. Each a in Z(f) is a zero of f and Z(f) is the set of zeros of f· When S c U, U \ S is open and f E H(U \ S), then: P(f)
o
•
•
0
(the set of poles of I) and
E(f) �f { a : a E S and a is an isolated essential singularity of f} (the set of essential singularities of I). When S = P(f), then f is meromorphic in U ( even when S = 0) . The set of functions meromorphic in a region Q is M (Q) . .
f E M (Q), then P(ft n Q = 0. [Hint: a P(ft n Q, a is not an isolated singularity.] [ 5.4.5 Note. The number N  2 in the PROOF of 5.4.3 is the order of the pole s.
5.4.4 Exercise. If If E
232
Chapter 5. Locally Holomorphic Functions
Not all singularities are isolated. For example, if
0
z
if j. and sin otherwise
z j. 0 1

k E H [N(O)] , for all sufficiently large n in N, k E C ( D (0, �) \ D (0, � ) 0 ) . Since n 2 7r 1 1 ° Kn = D (0, n7r ) \ D (0, 2n7r ) E K(C)
and if, for some N(O) ,
clef
= S clef
1 {z : z= 2n7r , n E \ {O} } c P(k), !Z
k;
and a contradiction emerges. Zero is a limit point of poles of zero is a of In Chapter 10 the treatment of leads to the general definition of a of a function. The more detailed behavior of functions near isolated essential singularities is treated in Chapter 9.]
nonisolated singularity k. analytic continuation singularity (singular point)
J a Z(f), no co( a) 0 z a, {cn (a)}�=o J(z) = (z  a) no n=O L cn (a)(z  a) n : a is a zero oj order or multiplicity no. 5.4. 7 Exercise. If Q \ S is a region, J E H (Q \ S), and a E P(f) n Q, then for some least no in N there is a sequence {cn (a)}�=o such that co(a) j. 0 1 and for all z near a, J(z) = �=O cn (a)(z  at: a is a pole oj order ( z  a) no n'" or multiplicity no. 5.4.8 Exercise. If U is open and J is a non constant function in M (U), then {Z(ft U [P (fW} n U = 0 . [Hint: 5.3.52 and 5.4.4 apply. ] 5.4.6 Exercise. If E H(Q) and E for some least in N, there is a sequence such that j. and for all near 00
ex:>
233
Section 5.4. Singularities
a
5.4.9 THEOREM. IF 1 E H(Q) , AND IS A ZERO OF ORDER no, THEN FOR SOME NEIGHBORHOOD ( ) , WHENEVER b E I( ( ) \ { } )
Na
i.e., 1 IS AN noto1 MAP OF
Na a ,
N (a) \ { a} .
if E D( , r �f
a N1 ( a ) , then I ( z ) = ( z  a ) no L cn ( z  a t �f ( z  a ) no g ( z ) n=O and if z E N1 (a), g ( z ) i O. Thus !L E H [N1 (a)]' and since N1 (a) is convex, g g g' '(() if, for z in N1 (a), G(z) =  de then G E H [ N1 (a)] and G' =  . g [a, z ] g ( () PROOF. For some positive
r,
z
r
00
,
1 ' G (Jl... )  e g'  g (eG) ' e G g'  ge G G' = 0 and for some constant Thus G e 2G 2G K K + K, g = : G . It foU:ws that if h( z ) = ( z a ) exp ( � G ) , then _
_
K+G(a)
h'(a) = eno
 i 0, the argument in 5.3.41 shows that h is injective Since on some open subneighborhood ( ) of ( ) and that h ( ) ] is open. Hence, for some t in (0, 1),
n
s
If 0 i b E D (0, t O t, for some and some
={
If S clef
Wk
no exp [i ( fJ +n2k7r )]
clef ...L
=
S
[N2 a
N2 a N1 a
w . . . , wno 
O
fJ,
and # ( { o, d ) = no. Since h is injective on injective on the (open) neighborhood
Hence
}
: 0 :::; k :::; no  1 , then
N2 (a), h is also
N ( a ) \ { a } contains precisely no points Zo, . . . , zno  1 such that
234
Chapter 5. Locally Holomorphic Functions
D of
b, a
H(Q)
no N(a): ¢ H[N(a)]; b + ¢no; ¢ ¢' N(a). 5.4. 11 COROLLARY. a) IF a E Q, I E H(Q \ {a}), AND a IS A POLE OF ORDER no OF I, FOR SOME POSITIVE R, {I b l > R} :::;. {# [J  l (b) n Q] = no} . b) FOR SOME POSITIVE r AND AN INJECTIVE ¢ IN H [D( a, ) ] , I = ¢nu . PROOF. a) If g(z) �f (z  a) no / (z), 5.3.41 implies that for some positive s, g is injective on D(a,sr · Then I I D( a, s) O ' which is z (z g(z)a) no ' . an nofold map of D(a, s)O on { z I z l > � �f R } . 5.4.10 Exercise. If I E and for some is a zero of order a) E 1  for some function and some neighborhood b) 1 = c) is injective and is never zero in
b,
¢
r
H
b) 5.4.10 applies. 5.4.12 Exercise. If I E 1 1 E 5.4.9 applies.]
°
_
IS
:
D and l i n is injective, f' l n is never zero and
H(Q) H[/ (Q)]. [Hint: 5.4.13 Exercise. If I (z) = eZ, then: a) for each in C, 1'( 0' ) j. 0; b) for some positi ve r, II D ( <>,r ) O is injective; c) for the g in the PROOF of 5.4.11, g l f( D ( <> ,r) o ) E H (Q d; d) g (Q d c U Ln ( w ) ; e) g' (w ) = �; f) if h E H(Q), wEn, aH E[D(Q, ,rr and h( a) j. 0, for some positive r, D( a, r r c Q and for some L in a ] ' h(z) I D(a ,r) o = eL (z) . If r > 0 and I E H [D(a, rr]' the basic version of Cauchy's integral formula (5.3.20) leads to (5.3.24), i.e., the representation of I by a power series converging to I throughout D( a, r r. A global version of Cauchy's formula leads to a similar representation when a is an isolated singularity of I. If a is not a removable singularity, the representation of I cannot converge at a but at best in the region D( a, r r \ { a} . A generalization of this kind of region is an annulus, i.e., when 0 :::; r < R < 00 , a region of the form { z : r < I z  al < R} �f A( a; r, Rr (for which the closure is A( a; r, R) �f { z : r :::; I z  al :::; R}) . An annulus A( a; 0, R) resp. open annulus A( a; 0, Rr is a punctured disc resp. a punctured open disc at a and is denoted D( a, R) resp. D( a, Rr. (For a given annulus A( a ; r, R) when 0 :::; fJ < ¢ :::; 27r the open annular sector is A(a : r,R;fJ,¢r �f { Z : z = a + pei,p , r < p < R, fJ < '1/J < ¢}.) a
235
Section 5.4. Singularities
A reasonable approach to such a representation for an J holomorphic in A( involves, when < < < an integration over the two curves, and such that
a; Rr, r,
'Yl
(The set
s S R,
r
'Y2
'Y � U 'Y� is the boundary
8 [A(a; s,srl = { z : I z  al = s } u { z : I z  al = S} .) For any w not in A( a; Rr, direct calculation shows that r,
Ind 'Y! (w) + Ind 'Yz (w)
=
o.
These remarks motivate the following 5.4.14 THEOREM. (Cauchy's integral formula, global version) IF U IS AN OPEN SUBSET OF C, J E H ( U ) , ARE RECTIFIABLE CLOSED
CURVES,
{ydl :s;k «,K K U 'YZ C U, AND FOR EACH w NOT IN U,
k=l
FOR EACH
K
2:)nd 'Yk (w)
K
a IN U \ U 'YZ , J(a)
k=l
=
(5.4.15)
0,
k=l [2:) 1 = LK � dz. 2�l k= l J  a k= l K
nd 'Yk (a)
1
J( )
.
'Yk
�
{ w : t Ind 'Yk (w) = o } is open, contains C \ U,
PROOF. Because the left member of (5.4. 15) is z:.valued and depends con tinuously on w, V �f and so V U U = C. For the function
G : U x U '3
k=l
(z, w)
H
{ J(w)w  zJ(z) if z j. w I'(z)
_
if = w
z
Chapter 5. Locally Holomorphic Functions
236
introduced in the PROOF of the Inverse Function Theorem, (5.3.41), the hypothesis (5.4.15) implies that the formulre
G(z) dgf
1
K
LJ
G(z,w) dw if z E U t (w Z du if z E V k= i 'Yk � �
are consistent on U n V: G is welldefined throughout C. Since G E H (U), E . Because
C E H(V) , and U n V =I 0, it follows that C E K V ::) n Ind 'Yki ( O ) , k=i { z : I zl
I z l , I C (z)1
> R }. For large for some positive R, V ::) is small and thus 5.3.29 implies = 0. The promised conclusion follows when the equation == ° is written in terms of the defining formulre for 0 [ 5.4.16 Note. The hypothesis (5.4.15) is satisfied if, e.g., as in Figure 5.4.1, for some positive R,
C(z)
C
C.
{ad i �k � K C D(O, Rr , K 27r t , 1] , 1 :::; k :::; K, and U 'YZ C D(O, Rr · 'Yk (t ) = a k + rk e i , t E [0 k=i U = D(O, Rr ,
In Figure 5.4.1, the dashed lines together with the small circles themselves may be construed as the image of a single rectifiable
Figure 5.4.1
Section 5.4. Singularities
237
closed curve, say r. Integration over r can be performed so that the integrations over each dashed line are performed twice (once in each direction) with the net effect that those integrations contribute nothing to The validity of 5.4.14 for the configuration just described follows directly from the basic version of Cauchy's Theorem. The approach in 5.4.14 permits a very general result free from appeals to geometric intuition, v. Figure 5.4.l .
1.
]
5.4.17 Exercise. In the context above, K
K
k=L1
1 J dz 'Yk
= o.
[Hint: If a E U \ k=U 'YZ and 1(z) �f (z  a)J(z) Cauchy's formula 1 applies to T]
bk} �= 1 and { Jj};'= l are two sets of rectifiable closed J curves such that U 'YZ U U J; c U and for each w not in U, k= 1 j = 1 5.4.18 Exercise. If K
J
K
j=1
k= 1
L Ind 8; (w) = L 1nd 'Yk (w),
1 J dz k=L1 1 J dz. jL= 1 J
then
8;
[Hint:
=
K
'Yk 5.4. 17 applies to the calculation of J
K
fdz  k=Ll'Yk Jdz.] jL= 1 l8) 1 The following is a useful consequence of 5.4.14.
z A(a; r, Rt, J E H [A(a; r, Rt]' FOR SOME {cn (a)} nEZ IN C AND
5.4.19 THEOREM. IF ALL IN
00
n=  (X)
(5.4.20)
Laurent series
J
THE RIGHT MEMBER OF (5.4.20) , THE FOR IN THE AN NULUS A( CONVERGES UNIFORMLY ON EACH COMPACT SUBSET OF A( R) 0
a; r,
a; r, Rt, •
Chapter 5. Locally Holomorphic Functions
238
PROOF. If Z E A(
a;
r,
Rr, for some [s, S] contained in ( R) , r,
A(a; s, Sr. If 'Yl (t ) = a + se 2 7ri( 1  t ) and 2 (t ) = a + Se 2 7rit , 5.4.14 implies J (w) J (w) dW J(z) = � 27rl [1 W  Z dw + 1 W  Z ] . For w in the first resp. second integral of (5.4.21), I wz  aa I < 1 resp. I � w  a I < 1. Hence � (w  a) n 1 1 . 1 = � (z  a) n + l w, a: n=O w  z z a :1  za resp. � ( z  a) n . 1 1 1 � z a w  z w  a 1 w  a n=O (w  a) n+ l If s < t < T < S, both series converge uniformly in A( a; t, T) . If � J (w ) (w  a) n dw if  1 27rl 1 Cn ( a) = 1 J(w) dw if 2': 0 27ri J ( w  a ) n + l Z
1'
E
')'2
')'1

_
{
(5.4.21 )
n
')'1
n
')'2
:::;
(5.4.20) obtains. D is not assumed to be 5.4.22 Remark. Owing to the fact that [
a, J <
holomorphic in some neighborhood of none of the coefficients ( ) in particular those for which n 0, need be zero.]
cn a ,
z
5.4.23 Exercise. If, for all near but not equal to
and ( v. 5.4.3) a ) Cn = 0 when n 0, then b ) for some negative no ,
<
00
a,
n=(X) a is a removable singularity of J; if n = n o if n n o
<
'
239
Section 5.4. Singularities
a {n : n f.
no;
is a pole of order c) if inf E ;2;, i O } =  00 , is an isolated essential singularity of
Cn
a
=
00
f z n=L(X) cn( z  a)n IS VALID FOR ALL z NEAR BUT NOT EQU AL TO a, THE residue of f at a IS 1 �f Res a (f). WHEN a IS A POLE OF f, THE principal part of f at a IS I Pa (f) �f n=(X) L cn (z  a ) n
5.4.24 DEFINITION. WHEN ( )
c
( A SUM INVOLVING ONLY FINITELY MANY TERMS! ) .
a
f, then Res a ( f ) Fm (z  a)f(z). 5.4.26 Exercise. If no < 0 and a is a pole of order n o of f, then  no  I [( z  a ) no f(z)] d Res a (f) !1E1 :( _n _ :C) no I 5.4.25 Exercise. If is a pole of order one of
=
=
1
.
0
z of: a
1 !
dz 

a [Pa (f)] = Res a(f). 5.4.28 THEOREM. (Residue Theorem) IF f E M (Q) AND hd l SkSK IS A K SET OF RECTIFIABLE CLOSED CURVES, S �f U 'YZ c [Q \ P(f)], AND FOR k=
5.4.27 Exercise. Res
1
EACH w NOT IN Q,
K
L Ind T'k (w) = 0,
k
THEN
=1
(5. 4 . 29)
(5.4.30) fdz L Res a(f) · [t 1nd T'k (a)] . 2�i t k1 k= 1 1T'k aEP(f) PROOF. The set S is compact; hence, for some positive r, S ¥D(O, r). If F � { a : a E P(f), t, Ind T'k (a) i 0 } is unbounded, F n [ C \ D(O,r)] i 0, =
=
240
Chapter 5. Locally Holomorphic Functions
and 5.2 .11b) implies that if lal is large
K
L Ind ,),k (a) = 0, a contradiction
k =l of the definition of F: F is unbounded. If U is a component of C \ S and U is unbounded, for each b in U, K
L Ind ')'k (b) = O.
If b E 8(Q) , then b E U C , whence the continuity of the
k =l
K
K
L Ind ')'k implies L Ind ')'k (b) = o. Hence, if F, which k =l k=l is bounded, is not finite, then Fe i 0.
Zvalued function
K
K
If b E Fe , then L Ind ')'k (b) i 0, because L Ind ')'k is a continuous Zk =l k =l valued function. Since 1 E M (Q), Fe n Q = 0 (v. 5.4.8) , whence b E 8(Q), a contradiction. In sum, F = 0 or F is a finite set, say F = { a l , . . . , am } . Thus the sum in the right member of (5.4.30) contains at most finitely many nonzero terms. If F = 0, the global Cauchy Theorem implies both members of (5.4.30) are zero. m If F = {a i , . . . , am } , then h �f 1 L P (f) has only removable sini=l ai

K
J
h dz = 0 and 5.4.14 applies. gularities in Q \ (P(f) \ F) and thus L k =l ')'k l'
5.4.31 Exercise. If U E 0(((:) and 1 E M ( U ), then f E M ( U ).
o
[Hint: If a E U, for all ,z near a, and for some nonzero C I , the l' Laurent series for f takes the form if a rf. [ Z ( f ) U P(f)] otherwise
.]
5.4.32 Example. a) If n E ;2;, 1 (z) = (z  a) n , and "( is a rectifiable closed 1 lI' dz = n · Ind ,),(a). If Ind ,),(a) = 1, curve such that a rf. "( * , then . 27rl ')' the formula above may be interpreted as a means of calculating 
J
{ theI order of a (the order of a) 
x
if n > 0 [ a E Z (f)] if n < 0 [ a E P(f)] .
241
Section 5.4. Singularities
{m,n } C ;2;+ , I (z) = (z  a) Tn + ( z 1 b) n , ,,( is a rectifiable closed curve such that { a, b} and 0 < 1 z  bl < 1 a  bl , then: bI) the Laurent series for 1 takes the form b) If
r:t.
* "( ,
{dp }pEZ
j
in the Laurent series for can be calculated b2) the coefficients by comparison of coefficients of like powers of in the two members of
zb
Tn k  b) k l �n l + L _ (z b + k= l Ck (Z 
Cz � b)n � Ck (Z  b) k) . Ct dp (z  b)P) .
=
+
A similar calculation when 0 <
f eq(z  a )
q= oo
q
for
(a
oo
I z  a l < l a  bl provides the Laurent series
j. The Residue Theorem 5.4.28 implies
1
If Ind ')' ) = Ind ')' the formula above may be interpreted as calcu lating the order of (a zero of I) minus the order of (a pole of For a function 1 in M (Q) and an in U Ord f denotes is a the order of (as a zero or a pole of I). If Ord f ) = of I. The preceding formulre have the following generalization.
(b)a
a
or pole
=
f) . (simple a) zero
b
a Z(f) P(f), (a 1, a
5.4.33 THEOREM. IF 1 E M(Q) , "( IS A RECTIFIABLE CLOSED CURVE SUCH THAT "( * c n D(O, U AND Ind ,),(w) 0 WHENEVER W tJ Q n D(O, ' THEN
[
{[Qrr] rrl \ [Z(f) P(f)]}, 1 f'   dz = �
•
J
27ri ')' 1
"
aE[Z(f)nD(O,r)D]
b E [P(f)nD(O,r)D]
[ 5.4.34 Remark. For the map ¢ of 2 .4.18, ¢ {[/ b (t)]} E Arg { /b (t)]}.
=
242
Chapter 5. Locally Holomorphic Functions
The formula in 5.4.33 may be used to calculate the total change in 4> as "( * is traversed. Owing to 2.4.18, the change is a multiple of and is independent of the choice of the map 4>. The formula is known as the or the It is most useful when
{f b(t)]} 271" Argument Principle .
Principle of the Argument
"((t) re2rrit , t
For example, if Q = C and = 0 ':::; .:::; 1 , the formula provides the difference between the sum of the orders of the zeros of in and the sum of the orders of the poles of in
f D(O, rt D(O,rt· A detailed discussion is given in 5.4.37.]
PROOF. At each point meromorphic function
f
a in Z(f) resp. b in P(f), the Laurent series for the j takes the form
The Residue Theorem 5.4.28 applies. D 5.4.35 Exercise. If E H(Q) , E Q, and i 0, for some positive �f Q I , and for some L in H (Qd then c Q, ° tJ. Furthermore, if "( is a rectifiable curve and = f.and (L I)' = * "( C then L "( E 1] and
f a f(a) r, D(a,rt f(D(a,rt) e Lof j. D(a,rt, f BV([O , ) 1 ff' dz = 1 1 d[L f (t)] (RiemannStieltjes integral!). [Hint: 4.10.30 applies.] 0
'Y
0
°
0
0
0
"(
5.4.36 Exercise. Under the hypotheses of 5.4.33, if g E H(Q) , then
1 f " g( a)Ord f( a )Ind ( a) � 271"i 1 gf' dz = aE(Z(J)nD(O,r)O ) g ( b)Ord f( b)Ind 'Y(b). b

'Y
'Y
E ( p(J)nD(O,r)O)
[Hint: The Laurent series for g j at the points in Z(f) and P(f)
are useful.]
Section 5.4. Singularities
243
5.4.37 THEOREM. (Argument Principle) IF: a) I E M (Q); b) "( IS A REC TIFIABLE CLOSED CURVE; AND c) "( * c {Q \ [Z(f) u p(fm , THEN:
= r IS A RECTIFIABLE CLOSED CURVE; A ) I 0 "( clef B) ind r (O)
=
b E [p(J)nfl]
aE[Z(J)nfl]
[ 5.4.38 Remark. The left member of the formula in b) is the winding number of the curve 1 0 "( about O. The result is most useful when, for each c in Z(f) U P(f), Ind ,,(c) 1.] =
11
PROOF. a) Since 1 1' 1 is bounded on stant M, I r(s)  r (r ) 1
=
h(r) ,,,(s)]
* "( ,
if 0 .:s; r < s .:s; 1 , for some con
l
I' (z) dz ::; M h(s)
 "((r) l , v. 4.8.6.
Because "( is rectifiable, r is rectifiable; because "( is closed, r is closed. b) By virtue of 5.4.35 and the compactness of "( * , for some Riemann part ition {tk L � k�n of [0, 1] and positive numbers {rd l � k � n : bl) "(
*
n
c
U D b (tk ) , rk t ;
k= l
b2) D b (tk ) , rkt n D b (tk + l ) , rk +l t j. 0, 1 .:s; k .:s; n  1; b3) there is an £k in H {D b (tk) , rkt } and such that e £ k o j = I ; b4) £k + l 0 1  £k 0 I is constant on D b (tk) , rkt n D b (t k + d , rk + lt, and in 27ri . Z. Finally, 5.4.28 applies. D
}
5.4.39 COROLLARY. (Hurwitz) IF { In n E N C H(Q), 0 tJIn � I
U In (Q) ,
nEN
AND
ON EACH COMPACT SUBSET OF Q, EITHER 1 == 0 OR 0 tJ I (Q ) . PROOF. If I t o, since I E H (Q), Z(ft n Q 0. Hence, if E Q, for Furthermore, 0, some positive r, 0 tJ J [ 0, r r] and E H
1 n
A( a;
1
j
[ A (a; � fJ . =
a
fn dz = . lim .  dz. The left member above is 27rl Iz  a l = � f 27rl I z a l = � I zero, whence so is the right. If I t O , for each in Q, I is not zero in some neighborhood of n+=
1
o tJ I(Q).
a
1
1'
a:
D
Chapter 5. Locally Holomorphic Functions
244
5.4.40 THEOREM. (RoucM) IF { f, g} c H (Q) , D(a, r) c Q, AND
(5.4.41 ) THE SUMS OF THE ORDERS OF THE ZEROS OF f AND OF f + g IN D( a, r t ARE THE SAME: aEZ(f+g )nD(a,1') O
aEZ(f)nD(a,r)O
PROOF. For t in [0, 1] ' the hypothesis (5.4.41) implies that the integral 1 (f + tg)' �f N ( t ) dz Iz a l =r f + tg
27ri 1
is welldefined. According to 5.4.33, N(t) is Zvalued. On the other hand, the left member above is a continuous function of t and must be a constant. Moreover, N(O) = aEZ(f)nD(a,r)O
N(l)
=
aEZ(f+g )nD(a,r)O
and, since N is a constant function, N(O) = N(l). D 5.4.42 Exercise. If, for the polynomials p, q, deg(p) = M < N = deg(q) and R is sufficiently large, then { I z l 2: R} :::;.. { lp(z) 1 < Iq(z) I } · 5.4.43 Exercise. If aN i 0,
f( z ) cl=e f
N"" l n=O �
NN N
n
a z , g ( z ) cl=ef a z , a n
Ir
0 , and h = f + g,
5.4.40 and 5.4.42 imply that for R sufficiently large,
L aEZ(h)nD(a,R)O
Ord h(a)
=
N,
i.e., the strong form of FTA is valid: if p is a polynomial of degree N and multiplicities of zeros are taken into account, p has N zeros. 5.4.44 Exercise. a) If a is a simple pole of f, in some nonempty open N( a) \ {a}, f is injective. b) If a is a simple pole of both f and g, some linear combination h �f of + (3g is holomorphic in some nonempty open neighborhood of a.
Section 5.5. Homotopy, Homology, and Holomorphy
245
5 . 5 . Homotopy, Homology, and Holomorphy
The close connection between ind 'Y and Ind 'Y when "( is a rectifiable closed curve suggests that there is a topological basis for many of the results about complex integration. An approach that reveals this basis is found in the next paragraphs. The fundamental material about homotopy is given in Section 1 .4. 5.5.1 DEFINITION. A CLOSED CURVE "( : [0 , 1] H Y IS null homotopic in A IFF FOR SOME CONSTANT CURVE 15 : [0 , 1] '3 t H r5(t) == y E Y: "( AND 15 ARE homotopic in A. 5.5.2 DEFINITION. FOR TWO CURVES "( AND 15 IN A TOPOLOGICAL SPACE Y, WHEN "((I) = 15(0) THE product "(15 IS THE CURVE
"(15 : [0 , 1] '3 t H
{
1
"((2t)
< t
r5 (2t  1)

2
The end "((I) of "( * is the start 15(0) of 15 * . The curveimage ("(15)* : a) connects "((0) to 15(1); b) is the union of the two curveimages, "( * and 15 * . If 1] : [0 , 1] H Y is a curve such that 1](0) = Yo (a curve starting at Yo) there is an associated curve 1] 1 : [0 , 1] '3 t H 1](1  t) starting at Y 1 �f 1](1). 5.5.3 Exercise. The product ( �f 1]1] 1 is a loop starting at Yo and null homotopic loop.
(
is
5.5.4 Exercise. a) The product operation for loops starting at Yo induces a binary operation on the set 7r 1 (Y, Yo) of homotopy equivalence classes of all loops starting at Yo. With respect to this binary operation and its associated inverse, 7r 1 (Y, Yo) is a group, the fundamental group of Y. The identity of 7r 1 (Y, Yo) is denoted 1. b) The fundamental group is independent of the choice of Yo, whence may be denoted simply 7r 1 (Y) . [Hint: If Y 1 E Y and is a curve connecting Yo to Y 1 , then a
[ 5.5.5 Note. When, as in Section 1.4, F provides a homotopy, the set F(·, S ) SE [O,l] may be viewed as a oneparameter family of graphs in [0 , 1] x Y. Then (1.4.2) expresses the circumstance that the graph of "( is continuously deformed into the graph of 15. In what follows Y is some region Q while "( and 15 are curves such that "(* U 15* c Q ; as s traverses [0 , 1] the curves F(·, s) constitute '
Chapter 5. Locally Holomorphic Functions
246
a family that begins with "( and ends with 15. Furthermore, for each s, F(·, s)* e n. Unless the contrary is stated, when "( and 15 are closed curves, i.e., loops, and "( "'F,rI 15, the condition
(5.5.6)
F(O, s) == F(l, s)
is imposed: each curve F(·, s) is assumed to be a loop, F is a loop homotopy in n, and F(·, 0) and F(·, 1) are loop homotopic in n.]
{� },
5.5.7 Example. Two loops can be homotopic in a region n via an F that is a homotopy in n but is not a loop homotopy in n. If n �f C \
"((t) = 3e 2 7rit , r5(t) = 2e4 7rit , and F (t, s) = (3 s)e 2 7r( I + S) i t : 5 . s = "21 and F(t, s) j. "2 (otherwIse, _
•
5 2' •
•
a contradiction; "( "'F,rI 15; both "( and 15 are loops, although if 0 < s < 1, the curve F(·, s) is not a loop: F(O, s) = (3  s) j. F(l, s) = (3  s)e2 7r (l + s) i .
5.5.8 DEFINITION. A REGION n IS simply connected IFF EACH LOOP "( SUCH THAT "(* e n IS LOOP HOMOTOPIC IN n TO A CONSTANT MAP. 5.5.9 Exercise. If n is simply connected, "( is a loop such that "(* e n, E n, and r5 (t ) == for some F, "( "' F,rI 15.
a
a,
5.5.10 Exercise. A convex region is simply connected. 5.5.11 Exercise. A starshaped region is simply connected. 5.5.12 LEMMA. IF: a) "( AND 15 ARE LOOPS SUCH THAT "(* U 15 * e n; b ) VIA SOME LOOP HOMOTOPY F, "( "' F,rI 15; AND c ) C IS A COMPONENT OF U �f C \ F ([0, 1] 2 ) , THEN ind F ( . , s ) ( a ) IS CONSTANT AS s TRAVERSES [0, 1] AND a REMAINS IN C.
PROOF. Owing to 2.4.20, ind F e ,s) ( a ) is defined and is a continuous ;2; valued function on the connected set [0, 1] x C. D
Section 5.5. Homotopy, Homology, and Holomorphy
247
[ 5.5.13 Note. Although F in 5.5.4 is not a loop homotopy in of "I and 15, via some G : [0, 1] 2 H C, "I and 15 might be loop homotopic in n. However, the inequality n
ind �
(� ) = 1
# 0 = ind 8
and 5.5.9 deny the possibility.]
(� )
5.5.14 LEMMA. IF "I AND 15 ARE RECTIFIABLE LOOPS, a E C, AND Ih  1511 00 < 11"1  all oo ,
( 5.5.15)
THEN Ind �(a) = Ind 8 (a). PROOF. (A pictorialization of (5.5.15) provides an intuitive argument for 15  a the conclusion.) Owing to (5.5.15), a tJ ("! * u 15* ), whence, If. 1] clef = , "I  a then 1 1 1  1] 11 00 < 1, i.e., 1] * C D ( I , I t : 1] is a loop contained in D( I , lr and so Ind 1J(O) = O. On the other hand, 
11 , 11
15
' dt us:  a
11 '
"I  dt = Ind 8 (a)  Ind 8 (a). "I a o D 5.5.16 Exercise. a) If "I is a rectifiable curve, "1* e n , and E > 0, for some polygon 7r* and some homotopy F, "I F, n 7r and 0 = Ind 1J(O) =
'!L dt = 1]
0
0
rv
t Esup[O,! ] h(t)  7r(t) 1 < E. b) If "I and 15 are rectifiable curves, "1* U 15* c n, and for some homotopy F, "I F, n 15, there is a finite sequence {7rZ L
and
Ih(t)  7r I (t) ll oo < E, II7r l (t)  7r2 (t) 11 00 < E,
248
Chapter 5. Locally Holomorphic Functions
Q is simply connected, "I is a rectifiable loop such that "1* C Q, J H(Q), then i J dz = o. (In particular, the result applies when 1z  o' Ind ( ) = 0.) tJ Q and J(z) cle f d ) If J E H(Q), "11 and "12 are rectifiable curves such that "I � U "I� C Q, "I I ( 0 ) = "12 (0) , "1 1 ( 1 ) = "12 (1), and "1 1 "' n "12 , then 1 Jdz = 1 Jdz . e) If J E H(Q), "11 and "12 are rectifiable loops such that c) If and E
=
a
/, 0'
/'2
/'1
then
:
1 J(z) dz 1 zJ(z) dz. /'1
Z
=
a
[Hint: 5.5.11.]
a
/'2
Q is simply connected and J E H(Q), for some F in [Hint: If {a, z } C Q and "I is a rectifiable curve such that "1 * C Q, "1 ( 0 ) = a, "1 ( 1 ) = z, the value of F(z) �f i J dz is independent of
5.5.17 Exercise. If
H(Q), F' = J.
the choice of "I.]
H(Q), J egoQ is simply connected, J E H(Q), and 0 tJ J(Q), for
5.5.18 Exercise. If some g in =
f'
[Hint: For some F, F' = j .]
If Q is simply connected, J E H(Q), 0 tJ J(Q), and H(Q), J gn . If "I : [0 , 1] Q is a rectifiable loop, then ;Y H(Q) '3 J i J dz is a linear functional defined on the vector space H(Q). The set of all finite . 1s I· S an ab e1I· an group Z . sums �r cle=f "1�1 + . . . + "In f such 1 ·mear functlOna n If a E Q and "I (t ) == a h is a constant map), then ;Y �f 0 , the identity of Zn . 1 If tJ Q the function Ja : Q '3 z �__ is in H(Q). 27rl z In Z� there is a subset::actually a subgroupBn consisting of all r such that for all not in Q, r (fa ) o. The quotient group 5.5.19 Exercise.
=
n E N, for some g in [Hint: 5.5.18.] H
:
0
H
a
a
=
a
H
Section 5.5. Homotopy, Homology, and Holomorphy
249
z.
is the first homology group of Q over Sums r and l1 are homologous in Q (r R:: n l1) iff r l1 E Sn . All constant maps are pairwise homologous. When ;Y E Zn , "(* is a cycle in Q. When ;:y E Sn , "( * is a bounding cycle in

Q. 5.5.20 Exercise. a) The groups Zn , Sn , and H I (Q2 Z) are uniquely � determined by Q alone. b) If F E H (Q), Q = F(Q ) , and r E Zn , there is a � corresponding (J ( r d�f F("( I ) + . . . + F ("(n ) in Zn . The map

)
is a group homomorphism. c) if 0 is the identity of Zn ' (J (Sn ) = O. d) If E Sn , then (J (J is a homomorphism of H I (Q, Z) into = (J
rH HI (fl, z) .
[Hint: d) If
(r) ( H ) :
1 (F(z)  ) r � (3 tJ fl, then iFh)  (3 = F(z)  (3(3 ' dz.
Argument Principle applies.]
W
'Y
5.5.21 Exercise. If Q is simply connected, HI (Q , Z) 5.5.22 Exercise. If Q � A(O; 1, 2r, then HI (Q, Z)
=
=
The
{O } .
Z.
[ 5.5.23 Note. The definition given of HI (Q, Z) is tailored to the developments of this book. There are connections between the approach above and De Rham's Theorem [SiT] . More general definitions of homology groups are given in [Sp] .] 5.5.24 Exercise. If "( is a loop such that "(* C Q and "( IS null loop homotopic in Q (5.5.1), then ;:y E Sn , i.e., ;:y R:: n O. Loop homotopic loops are homologous. 5.5.25 Example. The sets L� �f IDEFGBCDI, L; �f IAHGFCBAI in Figure 5.5.1 (p. 248) correspond to loops L I , L2 , neither of which, nor their product L 3 �f L I L2 , is null loop�omotopic in Q �f C \ ( {p } u {q} ). On the other hand, although neither L I nor L2 is null homologous, L 3 is null homologous.
Chapter 5. Locally Holomorphic Functions
250 H
F
G
E
p A \;:....I...'J D B C
I
Figure 5.5.1.
Homologous loops need not be loop homotopic.
"/
5.5.26 Exercise. If * C Q, for some m in N, {"/} contains a closed polygon 7r* such that 7r * C Q and each side of 7r* is a vertical or horizontal complex interval of length not exceeding [Hint: If E b} E 7rl (Q) , for some m, the distance between any point of * and any point of C \ Q is at least For some N in N, if I t  t' l < ' then h (t)  ,,/ (t') 1 < The polygon 7r * N k consisting of the segments , ,,/ , 0 :::; k ::;; N lies in Q and 7r is homotopic to Discussions of the fundamental group 7r 1 (Q) can be conducted in terms of rectangular polygons described above.
"/ "/
2  Tn .
2 m . T Tn .
1
["/ (�) ( � 1 )] "/.]
.
 1,
, "/n
5.5.27 Exercise. For a set ,,/ 1 , . . of rectifiable loops in Q and a E Q: a) there are rectifiable curves CI , . . . , Cn such that the product
is the homotopy equivalence class of a loop beginning and ending at a; b) if each begins and ends at a, 'h + . . . + 'h i c) 7 r C then a (3; d) h : if {a, ,6} b}, 1 (Q, a) '3 b } H ;;;jBn E HI (Q, Z) is a welldefined surjective homomorphism; e) if {"/} in 7r 1 (Q, a) is the product of commutators, i.e., if
"/i
=
hi · "/2 · . . . . "/n f=
Section 5.6. The Riemann Sphere then h ( h
})
= o.
251
(10)
(10)
[ 5.5.28 Note. Although HI � Z ( [Arm, Lef] � 7r1 7r1 (Q) is not and 5.9.23), for some Q, e.g., U abelian ( 10.5.13). On the other hand, for the commuta tor subgroup C of 7r J (Q) , HJ (Q, Z) � 7r I (Q)/C: HI (Q, Z) is the abelianization of 7r1 (Q) [Arm, Lef] .] v.
v.
C \ ({ I } { I}),
5 . 6 . The Riemann Sphere
The Riemann sphere is the compact subset
L: 2 �f
{ (�, 1], () : (�, 1], () E
ffi.3 , e +
( { B [ (0, 0, � ) ,� ) ] } =
a
1]2 (( +
�) � }
of ffi.3 . For the injection (stereographic projection)
G:
L: 2 \ {(O, O, I)} '3 P �f
illustrated in Figure 5.6.1,
l:2
(�, 1], ()
f+
G (P) �f z
e
(P)
2
=
E C,
Figure 5.6.1. The Riemann sphere and stereographic projection.
252
Chapter 5. Locally Holomorphic Functions
there is defined the function f : L? 3 P H f 0 8(P) . Behavior of f(z) for large I z l is the same as behavior of f(P) for P near (0, 0, 1) or alternatively as behavior of f for z near and different from 0. Similarity of L[N08(P)] and L(NQP) implies that
( �)
8[(�,
1],
()] =
1
� +i1=( : (
8 is a homeomorphism. Furthermore :[ 2 itself may be viewed as (the homeomorphic image of) the onepoint compactification Coo �f O:J{ oo} of C. In that context, 8 has a unique extension e to :[ 2 and e : :[ 2 H Coo is a homeomorphism. Regions, curves, etc., in Coo are 8images of regions, curves, etc., on :[2 . When Q is a region in C, then 8(Q) is by definition a subset of C. When Q is viewed as a subset of Coo , 00 may well be a boundary point of Q. Thus if Q is bounded otherwise When {a, b } c Coo , J (a, b) is the Euclidean distance between 8  1 (a) and 8  1 (b) (points in :[ 2 , a subset of ffi.3 ) . 5.6.1 Exercise. The function J i s a valid metric in Coo . In the context described, a neighborhood of 00 in C is equivalently described as a set containing, for some positive R, { z : I z l > R } or, for some positive r, as the image under 8 of { p : P E :[ 2 , 0 < J(P, N) < r } (a punctured neighborhood of (0, 0, 1) on :[ 2 ). A neighborhood of 00 in Coo is for some positive r the set { z : J ( 8  1 ( z ) , N ) < r } or equivalently, for some positive R, { z : I z l > R }. The behavior of f for large I z l is that of f(P) for P near but not equal to N. Thus the locutions f has a pole at 00, f has an essential singularity at 00, etc., may be construed equally well as descriptions of the behavior of f for z near but not equal to zero or of the behavior of f(P) for P near but not equal to N or of the behavior of f(z) for values of z in Coo and near but not equal to 00. 5.6.2 Exercise. a) If f is a polynomial of positive degree, f has ,a pole at 00. b) If f is entire and is not a polynomial, f has an essential singularity at 00. c) If 00 is a removable singularity of an entire function f, then f is a constant. 5.6.3 Exercise. If f ( z ) = e Z , a j. 0, and R > 0, for some z such that I z l > R, e Z = a . [ 5.6.4 Note. The result above and illustrates the two famous the orems of Picard. In summary form, they say that if a is an isolated
(�)
253
Section 5.6. The Riemann Sphere
Coo
essential singularity of I in and for the punctured neighbor hood N(a) \ { a , e.g. , A(a; 0, R), # { \ I[N(a) \ { a ] > 1, then I is a constant. In the neighborhood of an isolated essential singularity the range or image of a nonconstant function omits no more than one complex number. Picard's Theorems ( v. Chapter 9) are substantial strengthenings of the WeierstrafiCasorati Theorem, 5.4.3c ) . On the other hand, FTA says that the range of a nonconstant as a polynomial, a special kind of entire function having only pole, omits no complex number.]
}
C
}}
00
5.6.5 Exercise. If a is a pole of I and N (a) is a neighborhood of a, for some positive R, I(N(a) \ {a}) ::) > R for near but not equal to a pole of I, I(z) omits no complex number of large absolute value.
{ z : Izi
}
z
:
5.6.6 Exercise. The Maximum Modulus Theorem (5.3.36) may be re formulated for as follows. If Q is a region in I E H ( Q ) and for each a in 800 ( Q ) and some sup M, for all in Q, M, inf :::; M.
Coo
C,
N(a) EN(a) z EN(a)nn
I /(z) l :::;
z
I /(z) 1
[Hint: If Q is bounded 5.3.36 applies. If Q is not bounded and > M, for some positive E, each m in N, and some in
sup n Q,
Zm
I /(z) 1
I Zml > m and II (zm ) 1 > M + E.] Coo )
5.6.7 Exercise. ( Minimum Modulus Theorem for If I E H ( Q ) and o tJ I(Q), for a in Q, Equality obtains iff I is a constant 2': inf
I /(a) 1
function.
I/( z) l . z En
[Hint: The function g �f
7 is in H (Q ) ; 5.3.36 applies.]
n +1 sn+l n + lC 2 contains the set of all n + Ituples (Zl, . . . , znn++dl such that L I Zkl > o. There is a relation among the elements of s : k= l {(Zl, . . . , zn + d (W I , . . . , wn + d } { 3 >. 3 /t{ { 1 >' 1 + I IL I > O} 1\ { >' Zk + ILW k = 0, 1 :::; k :::; n + I}}} . The set
rv
{:}
rv
5.6.8 Exercise. The relation
rv
is an equivalence.
Chapter 5. Locally Holomorphic Functions
254
Sn+ l
n
5.6.9 DEFINITION. p ( C) �f / "' . Of particular interest for the context Coo and the map e defined earlier is p i (C) , the complex projective line. 5.6.10 Exercise. The map
pi
is a bijection between
otherwise (C) and Coo .
5.6.11 Exercise. If a E P(f), for some N(a) and some N(oo),
f[N(a) \ {a}] v . 5.3.39.
::J
[N(oo) \ {oo}] ,
(1[)),
5.6.12 Exercise. If f E H then 0 is a removable singularity of f iff 1R(f) or CS(f) is bounded near o. [Hint: 5.4.3 and 5.6.11 apply.] 5 . 7. Contour Integration
The Residue Theorem finds application not only in the theory of Cvalued functions defined in some n, but also in the evaluation of definite integrals and in the summation of certain series.
1 1 2k dx, k E N, is, for positive R, reI. f 0 < t < 1 Re27rit 2 /'(t) ' 1 1 R + ( t  2) 4R if 2 < t < 1 for which is the union of a semicircle and an interval [ R, R] . Di 1 rect calculation shows that if R 1 , on + z2k I 2': R2 k  1 . Hence, 1 R , whence lim r 1 k dz = O. 1 dz l :::; R2: if R is large, I r k 2 1 1 J l + z2 J l+z in n + �f { z : "5(z) O } , Thus, if is the sum of the residues of l + z2k 1 dx 27riL The set P+ of poles of l +1z2k in n+ is then 1 IR I + x 2 k 5.7.1 Example. The integral Jr IR + x lated to the curve (contour)
=
/'
{
*
>
rR rR ,
R too r R
rR
L
>
=
Section 5.7. Contour Integration
z
255
z) > 0, 1 + z2k = 0 } .
1 The residues of 1+ P + can be calculated via the formulre in 5.4.24. {
'S(
:
z2k
5.7.2 Exercise. ( Jordan's inequality ) If 0 :::; t :::;
at the points of
� , then � t :::; sin t.
[Hint: The geometry of the situation provides the clearest basis for the argument.]
1
00 •
sln x 5.7.3 Example. The integral dx can be treated by an integration x o e iz of  over the curve "( defined in terms of the positive parameters E ( small ) and R ( large ) :
z
4
E + ( R  E)t Re4 7ri ( t  i )
4(
( �)
R +  E + R) t Ee i ( ( �  t )4 7r + 7r )

. < t < 1 1f 0 1 1
4
4
4
4

Thus "( * is the union of two semicircles, Cf and CR in n + and two intervals tZ [E, R] and [R, E] . Since e is holomorphic in the bounded component of
C\
z
*
"(
1
'Y
eiz dz = o. Z
(5.7. 4 )
Rewritten in terms of the four constituent integrals of the left member of the equation becomes
(5.7.4),
2i
1 f
R
. sin x e tz dz. e" Z dz 1 dx =  1 .
X
CR Z
C, Z
Jordan's inequality, sin x = sin ( 7r  x ) , and Euler's formula permit the conclusion lim
R+oo
Hence
00 .
11
CR
z eiz z dz O. Since I = 1 ei dz
sin x 7r  x = _2 . x o
1
d
C, Z
1
c,
eiz z dz +  i . 7r
=
i
01 . 7r
eue
i.
dO, as E + 0,
Chapter 5. Locally Holomorphic Functions
256
5.7.5 Example. For N in N, if
1 if 0 < < 
t 4 1 1 if  < t 4 < 2 3' 1 if  < t < 4 2 .If 3 < t < 1 4 then "I� is the union of the four complex intervals
[(N + �) (1  i), (N + � ) (1 + i)] , [ (N + � ) (1 + i), (N + � ) (1 + i)] , [(N + � ) (1 + i), (N + � ) (1  i)] , [ (N + � ) (1  i), (N + � ) (1  i)] .
Direct calculation shows that for the sets
{Z { resp. 52 cle=f Z resp. 53 clef { Z 51 � f
=
there are Nfree constants CI , C2 , C3 such that on "I� n 51 on "I� n 52 on "I� n 53
I cot 7rZI :::; C1 , I cot 7rzl :::; C2 , I cot 7rZI :::; C3.
Hence on "I� , I cot 7rZI :::; max {CI , C2 , C3} �f C. Further direct calculation shows that if J E M (C) and P(f) n Z = 0, then Res n [7r cot 7rZ . J(z)] = J ( n ) , n E Z. If #[P(f)] is finite and 5 �f sum of the residues of 7r cot 7rZ . J(z) at the poles of J,
Section 5.8. Exterior Calculus
257
J"IN 7r cot 7rZ . I(z) dz L I(n ) + If 1 E M(C) and for Nfree constants K in (0, and k in (1, on "(� , I / ( z) 1 :::; :k ' then l i 7r cot 7rZ . I(z) dz l 7r�� ( 8N + 4), it follows I N that upon passage to the limit as N for large N, homotopy
� 27rl
=
:::;
N
s.
n= N
(0 )
(0 ) ,
+ 00
00
L
I(n) =  so
(5.7.6)
n =  (X)
If #[P(f)] = No, a second passage to the limit as #(P(f) + 00 validates (5.7.6) in general. 5 . 8 . Exterior Calculus
In Section 5.5, a curve "( [0, 1] H Q serves to define a linear functional ;
;Y H(Q) '3 1 H ;
"(
rv
�
"(.
i 1 dz ;
Since Q may be viewed as a subset of ffi.2 , 1 �f u + iv is a C oo map Q '3 (x, y) H (u, v) E ffi.2 . More generally, the curve
"( [0, 1] '3 t H [x(t) , y(t)] E Q defines on the vector space C oo (Q, ffi.2 ) the linear functional ), ;
to the formula
),( 1 ) ; C oo (Q, ffi.2 ) '3 [u(x, y), v(x, y)] H
( 1 ) according
1 1 {u[x(t), y(t)]x' (t) + v[x(t), y(t)]y' (t)} dt �f
i u dx + v dy.
( 5.8.1 ) ( 5.8.2 )
The notational definition of the right member of (5.8.1) as the right member of (5.8.2) is the origin of a formalism that is extended in a manner described below. When (a, b) E Q, the evaluation map
(a, b) ; C oo (Q, ffi.) '3 k H k(a, b) E ffi.
is a linear functional, an element of [C OO (Q, ffi.)] * . The linear span of the set of all evaluation maps is a subspace VO of [C OO (Q, ffi.)] * . The elements
Chapter 5. Locally Holomorphic Functions
258
A ( O ) of VO are considered to be Odimensional functionals since they are determined by finitehence Odimensionalsubsets of Q.
The linear span of the set of all functionals ), (1) described in is a subspace Vi of C oo (Q, ffi? ) * . The elements A ( l ) of Vi are considered to be Idimensional functionals since they are determined by finite sets of differentiable curves, i.e., Idimensional subsets of Q. When E E S )' 2 (Q) and ), 2 (E) < 00, according to the formula
(5.8.1)
), ( 2 ) : C(Q, JR.) '3 h H
Ie h(x, y) dx dy
E defines a functional ), ( 2 ) on C(Q, JR.) . The linear span of the set of all ), ( 2 ) is a subspace V 2 of [C(Q, JR.)] * . The elements A ( 2 ) of V2 are considered
to be 2dimensional functionals since they are determined by finite sets of essentially 2dimensional subsets of S),(Q). An element of f resp. (u, v) resp. h of C(Q, JR.) resp. C (Q, JR.2 ) resp. C(Q, JR.) determines an element w ( O) resp. w ( l ) resp. w ( 2 ) of (VO ) * resp. (V I ) * resp. (V 2 ) * according to the formulre
( ) ( ) ( )
w ( O ) A ( O ) �f A ( O ) (f) , resp. (u, v) w ( l ) A ( I ) �f A ( l ) [(u, v)] , resp. h w ( 2 ) A ( 2 ) �f A ( 2 ) (h). f
rv
rv
rv
In short, the pairings
serve to define two sequences
[C(Q, JR.)]* , [C (Q, JR.2 ) r , [C(Q, JR.)] * , ( V0 ) * , (V l ) * , (V 2 ) * , according as elements in the first resp. second half of a pairing are regarded as functionals defined on the second resp. first half. Functionals w ( o resp. w ( l ) resp. w ( 2 ) in (VO ) * resp. (V I ) * resp. (V 2 ) * are called Oforms resp. Iforms resp. 2forms. Thus a function f, according to its application, corresponds to a Oform or a 2form while a pair (u, v) of functions corresponds to a Iform . •
The result of applying a Oform w ( O ) by ( a, b) is f ( a, b) .
rv
f to the functional determined
259
Section 5.8. Exterior Calculus •
The result of applying a Iform w ( i ) by "I : [0, 1] '3 t H [x(t), y(t)] is
rv
(u, v) to a functional determined
1 {u[x(t) , y(t)]x' (t) + v[x(t), y(t)]y'(t) } dt cle=f
•
1
'Y
U dX +
(5.8.3)
v dy .
The result of applying a 2form w ( 2 ) an E in S), is
rv
h to a functional determined by
Ie h(x, y) dx dy �f Ie h dx /\ dy.
(5.8.4) (5.8.3)
The expression u dx + v dy in the right member of is the nota tion for {u[x(t), y(t)]x'(t) + v[x(t) , y(t)]y'(t) } dt in the left member. Thus the Iform (u, v) itself is denoted u dx + v dy. Similarly, the expression h dx /\ dy in the right member of is the notation for h(x, y) dx dy in the left member. The 2form h itself is denoted h dx /\ dy. The virtue of the notations just given becomes clear after the intro duction of the operator d defined on Oforms f, Iforms (u, v), and 2forms f according to the formulre:
(5.8.4)
d : (V O ) * '3 w (O) f H w ( l ) (fx , jy ) E (V I ) * , d : (V I ) * '3 w ( l ) (u, v) H w ( 2 ) Vx  U y E (V 2 ) * , d : (V 2 ) * '3 w ( 2 ) f H 0, rv
rv
rv
rv
rv
for basic forms. The formulre are extended by linearity to all (VO ) * resp. (V I ) * resp. ( v2 ) * . If f(x, y) == X, then df = (1, 0) ; if g(x, y) == y, then dg = (0, 1) :
dx = (1, 0), dy = (0, 1 ) .
Thus df = 1 dx + O dy
=
dx, dg = O dx + 1 dy = dy, i.e., dx = dx and dy = dy :
¢
the notation for Iforms is consistent. When and '1jJ are COO functions on which is based a change of variables T:
¢(�, 1])1]) ] �f ( xy ) ( 1]� ) [ '1jJ(�, H
the conclusions in 4.7. 18 and 4.7.23 lead to the following formulre.
Chapter 5. Locally Holomorphic Functions
260 •
•
For a Oform I, I( x, y) r+ 1 [4>( �, 1]), 'Ij!( �, 1])] �f F( �, 1]) (direct substi tution with no modification of the result) . For a Iform (u, v), the map is assumed to be bijectively C oo , i.e., is assumed to be a C oo map. Consequently, if
T
T 1
y) ] ( 1]� ) �f [ X(X, Y(x, y) ,
for each t, there are unique �(t) and 1](t) such that
�(t) )] = [ X(t) ] , T [( 1](t) y(t)
i.e., if J(t) � [�(t), 1](t)] , then
T maps "I to J. Furthermore,
u(x, y) = u [4>(�, 1]) , 'Ij! (�, 1])] �f U(�, 1])Y, v (x, y) v [4>(�, 1]) , 'Ij! (� , 1])] �f V (�, 1]) .
=
Hence
u[x(t), y(t)]x' (t) + v[x(t), y(t)]y' (t) �f [U[�(t) , 1](t)]x� + V[�(t), 1](t)]y�] ((t) + [U[�(t) , 1](t)]x1J + V[�(t), 1](t)]Y1J] 1]' (t),
1 {u[x(t), y(t)]x'(t) + v[x(t) , y(t)]y' (t) } dt
1 {U[�(t), 1](t)]x� + V[�(t), 1](t)]Yd ( (t) dt + 1 {U[�(t), 1](t)]X1J + V[�(t), 1](t)]Y1J } 1]'(t) dt, 1 u dx + v dy = 1 [Ux� + Vy�] � + [UX1J + VY1J] d1] : =
The Iform u dx + v dy in the variables {x, y} is replaced by the Iform [Ux� + Vy�] � + [U x1J + VY1J] d1] in the variables {�, 1]}. In the language of vectors and matrices,
[ U(X, y) ] 8(x, y) . [ U(�, 1]) ] v ( x, y r+ 8(� , 1] ) V (�, 1]) 1]) ] �f J . [ U(�, V(�, 1])
[ V (�, 1]) ]
� 8( 4), 'Ij! ) . U(�, 1])
8(� , 1] )
(the Jacobian matrix J modifies the result of substitution)
Section 5.S. Exterior Calculus •
for a 2form h, if H(�, 1])
261
�f h[
h(x, y)
r+
det (J) . H(�, 1]) ,
i.e., h dx 1\ dy r+ det (J) . H d� 1\ d1] (the determinant of J modifies the result of substitution).
cle�ff
�f cle 1 f
clef 1
When z x + iy, then dz dx + i dy, a complex Iform. By con. ventlOn, dZZ = dx  z. dy. Hence, 1' f a = "2 ( I  z.g ) an d b = "2 (I + zg ) , by polarization, I dx + 9 dy = a dz + b dZ. 5.8.5 Exercise. Both .
clef 1
clef 1
Iz dz = "2 ( Ix  Z.ly ) dz and fz dZ = 2 ( Ix + z. Iy ) dZ
are Iforms and dl = Iz dz + fz dZ. The Iform conjugate to w ( 1 ) I dx + 9 dy is * w ( 1 ) g dx + I dy. The form conjugate to dP = Px dx + Py dy is * dP = Py dx + Px dy. When "( : t r+ "((t) E C is a rectifiable curve and w ( 1 ) P dx + Q dy, 1 then W (l) P b(t)] + Qb(t)] dt = P dx + Q dy.
1 �f 1 {
�f
�:
�� } 1
�f �f
Forms (also known as differential forms) can be added, multiplied, differentiated, and integrated according to the following procedures. Addition. The sum of: two Oforms or two 2forms I and 9 is the Oform or 2form •
•
I + g;
•
two Iforms I dx + 9 dy and p dx + q dy is the Iform (I + p) dx + (g + q ) dy;
•
Multiplication. The product of: two Oforms I and 9 is the Oform I · g ; a Oform k and a Iform I dx + 9 dy is the Iform •
•
•
clef
a Oform k and a 2form I is the 2form k l ; two Iforms W I( 1 ) = I dx + 9 dy and w2( 1 ) = p d + q dy IS. the 2form w i l l 1\ W� l) (lq  pg) dx 1\ dy, the exterior (wedge) prod l) 1 ) uct of w i and W� ) . Differentials. The differential of: a Oform I is the Iform Ix dx + Iy dy; •
•
clef �f
kl dx + kg dy;
•
X
Chapter 5. Locally Holomorphic Functions
262 •
•
a Iform f dx + g dy is the 2form (gx  fy) dx 1\ dy; the product kw ( l ) is the 2form k · dw ( l ) + dk · w ( l ) . Integration. The integration of: •
•
•
a Oform f over a complex interval [a, b] is :
a Iform f dx + g dy over a curve "I [0, 1] IS
3
t r+ [a(t), ,6(t)] E
ffi.2
1 1 f b(t)] dx h' ) + g b(t)] dy b'(t)] = 1 1 f b (t)]a'(t) dt + g b (t)],6'(t) dt,
i.e., the line integral •
1 1 f((1  t)a + tb) dt;
2
1 f dx + g dy;
a 2form w ( ) over E is
Ie w ( 2) dx 1\ dy.
[ 5.8.6 Note. The preceding discussion deals only with two vari ables and 1 , and 2forms. The structure may be developed for n variables and nforms, n E N. For example, when n > 2, the product of a Iform and a 2form and the differential of a 2form are 3forms, etc. In a natural way, when n variables are involved and < k E N , all (n + k ) forms are zero. Excellent references are [Hic, Lan, SiT, Spi] .]
0,
0
5.8.7 Example. Green's Theorem ( Stokes's Theorem for the plane ) may be stated in terms of forms as follows. [a, b] X [c, d] c fl, then If
R �f
r f dx + g dy, + r + r + r i[( a ,c),(b,c) ] i[ (b,c),(b,d) ] i[ (b,d),( a , d) ] i[( a ,d),( a ,c) ] = r (5.8.8) (gx  fy ) dx 1\ dy, i[a ,b] x [c,d] r w ( l ) = r dw ( l ) . (5.8.9) iR ia ( R) A proof of Green's Theorem for wellbehaved functions defined over rectan gles in is given by direct appeal to the FTC. Indeed, Fubini's theorem implies
ffi.2
r (gx  fy ) dx 1\ dy = r ia i[a ,b] x [c,d]
b
(Jc d (gx  fy) dY) dx.
When FTC is applied to the right member of
(5.8.10)
(5.8.10), the result is (5.8.9) .
Section 5.S. Exterior Calculus
The equation
263
(5.8.9) itself can be related to FTC, e.g., h( q )  h(p) =
lq h'(x) d)",
according to the following interpretation. In JR., the boundary a ([P, q] ) of the interval [p, q] is {p, q}. For the signed measure space (a([p, q] ), !fj({p, q}), /l ) such that
{ I l
/l : a([p, q] ) 3 y r+
1
=
if y q '1f Y = p '
if h' exists at every point of [p, q] while h ' E L l ([P, q] , ).. ) , the burden of FTC is r h d/l = h ' d).. . Ja[p ,q] [p ,q] Stripped of all its qualifiers, Stokes's Theorem says that the result of integrating a function h over the boundary as of a set S is the same as the result of integrating some kind of derivative of h over S itself. 5.8.11 Exercise. For Iforms w P ) , W� l ) : a ) w i l l W� l ) = _W� l ) w P ) ; b )
1\
W I( 1 ) W I( 1 ) 0 [Hint: If T
1\
1\
( ; ) = (�), then det (J) =  1.]

•
[ 5.8.12 Note. Hence, by definition,
Iv F(x, y) dx 1\ dy = Iv F(x, y) dy 1\ dx.] 
(V, q, then = 2i r a dx 1\ dy = r a dZ 1\ dz. Jv Jv
5.8. 13 Exercise. If u E C OO
r
Jav
u
dz
u
u
V
5.8. 14 THEOREM. ( Pompeiu ) IF IS A NONEMPTY OPEN SET SUCH THAT FOR CONTINUOUSLY DIFFERENTIABLE JORDAN CURVES { 'Yih:'O: i :'O:IE N '
av = AND
u
I
U 1':
=
i l
Ve , FOR W IN V, a (5.8.15) dz 1\ dZ } . W
HAS CONTINUOUS FIRST DERIVATIVES ON (
u w
1. )=27rl
{ faav
(
u z
) dz +
W

z
1
v
u

z

Chapter 5. Locally Holomorphic Functions
264
(5.8.15)
[ 5.8.16 Remark. If u E H (V), then au = 0 and is Cau chy's integral formula (5.4.14). Thus the second term in the right member of may be regarded as an error term used, when u tt H (V), to compensate for the failure of u to be holomorphic and for the concomitant potential failure of Cauchy's formula.]
(5.8.15)
�f { z
PROOF. If E < d[w, (C \ V)] and U,
z r+
z E V, I z  w i
>
1

zw
is holomorphic in Ufo Stokes's Theorem applied to av 3 z r+
au
J zw u,
Oz
az A dz =
E } , then
u(z) dz ia v z  w
r
u(z) yields zw
l27r U (W + EeiO ) i dO. 0
As E + 0 the formula emerges.
[ 5.8.17 Note. The treatment above of Pompeiu's formula ap peals to Stokes's Theorem , which is derivable (v. [Lan] , [Spi] ) in the current context by methods related those used to reach, e.g., 5.3.11. Although Stokes ' s Theorem implies 5.3.11 and others (v. [Ho] ), the theory of differential forms [Lan, SiT, Spi] lies at the heart of the matter.
o
On the other hand, in the current treatment, appeal is made to Pompeiu's formula in only the simplest instances, e.g., 7. 1.16, where the validity of Stokes's Theorem is directly demonstrable, v. 5.9.15.]
5.8.18 Exercise. a) du * dv = dv * du. b) l1 = d (* d) . c) If Q I C Q, Q I is relatively compact in Q, and aQ I is a rectifiable curve, e.g., if r > 0 and Q I = D(a, rt C Q, there emerges Green's formula
r
ian,
u * dv  v * du =
r
in,
(u l1v  v l1v) dx A dy.
[Hint: Stokes's Theorem applies.]
Section 5.9. Miscellaneous Exercises
265
5 . 9 . Miscellaneous Exercises
5.9 . 1 Exercise. If {fn} nEN
then J E H
(1U).
12".
nlim teXJ 0
C
H
(1U), J E C (1U, C) , and for each r in (0, 1),
I Jn (re i O )  J (rei O ) I dO = 0,
[Hint: Cauchy's integral formula implies that if z E
1U, then
I (z) = J(z). nlim + = n Vitali's Theorem 5.3.60 applies.]
1Uc ,
5.9.2 Exercise. If Q � a E '][', J E H (Q \ {a}), a E P(f), Ord f (a) = = and in J(z) = L cn z n , then nlim � = a. + = Cn+l n=O [Hint: The CauchyHadamard Theorem applies to (z  a)J(z).]
1,
1U,
5.9.3 Exercise. If J E H [D(O, 2r] and the order of each zero of J in D(0, 2r is one, what are the zeros of J in if
1U
r I'(z) dz J(z )
J.lf
_ 
r Z I'(z) dz J(z )
J.lf
_ 
rZ
J.lf
2
J'(z) _ dz J(z )

.?
7n .
5.9.4 Exercise. If J E H(Q) and a E Q, the radius oj convergence ra of J C nl (a) the power series '" (z  a) n is at least inf { l a  w i : w E aQ } . � n! n=O 5.9.5 Exercise. Every region Q contains a maximal simply connected subregion, i.e., a simply connected subregion Q 1 that is properly contained in no simply connected subregion of Q. [Hint: If P is the nonempty set of simply connected subregions of Q, P is a poset with respect to {A < B} {:} {A C B } . Zorn's Lemma applies.] CXJ
5.9.6 Exercise. For some region Q , there is no unique maximal simply connected subregion of Q. [Hint: The region Q �f \ {O} merits attention.]
1U
5.9.7 Exercise. If K is compact, U is open, and K c U e Q, there are constants { Crn } rnEZ+ such that for each J in H (Q),
Chapter 5. Locally Holomorphic Functions
266
[Hint: If '1jJ is infinitely differentiable, vanishes off a compact subset of U, and '1jJ == in an open set V containing K, then
1
Pompeiu's formula yields '1jJ(w) f(w) =
0
f(z)8'1jJ(z) J 27r� z  w dz
�
u
1\ az .
j. 0,
(5.9.8)
Since 8'1jJ = in V, if z is such that 8'1jJ for some positive E, and 5.8.13 Iz  wi 2': E whenever w E K. Differentiation of yield the result.]
�f
(5.9.8)
5.9.9 Exercise. For the operators 8 and 8 and a differentiable function f P + iQ, there obtain the following formulre.
8f dx + dy = 8f dz + 8f az, 8x 8y 8 U 0 g) = [(81) 0 g] . 8g + [(8f )] · 8g, 8 U 0 g ) = [(81 ) 0 g] . 8g + [(8f )] . ag. 8f
[Hint: If f(z)
�f f(x, y) and g(z) �f p(x, y) + iq(x y), then ,
f 0 g(z) = f[P(x, y), q(x, y )].]
5.9.10 Exercise. a) If r > and f E H [ ( , r r ] there r) , � f. r) , of polynomial functions such that on is a sequence b) If Q lr \ and f(z) �, then f E H (Q) but no sequence z of polynomial functions converges to f uniformly on compact subsets of Q. [Hint: a) Fejer's Theorem (3.7.7e) ) and the Maximum Modulus Theorem (5.3.36) apply.]
0 {Pn} nEN �f D(O, {O}
{ Da
�f
n C (D(a, C] }, D( a, Pn
5.9.11 Exercise. In 5.3.43 a satisfactory value for p is:
5.9.12 Exercise. A net of functions is locally uniformly convergent resp. locally uniformly Cauchy resp. locally uniformly bounded on an open set U iff U is the union of open sets on each of which the net is uniformly convergent resp. uniformly Cauchy resp. uniformly bounded.
Section 5.9. Miscellaneous Exercises
267
5.9. 13 Exercise. (Montel) A subset F of H(Q) is normal iff F is locally bounded, i.e., iff for each compact subset K of Q and some in JR.,
MK
xEK JEF
sup I f(x) 1 :s;
MK .
[Hint: 1.7.21 applies.] 5.9. 14 Exercise. If: a) l'(t) = e 2 7ri t , O :S; t :s; 1; b) for some ¢ in C 1 (D(0, 1), q , ¢[D(O, 1) ]
�f QC and for l' as in a) ,
and ¢ 0 l' is a Jordan curve such that aQ = (¢ 0 1')* ; c) f and g in Q 3 (x, y) r+ [J(x, y), g(x, y) ] E JR.2 are continuously differ entiable Stokes's Theorem in the form (5.8.6) is valid.
5.9. 15 Exercise. If {u, v } c C OO (1U, JR.) , det
[��::�n
>
0, and
Q cle=f { [u(x, y) , v(x, y) ] : 0.5 < x, y < 0.5 } ,
r W ( l ) = r dw ( l ) . ln lo(n) [Hint: Both 5.9. 15 and the discussion in 5.8.5 apply.] [ 5.9.16 Remark. The result 5.9.16 provides extended circum stances where Stokes's Theorem applies. An argument based on patching together squares like Q �f { (x, y ) : 0.5 < x, y < 0.5 } and their images under maps like
then Q is relatively compact and for any Iform w ( l ) ,
T : Q 3 (x, y) r+ [u(x , y), v(x, y)] E Q C JR.2 above leads to very general forms of Stokes's Theorem [Lan, Spi] . If ),2 ( Q ) = 00,
In dw ( l ) need not make sense.
For example, if
I }, then ), 2 (Q) = 00 (and Q is not relatively com pact). For the Iform w ( l ) �f y dx + x dy, dw ( l ) = 2 dx 1\ dy and
Q �f { z : I z l
>
r
lo(n)
w ( l ) = 27r
j. lnr dw ( l ) = 00.]
Chapter 5. Locally Holomorphic Functions
268
( � ) ( � ) : such that g ( � ) = g ( � )
5.9. 17 Exercise. a) There is a function I in H (D(O, It) and such that =I = 2 ' b) There is no function g in H (D(O, 1) 0 ) and I =
:3 '
5.9.18 Exercise. If I E H (fl) and 5 is a constant or Z(f) n 5 ° j. 0.
f
� {z :
1
I I(z) l :S; I } c fl, either I
�

I Z) dz exists for a (z b) (z I z l=r all large r. b) If I I I is also bounded, then for each a, J'(a) = ° (a second 5.9. 19 Exercise. If I is entire, then: a)
proof of Liouville's Theorem.)
5.9.20 Exercise.
If
00
L an zn converges in 1U and:
n =Q
the recursion formula for the coefficients in
a) ao
00
j.
0, what is
L bn zn that represents the
n=O
17 b) a l j. 0, what is the recursion formula for the coefficients in L cn z n that represents the n= O ( inverse) function h such that near 0, h 0 I(z) z7 (reciprocal) function g such that near 0, g . I(z) 00
==
==
5.9.21 Exercise. If I(x, y) f u(x, y) + iv(x, y) and both . for some nonnegatIve . R, • ble at z clef · U" = x + zy, dluerentm I(z + h I(z) : h j. ° = 8[ D( a , R) ] .
{
�
�
u
and v are
r
o = pe ' , p > 0, ¢ fixed, the calculation [Hint: For z clef = re' and h clef I(z + h) I(z) as + ° applies.] p of h

°0
(10,z)
5.9.22 Exercise. HI � Z. [Hint: For k in Z and "Ik : [0 , 1 ] 3 t r+ ( 0.S ) e 2 k".it , "Ik = k::;i . If is a rectifiable loop and C 10 then for some k in Z, Ind 1J (O) k. then I is representable by a Laurent series.] If I E H
(10),
5.9.23 Exercise. a) If "Il (t) f 1 + e 2 7ri ( l  t ) , "I2 ( t )
and r
f
�
� ::;i  ,.y;, then r s: U
1]
1]*
=
f
f
� 1 + e27rit , fl � C \ ({I} U {I}),
1] 1 and 1]2 , 1 clef= 1]1 . "11 . 1]1 1 and u2 clef= 1]2 . "12 . 1]2 1 �n
O. b) For some curves s:
Section 5.9. Miscellaneous Exercises
269
are loops and 8 1 (0) = 82 ( 0 ). c) {
1
I l
and
G( z ) =
{ 1I
n+ =
1 zn  1

if I z l < if I z l >
1. 1
ex:> 1 1, then the Dirichlet series '" � nz converges n= 1 and defines a function f holomorphic in Q �f { z : �(z) > 1 } . 5.9.26 Exercise. If �(z)
>

[ 5.9.27 Remark. Riemann's zeta function ( is defined and holo morphic in C \ {I} and ( I n = f. Furthermore, P(() = {I} and Ord «(l) = It is known that Z(() C { a + it a .:s: I } and that Z ( ( ) is symmetric with respect to both a + it a =
1.
{ : O .:s: Riemann conjectured that Z (( ) C { a + it
:
:
�} a = � }.
and JR.. As of this writing, his conjecture remains unresolved, despite the efforts of some of the greatest analysts since Riemann's time. Rie mann's zeta function is of central importance in number theory, particularly in the study of the distribution of prime numbers, i.e., the cardinality 7r ( x ) of the set of prime natural numbers n such that n .:s: x. In J. Hadamard and C.J. de la Vallee Poussin, using properties of (, independently proved . THE PRIME NUMBER THEOREM. x+ hm (7r (xx ) = = ln x
1896,
) 1.

1948,
In P. Erdos and A. Selberg [Sha] proved the Prime Number Theorem without recourse to the methods of complex analysis.]
6
Harmonic Functions
6.1. Basic Properties
The subject of harmonic functions appears in 5.3.57. The conclusion to be drawn from 5.3.59 is that for some regions Q and some u in Ha lR (Q) , there is in Ha lR (Q) no function v that serves as a harmonic conjugate to u throughout the region Q. The next results explore other possibilities.
6.1.1 Exercise. a) If I �f u + iv E H(Q) , then
I E Ha(Q) and {u, v} C Ha lR (Q). b ) There are in Ha lR( Q ) functions u and v such that u + iv tt H(Q) . 6.1.2 LEMMA. a) IF u E Ha(Q) AND D(a, rr C Q, THEN THERE IS A v SUCH THAT I u + iv E H [D(a, rr] ' i.e., CONFINED TO D(a, rr , v IS A HARMONIC CONJUGATE OF u. b) IF, CONFINED TO D(a, rr, V I AND v2 ARE HARMONIC CONJUGATES OF u, THEN FOR SOME REAL CONSTANT C, V I  v2 = C. PROOF. a) If u E HaIR (D(a, rr) and a �f a + i(3, direct calculation, in view of the existence of Uxx and U yy and the validity of t:m = shows that x U y ( s , (3) ds is such the function v : D(a, rr 3 x + iy r+ ux (x, t) dt 
�
iY
I
0,
that T : D(a, rr 3 (x, y) r+ [u(x, y) , v(x, y)l has a derivative and further more that Ux = Vy and u y = vx . Thus I u + iv E H [D(a, rrl . If u = �(u) + i<S(u) �f p + i q E Ha [D(a, rt] ' then
�
{p, q} c HaIR [D(a, rn . The previous argument implies that for some � and 1] in HaIR (D(a, rr), {p + i�, q + i1]} C H [D(a, rt] ' whence p + iq + (i�  1]) �f u + iv E H [D(a, rtl . b) If Ij �f u + iVj E H [D(a, rtl , j = 1, 2, then i (II  h ) is JR.valued in ( D(a, rt) and the CauchyRiemann equations imply that for some real constant c, i (II  h) = c, i.e., V I  V2 = C. 0 270
Section 6.1. Basic Properties
271
6.1.3 COROLLARY. IF u E Ha lR (Q) , a �f a + i(3, AND D ( a , rr c Q: a) THERE IS A SEQUENCE {PTnn } :, n=o OF CONSTANTS SUCH THAT IN D( a, r r , u ( x, y ) =
00 , 00
L
PTnn (X  a ) Tn (y  (3) n ; b) THERE ARE SEQUENCES {Cn } nE Z '
Tn,n=O { an } nE Z+ ' AND {bn } nE Z + SUCH THAT IF 0 :::; R < r, THEN u (a + Re i O ) =
00
00
L
cn Rn e in O = L an Rn cos n O + bn Rn sin nO. n==  (X) n=O o
PROOF. The argument in the Hint following 5.3.25 applies. 6.1.4 Exercise. If r > 0 and u E HaIR [D ( a , rr ] ' for some v, v + iu E H [D ( a , rr ] .
6.1.5 LEMMA. IF u E Ha(Q) , D ( a , rr c Q, AND 0 :::; R < r, THEN
1 1 2". u (a + ReiO ) dO,
27r 0
(6.1.6)
u( a ) = 
i.e., u ENJOYS THE Mean Value Property MVP AT EACH POINT OF Q. [ 6.1. 7 Remark. Customarily the symbol MVP(Q) is reserved for the set of functions continuous in Q and enjoying the Mean Value Property at each point of Q; there is a corresponding meaning for MVPIR(Q). Thus 6.1.5 may be viewed as the assertion: Ha(Q) c MVP(Q) . The reversed inclusion is the burden of 6.2.16 below.] PROOF. For the harmonic conjugate v that serves in D ( a , rr ,
f �f u + iv E H [D ( a , rr] . o
Cauchy's formula applies.
6.1.8 THEOREM. (Maximum Principle) IF u E MVPIR (Q) , a E Q, AND
(6.1.9)
{
u( a ) 2': sup u ( x, y ) : ( x, y ) E Q } ,
THEN u IS A CONSTANT FUNCTION. PROOF. The MVP asserts that if D(a, R) C Q, the value of u at the center of D( a, R ) is the average of its values on 8[D( a, R)] . Hence, if m R resp. MR are the minimum resp. maximum of u on 8[D ( a, R)] , then implies
(6.1.6)
Chapter 6. Harmonic Functions
272 m R :::;
(6.1.9)
u(a) :::; MR and implies that for all R as defined, u(a) = MR . :::; R2 and u (xo, Yo) < u(a ) , then
If Xo2 + Yo2 clef = s2
{ (x, y) : X2 + y2 = S 2 , U(X, y) < u(a) } is a nonempty open subset of 8[D(a, s) ] . Thus
1 1 2". u (a + seiO ) de < u(a),
27r 0
u(a) = 
a contradiction. Hence, if u is not constant, for any a in fl, u(a) cannot be a local maximum value of u: is denied. 0 6.1.10 Exercise. If u E HaIR [D( a, r] n C[D(O, ) C] , then is valid when R = [Hint: When R < 6.1.5 applies. Passage to the limit as R t is justified by the Dominated Convergence Theorem (2.1. 15) and 6.1.8.]
(6.1.9)
r.
r
r
,
(6.1.6)
r,
r
6.1.11 THEOREM. (Maximum Principle in Coo ) IF fl c C, u E M V P IR(fl) , AND
{
u(x, y) sup {(a, b) E 8oo ( fl ) } '* N a b inf [( , ) ] EN[ ( a , b ) ] ( x , y) E N [( a , b) ]nn
:::;
THEN u 0 OR u(fl) c (  00 0) PROOF. If the result is false there are two possibilities: a) for some (xo , Yo) in fl, u (xo , Yo) > 0; b) for some (xo , Yo) in fl, u (xo, Yo) = 0 while u(fl) C (00, 0] . If a) is true, for some positive E, =
,
o} ,
.
If K is unbounded, then 00 E 8oo (fl) and for a sequence {(xn , Yn ) } nEN in K, (xn , Yn ) + 00 as n + 00. Hence lim u(x, y) 2': E > 0, a contradiction of ( x , y) + oo the hypothesis. Hence K is bounded and since u is continuous, K is closed: K is compact. Thus, for some (p, q ) in K, u(p, q) = max { u(x, y) : (x, y) E K } 2': E. If (x, y) E fl \ K, then u(x, y) < E , whence
u(p, q) = max { u(x, y) : (x, y) E fl } ,
and 6.1.8 implies u = u(p, q ) 2': E, a contradiction of the hypothesis.
in a Disc
Section 6.2. Functions Harmonic
If b) is true, 6.1.8 implies
u
273
u == o.
o
6.1.12 Exercise. If { , v} c MVPIR( fl) and for each point (a, b) in 8oo (fl) sup
inf
N [ ( a , b ) ] EN[ ( a , b) ] ( x , y ) E N [( a,b ) ] nn � on fl, either
u(x, y)
sup
inf
N [ ( a , b) ] EN[( a , b) ] ( x , y) EN [ ( a , b) ]nn
u < v or u == v.
[Hint: 6.1.11 applies to
u  v .]
3x
x
v( , y),
r+ x
6.1.13 Example. a) Although f : C is in Ha lR (C) , f 2 is not + iy 2 harmonic. a) both g : C z e Z and g are harmonic. c ) The map h:C + i y y is in Ha lR (C) ; h 2 is not harmonic but f 2  h 2 E Ha lR (C) .
3x
3 r+
r+
6.1.14 Exercise. If f E H (fl) and I f I E Ha lR (fl), then f is a constant function. [Hint: Since bo l f l = if f + iv, then Ux + Vx = U y + Vy = o. The CauchyRiemann equations imply ! , 0.]
�f u
0,
==
u
6.1.15 Exercise. If f E H (fl) and E Ha [J(fl)], then [Hint: If a E fl , for some positive r and positive s ,
u
0
f E Ha( fl ) .
D(f(a), s t C J [D(a, rn c f(fl), and 6.1.2 implies for some v, g g o f E H ( fl).]
�f u + iv E H {[D(f(a), st] }, i.e.,
6.2. Functions Harmonic in a Disc
00
00
1 = ( 1 + z) � '" z = 1 + 2 � '" z is in H (1U). 1U 3 z r+ � 1z Hence 'iR(f) E HaIR (1U). Customarily, polar coordinates are used to discuss 'iR(f) in 1U, and when z = re i O , 0 :::; r < 1, fJ E JR.,
The function f :
n
n
n= l
n=O
1  r2 2 sin fJ 1 + re iO ,."... ;,f ( z ) = 1 reiO = 1  2r cos fJ r2 + i .... 1  2r cos fJ + r2 ' + l re i O 1  r2 'iR + i = Pr(fJ) . 1  2r cos fJ + r 2 1  re O
{
The map
}
�f
1U 3 (r, fJ) r+ Pr (fJ) is the Poisson kernel.
Chapter 6. Harmonic Functions
274
6.2. 1 Exercise.
00
n=  (X)
�f
0
6.2.2 LEMMA. IF :::; r < R AND Ca(R) { z : I z  a l = R }, FOR THE COMPLEX MEASURE SPACE (Ca(R), 5(3 [Ca(R)] , p,) ,
h : D(a, r t
3 re iO
1
r+
Ca r r)
IS IN Ha [D(a, Rt] . PROOF. Since PI (8  t) R
=
�f
Pfi ( 8  t) dp, (a + Re it )
{ �},
�f h(r, 8)
eit + PI is in HaIR (D(O, Rt). Be. re ' e,t  R a + i(3, when z = re iO , h is a complex linear
�
R
cause p, = �(p,) + iSS(p,) combination of the real and imaginary parts of
cl f r27r
.
r27r
. Re it + z Re it z dO' (a + Re't ) + i J Re i + d(3 (a + Re't ) . I(z) � J Rei t z t z o o By virtue of 5.3.25, each integral above represents a function holomorphic in D(a, Rt · 0 [ 6.2.3 Remark. On Ca( R) arc length may be used as a basis for a measure space [Ca(R), 5(3 [Ca(R)] , �] that is the analog of T in 4.5.2. The group '][' acts on Ca (R) according to the rule '][' x Ca (R) {e iO , z } r+ a + e iO (z  a). _
_
3
With respect to the action of '][' on Ca(R), � is actioninvariant : if E E 5 (3 [Ca(R)] and :::; 8 < 271", then
0
a + e io (E  a) E 5(3 [Ca (R)] , � [a + e i o (E  a)] = �(E) .
clef dp,
If p, « � and g = � ' by abuse of language, 6.2.2 says the convolution Pfi g is harmonic. *
When k defined on Ca(r) is such that
r27r
Jo Pfi (8  t)k (a + re it ) dt
Section 6.2. Functions Harmonic in
a Disc
275
exists, the result is the Poisson transform of k and P(k) . It is a harmonic function defined in D ( a , r r . The role of the family {PI.. } R
0< _ 1'<
R
IS
denoted
is discussed in Section 6.4.]
6.2.4 Exercise. The following are alternative formulre for PIi ((J  t) when z = Re iO and a = re it :
(
)
z+a R2  1 a 1 2 PI.. ((J _ t) = 1R Za I z  al 2 2 2 R r R2  2rR cos((J  t) + r2 ' R
=
_
(1U) n C [(1U)C, C] AND, FOR = ret"0 , 0 :s; r < 1, 0 :s; (J < 27r, z clef
6.2.5 THEOREM. IF f E H
00
00
THE REPRESENTATION f(z) = L Cn Z n OBTAINS THEN:a ) L I cn l 2 < 00; b) n=O n=O ( POISSON ' S FORMULA )
PROOF. a) Since f l T E L2 ('][', T ) and { e i nt } n EZ consists of pairwise orthogo nal functions of absolute value one, Bessel's inequality (v. 3.2.14c)) implies
Cauchy's formula for the Cn applies. b) In particular, the argument in a) implies
(6.2.6) According to Cauchy's formula, if I z l <
1, then (6.2.7) (6.2.8)
Chapter 6. Harmonic Functions
276
(6.2.7) and (6.2.8) imply � 2 ". 1 (e it ) eitz dt, � 2". 1 (eit ) . dt I (z ) + 1  e,tz. 27r 1  e  ,tz 27r 2 1 + re i( t  O) ) = 1 r ". 1 (e'. t ) ( 27r Jo 1  rei ( Or) 1 reit  O) � r2". (} = 27r Jo Pr(  t)1 (eit ) dt. 6.2.9 THEOREM. IF u E HaIR [D(O, Rn n C[D ( O, R), C] AND l a l < R, r 2". R2  l a 1 2 U (Re i  a l 2 Then
1
1
_
0
0

_
c
o
+ a) cle=f z, (1U) 3 ( I(() cl=ef R(R( R + a( 1 (1I') = Ca ( R ) , 1 [(1U)C] = D(O, R), and 1 (0) = a. Thus u o l E HaIR (1U) n C [(1U)C, C] ( 6.1.15) and 6.1.10 implies that if PROOF. For the map I :
(
=
r+
v.
ei O and z = 1 (eiO ) �f Re i , then
. (= Smce
d()
R(z  a) and R2  az
. d( , d() = l(
( z � a + R2a� az ) d¢ = ( R::i� a + R2a�:;ei ) d¢ 2 2 = Rz  l a2l d¢ ( = � (�) d¢) za I  al
=
.
D(O, ¢)u
0
6.2.10 Exercise. a ) In the context of 6.2.9 with Rt replaced by 2 ". . D(a, Rt , if r < R, ( a + re' O ) = Pf; ((} b) If (a + Re '
D(
u
.
( ) 0,
1 2 D ( 7r
1 0
d¢.
(6.2.11)
Section 6.2. Functions Harmonic in
a Disc
277
[Hint: For some {rn } nEN and all fJ, 0 < rn t r and f (rn e iO ) j. O. Poisson's formula applies.] [ 6.2.12 Remark. The result (6.2.11) is the PoissonJensen for mula.]
In Section 6.4, the properties a )c) listed below for the Poisson kernel are explored in a general context. Analogies are drawn between Fejer's kernels (3.7.6) and approximate identities, which play an important role in topological algebras.
6.2.13 THEOREM. FOR THE POISSON KERNELS THE FOLLOWING OBTAIN: a) r fJ ) > 0, < fJ E JR. ; 2". b) IF < THEN r fJ ) fJ = 0 c) IF 0 < fJ < THEN lim Pr (fJ) = 0 AND THE LIMIT IS UNIFORM IF rtl o < r5 :s; fJ :s; r5.
P( O :S; r 1, 11 O :S; r 1, 27r P ( d 1; 27r, 27r PROOF. a) Since 1  2r cos fJ + r2 = (1  r ) 2 + 4r sin2 �, Pr(fJ) is the quo tient of two positive functions. b) The result in 6.2.5b) applies when f ( ) 1. 7r ' then 0 < 1  r2 < 2 ( 1  r) and, owmg to c) If 0 :s; r < 1 and I fJl :s; "2 2 Jordan's inequality (5.7.2), (1  r ) 2 + 4r sin2 � > (1 _ r ) 2 + 4r� . Thus 2 7r z
==
•
7r
if "2 < IfJl <
7r
[ 6.2. 14 Note. If sin � = 0, e.g., if fJ = 0 or fJ = 27r , Pr (fJ ) t oo as r t 1 .] 6.2.15 COROLLARY. IF
THEN
u
f E C (1I', q AND
IS THE UNIQUE FUNCTION SUCH THAT u
E Ha
(1U)
AND
u
(r, fJ) � f (e iO )
r 1.
AS t
o
Chapter 6. Harmonic Functions
278
PROOF. By virtue of 6.2.2, u E Ha Furthermore, 6.2.13b) implies
(1U).
II (eiO)  (r, O ) 1 = i 2� 127r {I [eiO]  I [ei (O  t) ] } Pr(t) dt i � ('s � 1 <  27r Jo + 27r 27r + � [eiO]  I [ei ( O t ) ] } Pr(t) 1 dt 2 u
8 1:�8 1{ 1
8
�f I + II + III.
0 r5 :s; t :s; 27r r5, r 1, negativity of Pr imply that if r5 is small, I + III :S; E � r27r Pr(t) dt = E. 27r Jo
If <  since I is continuous, 6. 2. 13c) implies, that for pos itive E and near and below II < E. The continuity of I and the nonThe uniqueness of u is a consequence of the Maximum Principle. 0 The preceding developments lead to the following characterization of harmonic functions.
6.2.16 THEOREM. MVP(Q) = Ha(Q). PROOF. According to 6.1.5, Ha(Q) c MVP(Q). If I E MVP(Q), C Q, < and
D(a, R)
0 :s; r R,
(D(a, Rt). Furthermore, 6.2.15 implies F ( a + Re it ) = I ( a + Re it ) , i.e., 'iR(F  f) and 'J(F  I ) are in MVPIR (D( a, Rt) and on 8[D( a, R)] each is zero. The Maximum Principle implies F  I = 0 on D( a, R ). 0 then 6.2.2 implies F E Ha
6.2.17 LEMMA. IF �
Q1 = Q U
(a,
a
=a ]
Q c n+ , < b, 8(Q) n ffi. [ , b , b) U Q, I E H(Q) n C[Q U b), q , AND
(a,
CS(f) I (a, b) = 0,
IN H (Q I ) THERE IS A UNIQUE II SUCH THAT II I nu ( a b) = I· , [ 6.2.18 Remark. The result 6.2. 17 is a version of the A variation on the theme is explored in Chapter 8, where the geometry of reflections and inversions in
Reflection Principle.
Schwarz
Section 6.2. Functions Harmonic in
a Disc
279
circles is discussed, and in Chapter 10, where the relevance of the Schwarz Reflection Principle to the process of analytic contin uation is illuminated.
]
PROOF. The Identity Theorem (5.3.52) implies there is at most one func tion h as described. If
h (z) =
{
b
I(z ) if z E Q u (a , ) _
1 ("2 ) if z E n
then h E C (Q I , q n H ( Q u n) . If a <
for large n,
1
r
Ja(Sn, ± )
Iz x l =r
x < b, r > 0, D (x , rt c Q I , and
h (z) dz = 0 and
h (z) dz = }�+�
(Jra(Sn,+) + Jra(Sn,_ ) h (Z) dZ)
=
O.
Morera's Theorem (5.3.48) applies. 6.2.19 Exercise. For Q in 6.2. 17, if E (a,
D(x, rt n Q
=
x b), for some positive r, D(x, rt n n + .
o
(The preceding positive statement is an implicit ingredient of the PROOF above. ) [ 6.2.20 Note. The conditions
lR. = [a, b] and 1 E H(Q) n C[Q u (a, b), C] are intended to preclude the possibility that [a, b] is, for I, part of 8(Q) n
its natural boundary (v. Hint following 7.1.28).]
The interplay between the theory of locally holomorphic functions and the theory of harmonic functions emerges in the next result, which is a considerable strengthening of 6.2.17.
b,
6.2.21 THEOREM. FOR Q, a, AND Q I AS IN 6.2.17, THE CONCLU AND SION OF 6.2.17 IS VALID IF 1 E H (Q), 2s(f) � v E C[Q u v l a,b = O. ( )
(a, b), lR.] ,
280
Chapter 6. Harmonic Functions
x (a, b), for some positive r, D(x, rt c Q l . If
PROOF. If E
v( z) �
{ v(v o
(z) if z E Q
z)
if z E Q , if z E b)
(a,
the integral formula that defines the Mean Value Property implies that v E MVPIR (Q l ), whence v E Ha IR (Q 1 ). Hence, if < < b, for some nonempty there is a u that is a harmonic conjugate of v in u + iv E H Moreover, 1 (u + iv) is JR.valued in n + , whence, for some real constant (u + iv) == C in C, 1 n + . Thus u + c is also a harmonic conjugate of v. Hence hx u + c + iv is holomorphic in and coincides Since hx is JR.valued on with 1 in if, for z in hx(z) = L cn ( z  x) n , each Cn is real. Hence in
a x
D(x, rt D(x, rt: (D(x, rt). D(x, rt n D(x, rt n �f D(x, rt D(x, rt n JR., D(x, rt n n + . D(x, rt, D(x, rt, hx (z) = L (z  xt = hx (z) . If a < y < b and D(x, rt n D(y, st j. 0, then h x = hy in D(x, rt n D(y, st n Q and thus, by virtue of the Identity Theorem (5.3.52), h x = hy in D(x, rt n D(y, st. Direct calculations

00
00
n=O
Cn
n=O
show that if
h (z )
{�f
I(z)
if z E Q
h x(z) if z E
D(x, rt ,
1 (z) if z E Q then h is consistently defined and meets the requirements. 6.2.22 Note. The continuity of I on Q u b) is not part of the hypothesis of 6.2.21.]
[
(a,
0
6.2.23 Exercise. a) (Schwarz's formula) If
[D(a, Rt] n C[D(a, R) , JR.] , w + z h(w) dw E H D l . then I(z) de =f Rt . b) (Harnack's mequa[ ] (O, w z w Iwl = R lities) If h E Ha IR [D( a, Rt] n C[D( a, R), JR.] and h[D( a, R)] C JR.+ , for z in D(a, Rt, h(a) � � I:: S h(z) S h(a) � � ::: . [Hint: Poisson's formula for h( a) applies to a), and a) applies to h E Ha IR
b).] An echo of the convergence phenomena that obtain for holomorphic functions is found in
Section 6.2. Functions Harmonic in
a Disc
281
{Un} nEN Ha(fl): ) ) fl, U E Ha(fl); ) un U a fl, {u n (a)} nEN {Un}nEN HaJR (fl ) , U n :::; U n+ l , E fl U, u n U U E HaJR(fl)). U E [ a, r)] ) D ( a , r) fl U E Ha (fl). vn �f U n  U l nEN ) {vn}nEN ' v, Vn v ) Vn :::; Vn+ l , E ) ) {vn}nEN " {un} nEN : D( c, R) fl, :::; r < R, :::; < R+ r. Rr R + r < PI.. (fJ) < Rr
6.2.24 THEOREM. (Harnack FOR A SEQUENCE IN a IF THEN b IF � ON EACH COMPACT SUBSET OF C n N, AND FOR SOME IN IS BOUNDED, FOR SOME � ON EACH COMPACT SUBSET OF ( WHENCE PROOF. a If C Poisson's formula implies and MVP D ( so } consists of nonnegative functions b The sequence { and t and b n N. If b obtains for for some obtains for it suffices to establish b for If C 0 and 0 fJ 27r, then R
Harnack's inequalities, 6.2.23b), and MVP imply
Rr R+r vn( R + r c) :::; Vn (C + re ie ) :::; R  r vn ( c), . R + r c). Rr v( R + r c) :::; v (c + rete ) :::; v( Rr The left sides of Harnacks's inequalities imply that if c E D( c, Rt e fl and lim= vn( c) = 00, for all z in D(c, Rt, v(z) = 00. Similarly, the right sides n+ of Harnack's inequalities imply that if n+ lim= vn( c) < 00, for all z in D( c, Rt, v(z) < 00. Those conclusions and the boundedness of {u n (a)} nEN imply 0 ¥ S �f { Z : z E fl, v(z) < OO } E O(C), T �f { z : z E fl, v(z) = oo } E O(C). Since fl = S u T, S n T = 0, and fl is connected, T = 0: v E MVPJR (fl ) = HaJR(fl) C C(fl, JR.). Dini's Theorem (1.2.46) implies Vn � v on each compact subset of fl. o 6.2.25 Exercise. For U in HaJR [D(a + ib, rt]: a ) if { m , n } C N, then a=+ nn u exists, is harmonic, and there is a power series a=xa y = L cpq (x  a)P (y  b) q p , q=O
Chapter 6. Harmonic Functions
282
D(a + ib, rt to 8rn+ nn u(x, y); 8rn x8 y
converging uniformly on compact subsets of
b ) (Identity Theorem ) if
SC
D(a + ib, rt, S· n D(a + ib, rt j. 0, and u l s = 0,
then u I D ( a+i b , r) o = O. Hint: 6.1.2 applies. ]
[
6.2.26 Exercise. If
{un}nEN c HaIR [D(a, rt] , I z al
[Hint: Both 6.1.2 and 5.3.60 applY. ]
10 z
I(z) �f I z l ,
6.2.27 Exercise. a) In 6.2.15, if then In function 3 r+ In is harmonic and nonconstant.
I zl
u
==
O. b ) The
u E HaIR (10) AND u IS BOUNDED IN 10, FOR SOME u. PROOF. If E JR., in H [D(0.5e is , 0.5tl �f H ( fls ) there is an Is such that Is = u + ivs. If flSIS2 �f flsi n flS2 j. 0, then � (fs l  IS2 ) I ns 1 s2 = 0: i.e., (fsl  IS2 ) I n8 1 82 is a constant. Thus, for any z in 10, 6.2.28 THEOREM. IF U IN Ha lR (1U) , U l u)= s
g(z) = L an zn . n=  (X) The region fl �f 10 \ ( 1, 0) is simply connected. For z in fl, there is a rectifiable curve "( : [0, 1] 3 t fl such that "((0) = 0.5, "((I) = z. If G(z) cl=ef 1 g(w) dw,
and g is represented by a Laurent series: r+
'Y
00
Section 6.2. Functions Harmonic in
a Disc
283
then G (z) is independent of the choice of "I and G E H (Q). For any G'  1� l ns = 0, whence G  Is l n s is a constant and
s,
�(G) = � ( f ) s + constant = u + constant. = 0' + l. (3 and Q 3 z clef If a � l clef = re '"0 , 0 < r < 1, 1 0 1 < 7r, then
� [G(z)l
u(z) + constant = O'(ln r  ln O.5)  (30 � (an ) cos(n + 1) 0 _ (an ) + '"' sin(n + 1) 0 rn+ 1 n + 1 n + 1 n EZ\{�l} =
[
�
]
'S
If r is fixed and (3 i 0, then lim u(z) i lim u(z), a contradiction since u is O f". Ol". continuous in 10. Hence (3 = 0, and there are constants Pn , qn , n E Z, such that u(z) = O' ln r +
L (Pn cos nO + qn sin nO) rn ,
n EZ
M in JR., lui :::; M and if n .:s:  1 and P� n i 0, then 7r [2 (Po + In r) ± (Pn rn + P� n r � n ) 1 :::; 47r M, ( IP�n l r � n + 20' In r) + 2po ± Pnrn �f I + II .:s: 4M. i 0, then lim I = 00, while II remains bounded, a contradiction: = O. dO Thus H �f e G E H (10), H = constant · eUh, and I h l 1. Since u is
For some
a
If a
a
==
bounded in 10, so is I H I , and 5.� .11 implies I G I is bounded. Hence 0 is a removable singularity of G. If G denotes the holomorphic extension to 1U f of G, then � u is harmonic and an extension to 1U of u. 0
(C) �
[ 6.2.29 No� e. The argument showing that u is the real part of a function G holomorphic near 0 depends on the fact that the
original domain of u is a punctured disc. For example, Riemann's Mapping Theorem (8.1.1) implies that there is a bijective holomorphic map I : n + r+ Q f { z : 1 � (z) 1 < 1, "5 (z) E JR. } : in n + , �(f) is bounded but III is not.l
�
Chapter 6. Harmonic Functions
284
6 . 3 . Subharmonic Functions and Dirichlet ' s Problem
The result 6.2.16 is related to a simpler situation for functions of a single real variable. The analog of the equation t:m = 0 for functions u : r+ is = 0 for functions r+ R Every such must be ( continuous and ) linear, i.e., of the form : 3 r+ b. Furthermore,
f"
f : JR. f f JR. x ax + f ( x ; y ) [J(X) ; f( Y )] ,
JR.2 JR.
==
an identity that can be construed as expressing the MVP in one dimension. Conversely, if is continuous,
f
g(x + y) g(x) + g(y). Hence g(O) = 0, g( x) = g(x), g(nx) = ng(x), n E Z, g (rx) = rg(x), r E Q, g(tx) = tg(x), t E JR., g(t) = tg(l), f(x) = g(x)  f(O) = g(l)x  f(O) �f ax + b. . f ( x + y ) [J(x) + f(y)] ' 1' f S·Iml1ar1y, when f IS' dpmnt convex, l.e., 2 � 2 k {a, ,B} C { t : t = 2n ' { k , n} e N } and a + ,B = 1, then f(ax + ,By) :::; af(x) + ,Bf(y) · If, to boot, f is continuous, then f is convex [Ge3]. A function f in C (JR., JR.) is convex iff either of the following obtains. a) The value of f at the midpoint of every interval does not exceed the average of the values of f on the boundary of the x y [J(x) ; f(y)] . interval: f ( ; ) :::; b ) For every interval [p, q], if y is a solution of the equation y" = 0 and y(p) = f(p), y(q) = f(q), i.e., the boundary values of f and y coincide, then for x on [p, q], f(x) :::; y(x). When Q c C, for a continuous function : Q JR., the following mod ifications provide analogs of ) and b) above. Intervals [p, q] are replaced g
==
then is continuous and
'
mz '
'
a
u
r+
Section 6.3. Subharmonic Functions and Dirichlet's Problem
285
0
by closed subdiscs of fl. The differential equation y" = is replaced the Laplacian equation t:m = Averages of the boundary values on an interval are replaced by averages of boundary values on the boundary of the disc. Below are the precise analogs a' ) resp. b ' ) for a) resp. b).
O.
a
D(a,
a' ) For every in fl and every disc R ) contained in fl, the value of u( does not exceed the average of the values of u on R(
a)
C a):
2 (6.3.1 ) a 217r 10 ". u ( a + Rei O ) de. b ' ) For every closed subdisc D( a, R) of fl, if l1v = 0 on D( a, Rt, i.e., if v E HaIR [D(a, rt] ' and if v I Ca ( R = u I Ca ( R) ' then u I D ( a,R) :S: v I D ( a,R) ' ) u( ) :S:

As 6.3.5 below reveals, these interpretations, carefully formulated, are equivalent.
:
6.3.2 DEFINITION. A CONTINUOUS FUNCTION u C r+ JR. FOR WHICH a' ) OBTAINS IS WHEN THE DOMAIN OF DEFINITION OF u IS A REGION fl, AND FOR EVERY R) CONTAINED IN fl, OBTAINS, THEN u IS fl: u E SH(fl). There is an intimate connection, developed below, between certain families of functions in SH(fl) and solutions of For a region fl and an f in (800 (fl), JR.) , to find a func tion u such that: a) u E (flC , JR.) ; b) u E Ha lR (fl); and c ) u l a� ( n ) = f · A region fl such that Dirichlet's problem has a solution for every f in (800 (fl), JR.) is a 6.3.3 Exercise. If 00, the disc ( t is a Dirichlet region. ( When = the disc is 0; when = 00 the disc is C. ) If < < 00, Section 6.2 applies; ( 00, JR.) consists of constants. ] Ramifications of Dirichlet's problem are central in the classification of Riemann surfaces (v. Chapter 10). 6.3.4 Exercise. a) Ha lR (fl) c S H (fl); b ) SH (fl) is closed with respect to addition and multiplication by nonnegative real numbers and closed with respect to maximization of finite sets:
subharmonic. D(a, subharmonic in
C
C
(6.3.1)
C
Dirichlet region. 0 :s: r :s: r r 0 [Hint: 0 r
{U i
E S H (fl),
1 :s: i :s: I}
Dirichlet's problem:
D a, r C
::::}
{U l
V · . . V UI E SH(fl) } .
The equivalence of a' ) and b ' ) above is proved and extended in the next result, which provides alternative views of subharmonicity.
Chapter 6. Harmonic Functions
286
uE D(a, rr. u E D(a Rr]. v [D( a, RrL u  v D( a, Rr). D(a, Rr. a, R �f u ICa ( R) ) [ 6.3.6 Note. The function U a,R is the Poisson modification of u.]
6.3.5 LEMMA. IF C (C, JR.) THE FOLLOWING STATEMENTS ARE LOGI CALLY EQUIVALENT FOR EVERY DISC a) SH [ , b) FOR EVERY IN HaIR IS CONSTANT OR FAILS TO ACHIEVE A MAXIMUM ( ON ' cf. 6.2.3, U :S; Ua,R O N c ) FOR U p(
PROOF. a) ::::} b): If U  is not constant on
v D( a, Rr and M �f max { u(z)  v(z) : z E D(a, Rr } ' SM �f { z : z E D(a, Rt, u(z)  v(z) = M } , since u  v is continuous, SM is closed in D( a, R) 0 . If b E SM and D(b, rt c D(a, Rr , since v E MVPIR (D(b, r)O), 2 M = u(b)  v(b) :s; 27r1 10 ". [u (b + re iO )  v (b + reiO )] de :s; M. Hence (u  v) l cb (r ) = M. Because r is arbitrary in [0 , R  I b  a l ), D(b, R  I b  a i r
D(a, Rt D( a, Rt, u  v E D(a, RtL w �f u  u a, R D( a, R) o. w u w w( a ) u( a)  u( a ) w D( a, R) O M �f SUp { W(Z) : z E D(a, Rt L there is a sequence, {zn} nEN contained in D( a, R)O, converging to some Zoe in D(a, R), and such that w (zn) + M as n + 00. If I zoo l < R, then w achieves a maximum in D( a, Rt, a contradiction. According to 6.2. 15, w I Ca ( R) = o. Hence, if I zoo l = R, then M = w (zoo) 0 as claimed.
whence is con c SM and so SM is open. Since nected and, by assumption, SM j. 0, SM = is constant, i.e., a contradiction. b)::::}c ): Since U a , R HaIR [ is either constant or fails to achieve a maximum on If is constant, then = = = 0, i.e., = Ua, R. If fails to achieve a maximum on and
=
Section 6.3. Subharmonic Functions and Dirichlet's Problem
(
) )
c ::::} a : Because U a, R is harmonic whence Ua,R E MVPjR
287
[D(a, Rr]),
u a ua, R( a ) :::; 0, i.e., 2 u( a) :::; U a, R( a) = 27r1 10 ". U a,R (a + Re iO ) de 1 12". U ( a + ReiO ) de. =
Hence ( )

27r
(
o
0

6.3.7 Exercise. Maximum Principle in C for functions in S H (Q)) If E Q, U E S H (Q), and ( ) 2: sup (z), then U is a constant function: zE
a
nu
ua
u ( ) = u ( a) . z
(
6.3.8 LEMMA. The Maximum Principle in Coo for functions in S H (Q)) IF c SH(Q) AND, FOR EACH IN 800 (Q),
{u, v}
a
u
sup (z) N ( a inf ) EN ( a ) z E N ( a) nn
:::;
sup
inf
N ( a) EN ( a ) z EN ( a) nn
v (z) :
)u v
) u ln < v ln ;
a OR b = E HajR (Q) . PROOF. An argument similar t o that in the PROOF for 6.1.11 applies.
[
6.3.9 Remark. A function such as i.e., is subharmonic.
v
]
v above is superharmonic,
The genesis of the Dirichlet's problem is to be found in physics. An electrostatic force vector E, arising from the presence of elec tric charges distributed in ffi.3 , is the of a scalar
tial function
gradient
(x, y, )
div E = div grad
=
poten (x, y, )
If the charge is distributed over the boundary 8(Q ) of a region Q in ffi.3 , then b.
B( n)
o
288
Chapter 6. Harmonic Functions
The twodimensional problem can be interpreted as that of find ing an elastic membrane described geometrically by the condi tions u = u(x, (x, E Q and statically by the requirement of equilibrium. The system b.u �f u xx + U yy = 0, u l a (n) = is the associated differential equation If the topological properties of Q are adequately restricted, Q is a Dirichlet region. However, not every region is a Dirichlet region, v. 6.3.11, 6.3.26, and the observations preceding 6.3.30. 6.3.10 Example. If Q = D(O, It and E Poisson's formula (v. 6.2.5 and 6.2.15) provides a solution of Dirichlet's problem .
y), y)
f
[Ab].
f C('][', ffi.),
.
6.3.11 LEMMA. THE REGION 1U cle=f 1U \ {O} IS NOT A DIRICHLET REGION. PROOF. Since r � 8(1U) = {O} U
'][', if
(10) , f(O ) = 1, f l 0, a E r, and z+lima u (z ) = f(a), then lui is bounded in 10. Hence 6.2.28 implies that for some U in Ha lR ( 1U) , u E HaIR
'][' =
U l v= u. The Maximum Principle implies that U
contradiction:
==
1. There results the
1 = lim U(z) = lim u(z) = O. I zitl
o
I z itl
For a Dirichlet region Q, a fundamental technique, due to Perron, provides a solution to Dirichlet's problem.
6.3.12 DEFINITION. FOR A REGION Q, ITS BOUNDARY r �f 8oo(Q) , AND A BOUNDED FUNCTION DEFINED ON r, THE CORRESPONDING E SH(Q), lim IS F(Q, �f
ffamily
f f) { v : v
f
z+ ,), E r
v(z) :::; fh) } .
Perron
6.3.13 Exercise. For any Q and any as described in 6.3.12, the Perron ffamily F(Q, is not empty. If M = sup and u(z ) = M , then u E F(Q,
[Hint:
f)
,), E r
f).]
I fh) 1 f
6.3.14 Exercise. For as in 6.3.11, F
(10, f ) = {O}. f) },
THEN Uf E Ha lR ( Q ). 6.3.15 THEOREM. IF Uf �f sup { u : u E F(Q, 6.3.16 Remark. No condition, e.g., continuity, etc., save bound edness is imposed on
[
f.]
Section 6.3. Subharmonic Functions and Dirichlet's Problem
289
M = sup ,), E r I lh)l , U E F(n, I), a E 0., and u (a ) > M, 6.3.7 ap plies when u ( ) M and leads to a contradiction. Hence u l n :s: M. If r > 0 and b E D(a, r) c 0., for some sequence {u n } nEN in F(n, I), n+lim= un (b) = Uf(b). The sequence { wn �f U l V . . , V U n } nEN is part of PROOF. If
==
z
F( n, I). Furthermore, the functions
n ( ) �f { wPn(w(z)n ) ( )
w
z
z
if z E D(a, r)O otherwise '
n
E N,
are in F(n, I), are harmonic in D(a, rt, and
Harnack's Theorem (6.2.24) implies that if z E D(a, rt,
exists and nition,
Wb E HaIR [D(a, rt] ' whence Wb (b) = Uf (b). Moreover, by defi
If C E D( a, r t, F(n, I) contains a sequence
If
Yn cle=f U n V Un , E N, and Yn (Z) �f { YnP ((z)Yn) ( ) 
(6.3.17)
{un } nEN such that
n
z
if Z E D( a, r)O ' otherwise
n
E N,
Ye
n+ = Yn
in the preceding paragraph, if Z E D( a, r t , then ( z ) �f lim ( z ) ex= Uf ( ). The analog of (6.3.17) ists and E HaIR [D(a, rtL whence
as
IS
Ye
Ye (c)
c
which is valid by virtue of the definition of the
Wb :s: Ye ,
Yn ' Hence
(6.3.18)
V �f Ub , T  Ue,T :s: 0 and so V(b ) = O. The Maximum Prin Since ciple implies V I D ( a , T) O == 0, i.e., = Uf ( ). Since is any point in D(a, rt, Uf E HaIR [D(a, rtl; since D(a, rt is any open subdisc of 0.,
Uf E HalR (n).
Wb(c)
c
c
0
Chapter 6. Harmonic Functions
290
6.3.19 Exercise. The function Uf (the Perron function associated with Q and f) is unique. Via 6.3.15 there can be constructed a function Uf harmonic in Q and, by abuse of language, majorized by f near r �f 8oo ( Q ). Since r is compact, the condition that I f I be bounded is satisfied if f is continuous. 6.3.20 Exercise. A Perron ffamily F �f F(Q, f) is a Perron family, viz., a set F contained in SH(Q) and such that:
1\
{{v E F} {D(a, Rr c Q}} ::::} {Va , R E F} , {{V l , V2 } c F} ::::} { :3 (V 3 ) {{V3 E F} {V3 2: max {v l , v2 }}}} '
1\
(6.3.21 ) ( 6.3.22)
6.3.23 THEOREM. IF F IS A PERRON FAMILY, V cle=f sup v, AND V < 00, F THEN V E HalR(Q) .
(6.3.22)
PROOF. The condition permits the replacement of convergent se quences selected from F by monotonely increasing sequences in F. Hence Harnack's Theorem is applicable. An argument similar to that in the PROOF of 6.3.15 applies. 0 Absent topological conditions on Q, Q can fail to be a Dirichlet re gion, e.g., if Q = 10 � 1U \ {O}, v. 6.3.1l. When Q = 10 the following are noteworthy features: •
•
�f 8oo ( Q) = {0}l:J1I';
r
HaIR (Q) n C ( Qc , JR.) contains no function (3 such that (3(z) is
{ positive zero
�f z E r \ {O} (= 1I') . If z = 0
P
(Indeed, for such a (putative) (3, ((31 r ) �f v E HaIR ( 1U) n C (U , JR.) . Since (3 is positive on 1I' and v E MVPIR(1U) , v(O) = v(O)  (3(0) > O . If, for some a in 10, v(a)  (3 ( a ) �f E > 0, for z in 1Uc and
l
p(z) �f v(z)  (3(z) + E l z l 2 ,
p T = E and p( a ) > Hence for some b in 1U, p( b) 2: max p(z) . Consez El[JC quently Px (b) = p ( b) = 0 and Pxx (b) < 0, p ( b) < 0: t:J.p(b) < O. On the other hand, since v  (3 is harmonic in 10, t:J.p = 4E > 0, a contra diction. Thus v  (3110= 0 and zlim v(z)  (3(z) is both v(O) (> 0) and, +o by virtue of continuity, 0, a contradiction: no (3 as described exists.) y
Eo
The preceding considerations motivate
yy
Section 6.3. Subharmonic Functions and Dirichlet's Problem
6.3.24 DEFINITION. FOR A REGION Q AND AN a IN r TION (3 IN HaIR (Q) n C (Qc , JR.) SUCH THAT
(3(z) is
positive { zero
291
�f 800 (Q) , A FUNC
IF Z E (r \ {a}) IF Z = a
IS A barrier at a. 6.3.25 Exercise. If Q is a Dirichlet region, there is a barrier at each point a of 800 (Q) .
�f { �, �}
6.3.26 Exercise. If Q 1U \ there is no barrier at either 1 or 2 . (Hence Q is not a Dirichlet region. ) For 6.3.25 there is a converse derived from
�
�f
6.3.27 THEOREM. IF Q IS A REGION, a E 800 (Q) r, AND THERE IS A BARRIER AT a, FOR EACH f IN C (r, JR.) AND THE ASSOCIATED UNIQUE PERRON FUNCTION UJ , lim UJ(z) = f(a) . zEn
PROOF. If E > 0, for some positive r, Z E D( a, r t n r implies
I f(z)  f(a) 1 < E.
�f
The hypothesis implies that (3 has a positive minimum m on the compact set r \ D(a, rt. If M Il f lloo the function :
u Q C 3 Z r+ f(a) + E + is in HaIR (Q) , and
(3(z) m
[M  f(a)]
{
u(Z) > f(a) + E > f(z) if z E [D(a, rt n r] M + E > f(z) if z E {[C \ D(a, rt] n r}. If v E F(Q, f), by virtue of the Maximum Principle in Coo , whence (UJ  u) I n :::; o. Therefore, inf
sup
N (a) EN(a) zEN (a)nn
UJ(z) :::; u(a) = f(a ) + E.
On the other hand, the function w
:
QC
3
(3(z) z r+ f ( a)  E   [M + f ( a ) ] m
Chapter 6. Harmonic Functions
292
is in HaIR(Q) , and
{
E < J(z) if z E [D(a, r) O n r] w(z) < J(a) M E < J(z) if z E {[C \ D(a, r) O ] n r} . 


Hence the harmonic function w is in F(Q, J) and so (UJ sup inf UJ(z) 2': w(a) = J(a) E.

w) I n 2': 0, i.e.,
0

N(a) EN(a) zEN (a)nn
6.3.28 COROLLARY. IF THERE IS A BARRIER AT EACH a IN r, FOR EACH J IN C(r, JR.) , DIRICHLET'S PROBLEM HAS A UNIQUE SOLUTION. PROOF. The result 6.3.27 applies. [ 6.3.29 Note. Aside from useless tautologies, there seems to be no general necessary and sufficient condition for the existence of a barrier at a point a in 800 (Q) .
o
On the other hand, a sufficient condition for the existence of a barrier at a in 800 (Q) is the following. If Q, a E 800 (Q) , and the component of Coo \ Q that con tains a is not a itself, there is a barrier at a, v. 8.6.16.] The following observations provide some orientation about Dirichlet regions and nonDirichlet regions. If Q is a simply connected proper subregion of C, then Q is a Dirichlet region. ( Owing to 8.1.8d ) , Coo \ Q consists of one com ponent and is not a single point. ) Although 800 (q = 00 ( a single point ) , nevertheless C is a Dirich let region. ( The (constant ) function u == J(oo) is a solution of Dirichlet's problem. If 0 < r < R :::; 00 , the region A( a; r, R) 0 , which is not simply con nected, is a Dirichlet region. The regions 10 ( = A(O; 0, It) and C \ {O} are not Dirichlet re gions, v. 6.3.11. 6.3.30 Exercise. If r > 0 , D(b, rt n Q = 0, and a E {[800 (Q)] n Cb( r) }, a is a barrier at a. then (3 : QC 3 z r+ In In z
(�)
are:

I

; bI
Alternative definitions of the notion of a barrier at a
m
r
�f 800 (Q)
Section 6.3. Subharmonic Functions and Dirichlet's Problem
293
(3 N(a) n Q, (N(a) n Q)C , �f Zz E [(N(a) n Q)C \ {a}] . (3(Z) (3r {(3r } r>O Qn ( (3r(z) (3r(z) EQn ( , (3r(z) (3 E SH(Q); (3 E 8(Q) \ {a}, z=b 6.3.31 Exercise. If Q is a Dirichlet region, for each a in 800 (Q), there is, in the sense a) or b) a barrier at a. 6.3.32 Exercise. If, in the sense a), (3 is a barrier at a, then lim U (z) f(a). z+ a J 6.3.33 Exercise. If, in the sense b), (3 is a barrier at a, then lim U (z) f(a). z+ a J 6.3.34 Exercise. If, in the sense c), (3 is a barrier at a, then at a there is a barrier as defined in 6.3.24. [Hint When f (3, 6.3.15 applies followed by a change of sign.]
a) ( [Re, Ts] ) A barrier at a is a function defined in some open neigh borhood a ) continuous on subharmonic on and such that is { positive zero If = a such that: a) is su b) ([Con] ) A barrier at a is a family perharmonic in D a , rt and 0 :::; :::; 1; b) lim = 0; c) if z+ a w Ca r ) then z+ limw = l. c) ([AhS, B e] ) A barrier at a is a function such that: a) then lim(3(z) < o. b) z+a lim (3(z) = 0; c) if b
N( ,
=
=
:
=
For some purposes the discussion of subharmonicity is carried out in the following more general context.
u u(Q) [00, (0); u E un ..l
6.3.35 DEFINITION. A FUNCTION IN JR.r1 IS subharmonic in the wide sense, i.e., U C b) usc ( FOR A IFF: a) SEQUENCE OF CONTINUOUS FUNCTIONS, U, v. 1.7.24) ; c) FOR EVERY a AND EVERY POSITIVE R SUCH THAT D( a, Rt C
E SHW(Q), {Un} nEN
1
1 271"
U ( a + Re iO ) dO, u(a) :::; 0 n
27r E
n
E N.
Q,
(6.3.36)
u(Q) c [00, (0), u E SHW(Q) IFF Q, { {v E HaIR (KO ) n C(K, JR.)} 1\ { v l a( K) 2: U l a ( K) } } ::::} { v i K 2: u l K } ' (6.3.38)
6.3.37 THEOREM. IF U usc(Q) AND FOR EVERY COMPACT SUBSET K OF
Chapter 6. Harmonic Functions
294
i.e., IFF DOMINATION ON 8(K) OF U BY A HARMONIC FUNCTION v IMPLIES DOMINATION OF U ON K BY THE SAME v. PROOF. If U E USC(Q), U (Q C [ 00 , (0 ) , and U E SHW(Q) , the averaging argument used for 6.1.8, 6. 1.11, 6.1.12, and 6.3.8 yields, for each compact subset K of Q and each v in HaIR(Q), the inequality For the converse, if r > and D( a, r) 0 C Q, for some sequence of functions continuous on D( a, r) , ..l Thus
)
0
P

(6.3.36) .
{un } nEN
Un u.
{ Vn �f (Un ) } nEN c HaIR [D(a, rtl ,
Thus U E SHW(Q) .
{un} nEN C SHW(Q), AND b ) { Z E Q } { lim Un (z) � C < oo} , n+=
o
6.3.39 THEOREM. ( Hartogs ) IF: a ) ::::}
FOR EACH COMPACT SUBSET K OF Q AND EACH POSITIVE E THERE IS AN no(K, E ) SUCH THAT ON K
(6.3.40) (For the significance of a ) , v. 6.3.42) . PROOF. For some M and all n, � M on K since otherwise, K contains a sequence such that for some Z= in K and some sequence > n. Then, because each is usc, n , lim Z = Z , UTn n
{zn} mn 2': n+=nENn =
Un (zn) n+lim= Un (z= ) = 00 (> C),
Un
{mn} nEN '
a contradiction. Hence, for the purposes of the argument , the assumption
Un I K < 0 is admissible. Since Un is usc, K c U cle=f { z : u n (z) < o } E 0(((:) , whence u nl u < o.
Section 6.3. Subharmonic Functions and Dirichlet's Problem
295
Because K is compact, 1.2.36 implies that for some fixed positive r,
U D(z, 2rt C U. If Z is fixed in K and un (z) > 00, since Un E SHW(Q), zEK
0,
Since u n l u < if v E U and Iz  v i < rS < r, then replacing r by r + rS in the last inequality leads to 7r(r + rS) 2 un (v) :::;
r
JD(v,r+W
un (w) dA 2 (W)
(6.3.41 )
:::; rD(v,r O un(w) dA 2 (W) . J
)
(6.3.40)
On the other hand, implies that if E is a measurable subset of D(z, 2rt, then lim r un (w) dA 2 :::; r lim u n (w) dAs :::; A 2 (E) Thus, n� = JE h n� = for some if > then u n (w) dA 2 < + E) . A 2 (E). then Consequently, if rS is sufficiently small (and positive) and >
Ie
nz, n nz ,
(C
(6.3.41) implies that if n > nz , then un (z) < C + E. for some finite set Z 1 , . · · , zm , K C U D (zm , rSt. If k= 1 nO ( K, E ) clef= 1 �k�rn max nZk
and
the desired conclusion follows. 6.3.42
Examp 1e.
[ = If K clef
.
C
n nz ,
Since r is zfree,
o
("\
0, 1] 2 , H clef= C,
if x > O , if x :::; °
1 n1 ' if x 2': n 1' f
X
<
Chapter 6. Harmonic Functions
296
Un
E C OO (C, lR.), lim
n + oo
Un
==
0, but Un
( 2� 'Y)
= n.
(6.3.36)
Hence on K, {un } n EN is unbounded. The condition serves to control the behavior of the sequence so that its pointwise boundedness is strengthened to uniform boundedness on compact sets. 6 . 4 . Appendix: Approximate Identities
Fejer's kernels (3.7.6 and 4.10. 17), the functions U v (4.10.13), and the Poisson kernels (Section 6.2) constitute sets of functions that are examples of as described below. The context for an approximate identity is a i.e., an A over a K and endowed with a Hausdorff topology T. The maps A x A 3 r+ AxA3 r+ and KxA3 are assumed to be continuous. r+
approximate identities algebm topological field (x, y) xy, (a , x) ax
topological algebm, (x, y) x  y,
6.4.1 Exercise. Every Banach algebra is a topological algebra. In a topological algebra A a net n : A 3 ), r+ n;. E A is an iff, for each in A, n;. converges to
identity
.x
x
x.
approximate
6.4.2 Exercise. For some topological algebra A resp. B the Poisson kernels resp. Fejer kernels constitute an approximate identity. 6.4.3 Exercise. If A is a commutative Banach algebra and n
:
A 3 ), r+ n;. E A
is an approximate identity, ii:).. (M) � l .
6.4.4 Exercise. a) If G is a locally compact abelian group, p, is Haar mea sure, and A L 1 (G, p, ) , and for each nonempty open neighborhood N of 1 . · · 1· dent he 1· dentlty nN = ( N ) X N , t hen { nN } n EN(e) IS an�proxlmate
�f cle e, f
p,
tity. b) For any approximate identity { n;. h EA in A , ( n;' ) [x] (M) + and the convergence is uniform with respect to the parameter
x.
6 . 5 . Miscellaneous Exercises
6.5.1 Exercise. If f E Ha(Q) and 0 tJ f(Q), then In I f I E HaJR (Q) .
[Hint:
If f = U +
iv, then If I = (u2 + v 2)
1
2"
.J
aM (x)
Section 6.5. Miscellaneous Exercises
297
6.5.2 Exercise. If Q is starshaped and u E HaIR ( Q), some v is a harmonic conjugate of u (in Q). 6.5.3 Exercise. If u has continuous partial derivatives of the second order in Q and l1u > 0 resp. l1u < 0 in Q, then u is subharmonic resp. superharmonic in Q; u is subharmonic in Q iff l1u 2: 0 in Q. 6.5.4 Exercise. If
u(z) clef =
{ �(z) lzl2 0
if z j. 0 otherwise
then u E HaIR (C= ) . 6.5.5 Exercise. Which of 6.3.4, 6.3.5, 6.3.10, 6.3.13, 6.3.14, 6.3.19, 6.3.23 remains valid and/or meaningful when is replaced by
subharmonic monic in the wide sense? 6.5.6 Exercise. If 0 < r < 1 and u E HaIR [A(O; 0, It]: a) * du = 0 Iz l =r
subhar
1
)
(v. Section 5.8 ; b) for Zo in A(O; 0, It, the map :
F 1U 3 z r+ u (zo) +
)
l z ( du + i * du) Zo
(cf. Section 5.8 is welldefined and F E H [A(O; 0, It]. For a), Stokes's Theorem applies, and a) applies for b).] The next two results are used in Chapter 10.
[Hint:
6.5.7 Exercise. If f E H [A(O; 0, 2t] and 0 < p < 1: a) for some up in HaIR [A(0 ; p, 2t] ' up I 1 z l =p = �(f) l l z l=p ; b) if P < I zo l , I z l < 2 and
))
(v. 6.5.6b , then Fp  f is represented by a Laurent series
s
L cnz n ; c) if
nEZ {an} nEZ and {bn}nEZ such that [Up  �(f)] ( re iO ) = L ( a n cos n() + bn sin n(}) s n ; nEZ
p < < 2, there are sequences
Chapter 6. Harmonic Functions
298
d) for m in Z+ ,
� �
t [u p  �(f)] ( re iO ) cos mO dO = am s m + L m S � m , Jo t [up  � ( f )] ( re i O ) sin mO dO = bm s m + b� m s �m ; 7r Jo
7r
s p
s 1,
a � m = _ p2m am and b� m = _ p2m brn resp.
e) when = resp. = then �f max + max 1 �(f) I , for
Mp
.
f) If 0 <
I z l =l
l up l
Izl=l
2 max { I a ml , I bml} :::; p < 1, mEN+ 1 2 Mp; g)  P
r_; i ) if 0 < r < � , then :::; max 1 �(f) 1 + 8Mp_ u p l l 1r 5 Izl=l I Z I =r 1  r ( 1 �(f) 1 + max 1 �(f) 1 : Mp :::; 1  5r max Izl=r Izl=l )
h) max
r 5
Conclusion: If 0 < < � , for some function
[Hint:
k (r),
For h) the Maximum Principle and f) apply.]
6.5.8 Exercise. If K(C) 3 K and fl �f C \ K is a Dirichlet region, in C (flc , lR.) there is a u such that: a) u a (fl) = b) u E HaJR(fl); c) in fl, O
1 [Hint:
l
a
v(a) 1.
1;
v, v l 1
Section 6.5. Miscellaneous Exercises
299
�f
4.
is a nonempty Perron family. c) W sup w E Ha IR (Q) . For F some relatively compact Dirichlet region Q K c Q 5. If g is a (harmonic) solution of Dirichlet's problem g(z) = then g  E SH (C) and g W serves for v
u.]
{ 0I
1,
1 .
if z E K if z tJ Q l '
v l lC\n :::; O. 5. g E F, g :::; W :::; v, and �f 0 :::; I :::;
6.5.9 Exercise. a) If K is compact, then Q C \ K is not relatively compact. b) The set of 0 of Dirichlet regions Q, that are relatively compact and contain K is nonempty. c) If E HaIR (Q) , M < 00, for each there are solutions and for Dirichlet's problems:
u,
1 v, {u" vd c HaIR (Q, n Q) , u, (z ) _ {I0 (z) ifif zz EtJ Q,8( K) ' 
v, (z ) d) U
=
t,
{ 0I andifif zz EtJ Q,8(K) .
�f sup u, E HaIR (QC ) and V �f sup v, E HaIR (QC ) . e) o
0
[I ( ] 0 , then
6.5.10 Exercise. If Q is convex and for each z in Q, � ' z) j. is injective on Q. If
1
[Hint:
. { clef + .(3, C cle=f "I + .S:} C H, i(J (3) �f + iq, t E [0 , 1] ' tc 1[(1 t)a ] �f u[x(t), y(t)] + iv[x(t), y(t)] , F(t) �f c p + lq 1 cle=f
+ lV, a = C  a = "1  a + +  a
then
U
a

l
lu
n
P
(p2 + q2 ) F(t) = pu[x(t), y(t)] + qv[x (t), y(t)] + i{pv[x(t), y(t)]  qu[x(t), y(t)]}.
300
H r onic Functions For some T and a in [0 , 1], � �f X( T ) + i Y ( T ) and 1] �f x(a) + iy( a ), Chapter 6.
=
a m
pUx [X(T ), Y(T )]p + pUy [X (T), Y(T)]q + qVx [X ( T ), Y ( T )]p + qVy [X( T ), Y ( T )]q + i {pvx [(x(a), y( a )]p + pvy [x(a), y( a )]q}  i {qux [(x(a ), y(a )]p + quy ([x(a ), y(a)]q} .
Owing to the CauchyRiemann equations, the right member of the display above is ( + ) +
p2 q2 {� [f'(�)] i s' [f'(1] )]}.] 6.5.11 Exercise. If z �f re i E 1U, E E S,B ClI') , for the harmonic measure f w ( z, E) � � r Pr ((}  ¢) d(} of E with respect to z: 27r JE i a) w(O, E) = T (E); b ) lim w (z, E ) � { I if e E E r + 1
"0
0 otherwise
[Hint: If w clef= e" , w' clef= e'w" , z E [W, w'] , then 1  I z l 2 = I w  z l ' I w '  z l , I W 'dw z l I w d(} z l dw ,B o :::; (31 :::; (3 :::; (3 :::; 27r } , the and = Pr. ((}  ¢). If E = { e" 2 d(} results are valid.]
7 Meromorphic and Entire Functions
7.1. Approximations and Representations
A locally holomorphic function I is defined in some Q throughout which it is differentiable. When Q = C, I is The results for entire functions have an importance that is reflected in the general theory of locally holomorphic functions, e.g., in the study of for representing functions I in H (1U) . If and z ut j. 0, the Identity Theorem 5.3.52 implies 1 == 0. Thus C (and 1 == 0) z ut ° and I =t= 0 . I' Consequently, if I =t= 0, then f M (C) , and the study of is thereby related to the study of M (C) . The dominating theorem for M (C) is that of His result is a consequence of a later and more general result of Runge. Thus, for concision and efficiency, Runge's Theorem is the first object of study in the current Chapter. 1 For in C, the function Ra C \ 3 Z r+  is an example of za the simplest type of meromorphic function. Not unexpectedly, a study of Ra leads to a more refined understanding of more general meromorphic
entire: l E E.
Blaschke products
lEE
=
{
E
E
MittagLeffler. a
:
{a}
00
functions, e.g., rational functions, convergent series L rn (z) in which each term is a rational function and
[U r ] · P ( n)
nEN
n=l =
0,
etc.
The following observation is reminiscent of the argument used in deriving a power series representation via Cauchy's integral formula. If > 0, w and I z then
r
a l 2: r,
E D(a, rt ,
Rw ( z )
= a  a) = ( aa ) _ a a 1 z   (w
Ra (z)
1
1
z
w
=� �
n=O
(w
(z

)
)n . n+l
( 7 .1.1) 301
Chapter 7. Meromorphic and Entire Functions
302
Convergence of the right member of (7.1.1) is uniform on the closed set 2: What follows exploits these elementary conclusions to reach a consid erable generalization. The global version of Cauchy's integral formula (5.4.14) may be viewed as a special case of the next result, which is the key to the derivation of Runge's Theorem.
{ z : I z  a l r }.
7.1.2 THEOREM. IF (O(C) \ 0)
3
U ::J K E K(C) , THERE ARE RECTIFI
N bn h <; n<;N SUCH THAT: A ) U 1'� C U \ K; B ) IF n= 1 J E H(U) AND z E K, THEN N J (w) (7.1.3) L 1  Z dw. J (z) = � 27rl n= l ABLE CLOSED CURVES
In
�
W
[ 7.1.4 Remark. As the proof below reveals, each 1';' is the union of a finite number of horizontal and vertical complex intervals.] PROOF. The argument is conducted in terms of closed squares clef
Qpqrn =
The
[ 2m ' � + 1] X [ m ' � + 1 ] , {p, q , m} 2 p
p
q
q
C
Z.
oriented boundary aOQpqm of Qpqm is the union of four oriented sides:
and if
a + 4 (b  a)t if 0 < t <  �4 b + 4(C  b) (t  � ) If. 41 < t < 21 C + 4(d  C) (t  �) if 21 < t < 43 ' d + 4 ( a  d) (t  �) 1. f 43 < t < l 
l'(t) =
Section 7.1. Approximations and Representations
303
Since K is compact (hence bounded), for any m , K is covered by a set Q consisting of finitely many Q ... each of which meets K. There is a second set consisting of finitely many Q ... not in Q but meeting Q ... . The set
S K �f ( U Q ... ) is compact and if m is such that
U Q
Qus
3 . < mf { l z  wl z E K, w E (C \ U) } , 2m then K c KO C U. all A side belonging to only one constituent square of K is other sides of constituent squares of K are If is shared by Q ... and Q ... and, as an oriented side of Q ... = as in Figure 7.1.1, then as an oriented side of Q ... = Furthermore, for any g,
shared. s , s [a, b]
, s [b, a].
r
Jra ,b] exists iff
r
J�, a]
g
g
unshared ;
dz
dz exists and Jrra, g dz =  J�r , a] g dz. The oriented bound
� ary r 8° K is a union of horizontal and vertical oriented complex closed intervals that are unshared sides of constituent squares Q . . . of K. Any configuration of intersecting unshared sides is, modulo rotations through 7r integral multiples of "2 radians, one of those illustrated in Figure 7. 1.1.
�f
w
w
u
Figure 7.1.1. Solid lines are unshared sides of constituent squares of K . ' Dotted lines are shared sides of constituent squares of K .
Chapter 7. Meromorphic and Entire Functions
304
vertex v
A (an endpoint) of an unshared side is the intersection of Or a As two or four unshared sides. Accordingly, v is a indicated in Figure 7.1.1, for a given v, there are in algorithmically defined vertices u and w such that {[u, v] , [v, w] } are oriented, with respect to the squares to which they belong, according to the pattern in Thus, if v? is a vertex in there is in an algorithmically defined vertex vt such that [v? , vt] is oriented according to the pattern in and an algorithmically defined vertex v� 1 such that [v� 1 , v?l is oriented according to the pattern in By induction there is generated a sequence
2vertex
r
r
4vertex.
(7.1.5). (7.1.5)
r
(7.1.5).
of algorithmically defined vertices to which no further algorithmically de fined vertices can be adjoined: 0" 1 is maximal. An examination of the possibilities: a) VIPl is a 2vertex and vil is a 2vertex; b) VIPl is a 2vertex and vil is a 4vertex; c) VIPl is a 4vertex and vi1 is a 2vertex; d) VIPl is a 4vertex and vi1 is a 4vertex; leads to the conclusion: Only a) or d) can be valid. Owing to the maximality of 0" 1 , a) implies V � P l = vil ; d) does not imply V � P l = vi l However, every vertex in
ql  l r 1 �f U [VIj ' VIj+ l ] j= Pl is either a 2vertex or a 4vertex, whence r 1 is an oriented cycle (v. Section 5.5), i.e. , r 1 is a closed curveimage. If v� E (r \ r d , the procedure just described may be repeated to pro q2  1 duce an oriented cycle, r 2 �f U [v�, V� +I] . After finitely many repeti j= P2 tions of the procedure, all the vertices in r are consumed and there emerge N oriented cycles r 1 , . . . , r N such that r = U rn . n= 1 For each n there is a rectifiable closed curve "In such that "In lin[Pn +j qn ' Pnj ++ qnl ] onto the (j + l)st interval of rn o Hence early maps "I� = rn, 1 :::; n :::; N .
If z is on no side of any Q ... , then
if z E Q� . otherwise
(7.1.6 )
Section 7.1. Approximations and Representations
305
Owing to cancellations that result from integration over shared sides, for z as described above and in K,
C�J
1 27ri
OQ
��� dw + �Jo Q ��' � dw
)
Q = L Ind ao Q (z)J(z) + L Ind a o Q (z)J(z) Q ES Q EQ ...
1 27ri
=_
(tn=l 1 lJ:!!l ) 'Yn
WZ
dW
..
..
(7.1.7) (7.1.8)
•
Because of ( 7.1.6 ) , the right member of (7.1.8 ) is J(z). Furthermore , since r n K = 0, the right member of (7.1.8) and J(z) are continuous functions of z on all K. Thus, if z E K, then J(z) =
�1 � 27rl �
n= l
In
�
J(w) dz. WZ
0
7.1.9 Exercise. The assertions about the possibilities a)d) are valid. [ 7.1.10 Note. As elaborated in [New] , the argument underlying 7.1.2 leads to Alexander's proof of the Jordan Curve Theorem.] 7.1.11 THEOREM. (Runge) IF: a) K 3 K c Q c C; b) 5 c (C= \ K) AND MEETS EACH COMPONENT OF C= \ K ; c) J E H (Q); THEN J IS UNI FORMLY APPROXIMABLE ON K BY RATIONAL FUNCTIONS R FOR WHICH P(R) C 5. PROOF. If R is the set of all rational functions r such that P(r) C 5, each such r has only finitely many poles, and thus each such r is in H (K). Furthermore M R I K is a vector subspace of C (K, q . The content of 7.1.11 is that J (in H(Q ) n C(K, q ) is in the II=closure of M. If the assertion is false, the HahnBanach Theorem and the Riesz Rep resentation Theorem imply there is a measure space (K, S ,B , p,) such that 5
�f
I
1 J(z) dp,(z) = 1 and {g E M}
'*
{ l g(z) dp,(z) o} . =
For this p" h C= \ K 3 z r+ r is holomorphic in C= \ K. If C is iK w  z a component of C= \ K, 00 E C, and 00 i E (5 n C), for some positive E, D(S, Et c C. If z E D(S, Et, for w in K,
dp,( w)
:
s
N
s n � _1 _ . s n+ l W  Z
" (z  ) � (w  )
(7.1.12)
Chapter 7. Meromorphic and Entire Functions
306
R(w) in the left member of (7. 1 .12) is in M, whence i R(w) df.L(w) = 0, and the Identity Theorem implies h l c = 0. When = 00, (7. 1 .12) is replaced by '"N wn n l 1 ' �z + wZ n=O and convergence obtains throughout K if I z l > sup { I w l w E K } . It follows that h llC= \ K = 0. Then 7.1.2 yields Each term
s
u
+
1
= }rK J (z) df.L(z) = }rK
[�tn=l 1 WJ(w) dW] df.L(v). � n=tl 1 J(w)h(w) dw = 27rl
"In
Z
Fubini's Theorem permits reversing the order of integration in the right member above. Since
h = 0, 1 = 27rl
"In
0, a contra
diction. [ 7.1.13 Remark. Since Coo is separable and the components of the open set Coo \ K are open, there are at most count ably many such components. For each component C of Coo \ K, 5 n C may be a single point, i.e., 5 may consist of at most count ably many points, exactly one in each component of Coo \ K. For such an 5, the available rational functions constitute a minimal set; the impact of Runge's Theorem is maximal.]
0
7.1.14 COROLLARY. ( Polynomial Runge ) IF Coo \ Q IS CONNECTED, POLY NOMIAL FUNCTIONS SERVE AS THE RATIONAL APPROXIMANTS PROOF. An admissible choice for 5 is If is a rational function and then is entire, i.e., a polynomial. C 0 D(O, 1 ) , and E H ( KI ) , 7.1.15 Example. a) When Q = D(O, 2t , for some positive E, is representable by a power series for which the radius
R.
{oo}. R Kl = J J of convergence is 1 + E: J ( z) = L cn z n . Hence, on K1 , J is uniformly n=O N approximable by the sequence of polynomials L cn z n , N E N" which are n=O functions in H [D(O, 2t] .
P(R) {oo},
R
00
Section 7.1. Approximations and Representations
307 1
b) On the other hand, if K2 = { z : I z l = I } and I(z) =  when z z j. 0, then I E H (K2 ). If g E H(Q) , then
r
iK2
g(w) dw = ° while
r
iK2
I(w) dw = 27ri.
Thus g cannot approximate I uniformly on K2 • What is an essential difference between K1 and K2 ? For the com ponent e2 D(O, It of Q \ K2 , e� n Q = D(O, 1) is compact. However, e1 { z : 1 < I z l < 2 } , is the only component of Q \ K1 and
�f
�f
which is not compact: e1 is not relatively compact in Q. Consequently, Runge's Theorem has a number of variants. In the con text of 7.1.11, the set R may be replaced by H(Q) . In particular, the following result, called Runge's Theorem by some, applies when K is a compact subset of an open set U. 7.1.16 THEOREM. (Rungevariant) IF K(C) LOWING STATEMENTS ARE EQUIVALENT:
3
K C U E O(C) , THE FOL
a) IF I E H(K), I IS UNIFORMLY APPROXIMABLE ON K BY FUNCTIONS IN H (U) ; b) IF e IS A COMPONENT OF ( THE OPEN SET ) U \ K , THEN e c n U IS NOT COMPACT, i.e. , NO COMPoNENT OF U \ K IS RELATIVELY COMPACT IN U; IF z E U \ K , FOR SOME I IN H(U) , I /(z) 1 > sup I /(w) l . c)
wEK
PROOF. The pattern of proof is: c)
::::}
b); a)
::::}
b)
::::}
a) ; b)
::::}
c).
c) ::::} b): If b) is false. for some component e of U \ K , e c n U is com pact. If p E Be, some sequence {Pn }nEN contained in U converges to p. Since e is open, p tJ e, whence p E e c n U c U, P E U \ e K: Be c K. The Maximum Modulus Theorem 5.3.36 implies sup I /(z) 1 :s; sup I /(z) l , a Cc K denial of c) .Hence ,b) ::::} ,c) (7. 1 . 17) =
and c) ::::}
::::}
b).
b): If b) is false and I E H(K), for some sequence {fn }nEN in H (U), In � I on K. Furthermore (7. 1 . 17) implies that for all n and m in N,
a)
Chapter 7. Meromorphic and Entire Functions
308
sup I /n (z )  Irn (z ) 1 � sup I /n (w)  Irn (w) l · Thus {fn } nEN converges uniwEK formly on Ge to some F. Hence F is continuous on G e , F = 1 on K , which contains BG (v. preceding argument for c) ::::} b)) , and F E H(G). If w E G, there are an open V containing K and an N (w) such that V n N ( w) = 0. 1 __ . Then A particular choice of 1 is given by the formula I(z) zw 1 E H(K) , (z  w)F(z) = 1 on an open set W containing K, and on the nonempty open set G n W. The Identity Theorem implies (z  w)F(z) = 1 on G. When z = w, there emerges the contradiction, 0 = 1 : a) ::::} b) .
z ECC
�f
b) ::::} a): If a) is false, I, viewed as an element of G(K, q , is not in the closure of H (U) viewed as a subset of G(K, q . The HahnBanach Theorem (3.3.8) implies there is a measure space (K, S ,B , p,) such that for every g in H(U), g dp, = 0, while 1 dp, = 1 . The following argument shows that K K the last equation is false. 1 The map ¢ : C \ K 3 w r+ r __ dp,(z) is in H(C \ K) (v. 5.3.25). z w iK If w E C \ U, since z r+ l k is in H(U), the HahnBanach Theorem (z  w ) + implies here that
i
i
1
k ¢ ( l (W ) = k!
1
r
k dp,(z) = 0, iK ( z  W ) + 1
00
1
k
E Z+ .
(7. 1. 18)
zn
1 '
If I w l > sup I z l , then  =  '" n + and the series converges unizW K n=O w formly for z in K. Since z n dp,(z) = 0, n E N+ , ¢( z ) = 0 if z is in the (unique) unbounded component of C \ K. If V is a bounded (open) component of C \ K and V n (C \ U) = 0, then V C U and the boundedness of V implies Ve n U is compact, a contra diction of b): V n (C \ U) j. 0. If w E V n (C \ U), (7. 1 . 18) implies that for some neighborhood W of w, ¢ I w= 0, whence ¢ I v = o. In sum, ¢ I IC\K = o. If N is a neighborhood of K, the compactness of K implies that N may be assumed to be the union of finitely many open squares Q� . of the form in 7.1.2. For some infinitely differentiable '1jJ defined on N, '1jJ I on K and '1jJ 1 1C\ N = 0, in particular, '1jJ = o. If 1 E H( N), al = o. According to the product rule for derivatives, aU · '1jJ) = (af) . '1jJ + I · a'1jJ = I · a'1jJ. Hence Pompeiu's formula (5.8.14) implies that if z E K, then
i
l
�
==
iJN
I(z) = l (z)'1jJ(z) =
1 27r l
.
1 I(w) B'1jJ(w) dw 1\ dw. N

WZ

Section 7.1. Approximations and Representations
309
Fubini's Theorem and the definition of ¢ imply
r
}K
J(z) dpJz) = �
=
1
(1
)
8'1jJ(w) dw 1\ dW dp,(z) J(w) WZ K } J(w)8'1jJ(w) . ¢(w) dw 1\ dw.
27r l
r
N
i
Since ¢ I IC\ K = 0 and 8'1jJ(w) = 0 on K , J dp, = 0, a contradiction. (A similar technique is employed in the PROOF of Runge's Theorem.) r,
r
b) ::::} c): If z E U \ K, for some positive D(z, ) c U \ K. If C is a component of U \ [K U D(z, )] either C or C U D(z, ) is a component of U \ K. Thus b) obtains for K U D(z , ) Since K n D(z , ) = 0, there exist disjoint open neighborhoods N(K) and N[D(z , ) ] . Hence there is in H {N(K) U N[D(z, r)] } a function J such that J I K = 0 = 1  J I D ( z , T) ' Since b ) ::::} a) , there is in H (U) a g such that r
'
r
.
r
r
r
o
from which c) follows.
7.1.19 Exercise. If K is compact, every function in H (K) is uniformly approximable on K by polynomials iff C \ K is connected. [Hint: a) In the current context, for U in 7.1.16 , C may serve. b) A component of C \ K is relatively compact in C iff C \ K is not connected. c) If J is entire, J is uniformly approximable on each compact set by polynomials.] The similarity of the techniques used in the argument for Runge's Theorem and its variant leads, by abuse of language, to the conclusion: Rungevariant ::::} Runge. 7.1.20 Exercise. For the set K in the argument for 7.1.2, no component of U \ K is relatively compact in U. (The condition 7.1.16b) may be in terpreted roughly as saying that part of the boundary of each component of U \ K meets the boundary of U, v. 7.1.15b) .) Since the components of Coo \ K play a role in the previous discussions, the following result is of interest, and proves central in 8.1.8. 7.1.21 THEOREM. IF Q IS SIMPLY CONNECTED, THEN F CONNECTED.
�f Coo \ Q IS
Chapter 7. Meromorphic and Entire Functions
310
oo J
PROOF. As a closed subset of C , if F is not connected, there are two dis joint, closed, and nonempty sets and such that F = 00 is in one, say and thus is compact. Furthermore O(q 3 U C \ = 1 If f 1 in 7.1.2, one of the summands, e.g. , __ dw, in (7.1.3) is
K
J,
==
K
Jl:JK; �f J Ql:JK.
1
/'1
wZ
J) K (Ql:JK) K Q Q.
K
\ = and not zero. On the other hand, U \ = (C \ \ = 1 According if Z E then __ , as a function of w , is holomorphic in
K,
to 5.3.14b),
1
wZ
/'1
1 __ dw = W  Z
0
0, a contradiction.
7.1.22 Example. The complement (in q of a nonempty compact subset of C is not simply connected. The region C \ (  00 0] is simply connected. Although the complement (in q of the strip S :  7r < ( ) < 7r is not a connected subset of C, S is simply connected. The presence of Coo rather than C is essential in the statement of 7.1.21. ,
�f { z
'S z
}
7.1.23 THEOREM. (MittagLeffler) FOR AN OPEN SUBSET U OF COO , IF S C U, S· n U = 0, AND FOR EACH IN S THERE IS A RATIONAL FUNCTION ( Z ) = FOR SOME f IN P(f) = S AND (Z P (f) = PRO O F. For as in 4.10.9, the sets
a
ra clef LnN=(a1) cn (a)a) n ' ra · {Kn } nEN _
a
M(Q),
Pn ( z ) � aEL ra ( z ) is a rational function holomorphic in an open set containing Kn  I . Since Sn is finite, Coo \ Sn has only one compo nent Cn and 00 E Cn . Thus Runge's Theorem (7.1.11) applies and yields a rational function Pn for which 00 is the only pole: Pn is a polynomial. 1 on Kn  I , Furthermore, the Pn may be chosen so that I Pn (z)  Pn (z) 1 < and f � P I + L�= 2 ( Pn  Pn ) meets the requirements. 0 are finite, and
Sn
2"
7.1.24 THEOREM. (Weierstrafi) IF
U E O(q , S c U, AND S· n U = 0 , FOR SOME F IN H(U), S c Z(F) . PROOF. If the set S = ° resp. S =
{anh � n� N < oo' then F
==
1
resp.
Section 7.1. Approximations and Representations
311
5 {an } nEN 5 {anh S nS N < oo , U Ie z C, I (z) �f !c(z) , I C U U. U 0 tJ 5. 0 5 fl �f U \ 5 a fl, Pn (z) �f z 1_ a n D(z, rt fl, a z fl,
requires meets the requirements. Thus only the possibility = motivates the attention. The paradigm for F when = argument below. The components of are pairwise disjoint regions. If is the solution for the component of and for in then is a solution for Thus , for ease of presentation, it is assumed that itself is a region and The situation for which E is dealt with in 7.1.26c) . The region is polygonally connected (1.7.11). In 7.1.23, when and is fixed in for a nonempty contained in and a polygon 7r; connecting to and contained in the function

__
In,7rz D(z, rt 3 z' r+ exp 7rz + [z ,z 1 Pn (Z) dZ) IS H [D(z, r)O]. If 1]; is a polygon like 7rz , then 7T"z 1]z Pr> (Z) dz is in 27riZ, whence In, 7rz (z') is independent of the choice of 7rz : In, 7rz �f In. ' If Fn �f exp ( fn ), then ( z  an ) = 0, and direct calculation shows Fn ( z) z  an . For appropriate polynomials {Pn } l < < , the series that Fn (z) a  an 00 PI + nL ( Pn  Pn ) � I E M(fl). Furthermore F �f exp(f) is welldefined =2 in fl, F E H(fl), and
(1 1
;
m
= ___
I
1 1
_ 11.
CXJ
z  a l II { z  an exp  Pn (W) dw] } . (7.1.25) (z) = a  a l n2': 2 a  an The right member of (7.1.25) is an infinite product. Owing to the continuity of exp, the infinite product converges uniformly on compact subsets of fl and as z + an F (z) + O. Hence the set 5 consists of removable singulari ties of F and if F (an ) �f 0, n E N, then F E H(U) and 5 C Z(F). 0 The discussion of infinite products is given in Section 7.2 where the development implies Z(f) 5, v. 7.2.11. [ 7.1.26 Note. a) If I is required to have a zero of multiplicity f.ln at an , the sequence {an} nEN may be modified so that each an appears f.ln times. F
.
__
=
[1
'Y
312
Chapter 7. Meromorphic and Entire Functions
Z(f) �f {an} nEN ' then n, I z l < l anl , then 1 1 zk '"'"  . = = Pn = � z  an an ( 1  z ) k=O a�+l an Kn )k The approximating polynomial Pn is, for some Kn ,  L ( : k=O ) n and the convergence inducing factor exp (1 Pn (W) dw takes Kn Kn zk + l k ) . The exponent L zk + l k the form exp ( L k=O (k + l)an+ l k=O (k + l)an+ l f
f(O) i 0, a 0,
b) If is entire, = and S· 0 and + 00. Hence, for each if
l anl
=
(Xl
( z  an )
is a partial sum of the familiar pOwer series representation of a de1 termination of  In _ . The MittagLeffler Theorem com bined with the argument above yields the _ _
WeierstrajJ product rep
resentation for the entire function f: f(z) = II n� l
{ (I  :n ) (tk=O (k zkl)a+l nk+l ) } .
f(O) 0,
exp
(7.1 .27)
+
k
c) If = for some in N, the product representation is pre ceded by a factor
zk .
Since the right member of (7. 1.27) converges uniformly on compact subsets of U and since the function exp is continuous, the validity of (7. 1.27) is automatic. Its derivation is independent of the theory of infinite products.]
[ f natural boundary f.
7.1.28 Exercise. If Z ( W = au, there is no function F such that: a) F is holomorphic in an open set V that properly contains U; b) F l u = Thus au is a for For such an F, if Z ( Ft = 0, then U = C . If Z ( Ft 0 5.3.52 applies.]
[Hint:
f.
i
f
7.1.29 Exercise. If U E 0(((:) , then H (U) contains an for which au is natural boundary. In U there is a sequence S such that S· = au. In H(U) there is an such that = S.]
a
[Hint:
f
�f {zn} nE N Z(f)
Section 7.2. Infinite Products
313
7.1.30 Example. If a sequence of holomorphic functions converges every where, need the convergence be uniform on every compact set? In [Dav] the following construction uses Runge's Theorem to produce a sequence of polynomials such that
{Pn } nEN
if Z = 0 (7. 1.31) otherwise · Although the sequence converges uniformly on every compact set not con taining the sequence fails to converge uniformly on every compact set properly containing For in N, if �f > while < + lim
n + oo
{O}, n
Pn (Z) = { 01
{O}. Un { a ib : a  �, I bl clef
clef
�} Fn l:.J [; , n] ,
1 Fn = D (O , n ) \ Un and Kn = each Kn is compact, Kn C K� + l ' and U Kn = C \ {O}. Furthermore, for nEN each n, C \ Kn is connected and there are disjoint Open neighborhoods Vn of Kn and Wn of o. Hence there is a function gn holomorphic in Vn l:.JWn that
and such
if z E Wn if Z E V;, . Polynomial Runge (7. 1.14) implies there is a polynomial 1 < . Hence (7. 1 .31) is valid.
Kn , I gn  Pnl n
Pn such that on
7.2. Infinite Pro ducts
The Weierstrafi product representation (7. 1 . 27) leads naturally to a discus sion of
infinite products.
7.2.1 DEFINITION. FOR A SEQUENCE f
00
{an} nEN OF COMPLEX NUMBERS,
infinite product P � II ( 1 + an) EXISTS IFF a) n= l N · PN 11· m II (1 + an ) = 11m N + 00 n N+ oo =l EXISTS, IN WHICH CASE P = 11· moo P N ; b) FOR SOME no, N+
THE
def
def
N
lim II ( 1 + N+ oo
n=:no
an )
Chapter 7. Meromorphic and Entire Functions
314
EXISTS AND IS NOT ZERO. 7.2.2 Remark. The condition b) has the following motivations. A product of finitely many factors is zero iff at least one factor is zero, whereas, e.g., if each factor 1 + is nonzero
[
•
an � an N 1 and yet lim II (1 + an ) 0. lim N+= 2N N+= n= 1 An infinite series converges iff every subseries arising from the ==
=
=
•
deletion of finitely many terms converges. The validity of the analogous statement for infinite products is assured by b). If
if n 1 an {I otherwise ' 1 N N then lim II (1 + an ) = ° but lim II (1 + an ) does not N+= n= 1 N+= n=2 = exist. The condition b) eliminates II (1 + an ) from considn= 1 eration as an infinite product. =
=
As the developments below reveal, the simpler definition requiring
N
only that lim II (1 +
N+= rL= l
an ) exist suffices in the context of repre
senting entire functions as infinite products.]
= 7.2.3 Example. If the series L bn converges and 1 + an exp (bn ), then n= 1 = lim bn 0, lim an 0, and the infinite product II (1 + an ) converges to n= 1 =
rL.t cx)
=
=
7.2.4 Exercise. If
{anL :SnSN <= C C:
)
qn �f g (1 + l anl ) � exp (� I anl ; Ip N 1 1 � qN  1 . [Hint:

If x 2: 0, the relation 1+
x2
x � 1 + x + 2T + . . .
=
exp(x)
Section 7.2. Infinite Products
315
applies for the first inequality. Induction leads to the equation
which deals with the second.
] ex:>
{an} nEN C eX ; c) L l an (x) 1 CONn= 1 VERGES UNIFORMLY ON X; d ) sup l an(x ) 1 < 00 , THE INFINITE PRODUCT 00 II [1 + an (x)] CONVERGES UNIFORMLY ON X AND DEFINES A FUNCTION n= 1 X f IN e FOR ANY PERMUTATION 7r N 3 n 7r(n) E N, 00 f(x) = II [1 + a7r(n) (x)] . n= 1 FURTHERMORE, f(b) = 0 IFF FOR SOME no , 1 + ano (b) = O. [ 7.2.6 Remark. The hypotheses c ) and d ) are independent. For )
7.2.5 THEOREM. IF: a X IS A SET; b) n E/'! xEX
ft
:
example, if X = [0, 1] and: •
x
if E (0, 1] otherwise
•
x and an (x) = n n 2: 2, c ) holds and d ) does not . 2 If an (x) 1 , n E N, d ) holds and c ) does not. ] ==
'
PROOF. For some M, sup n E/'!
xEX
l an (x) 1 :::; M < 00 .
If K E N, there is an N
depending on K and such that (1, 2, . . . , K) C [7r( I ) , 7r(2) , . . . , 7r(N)] . If 1  > > 0 there is a Ko such that sup < If 2 ex:>
E
L l an (x) 1 E. xEX n=K o K N K 2: Ko , PK �f II [1 + an (x)] , and nN �f II [ 1 + a 7r( n ) (x)] , n= 1 n= 1
owing to 7.2.4,
InN  P KI .:::; IPKI
( eC

1)
n :::; I P KI L E < 2 1 pKI E. n= 1 00
(7.2.7)
Chapter 7. Meromorphic and Entire Functions
316
Moreover, 7.2.4 implies that for some P and all K, I P K (X) I :::; P < 00 . If is the identity permutation, ( 7 . 2 . 7 ) implies that for some f, P K � f. Furthermore, if K > Ko , then
7r
I P K  P Ko I :::; 2 1 p Ko I E, I P K I 2': (1  2 E ) I P Ko I , {x E X} '* { I f(x) 1 2': (1  2 E ) I p Ko (x) I } .
Hence {J(x) = o } {} {p Ko (x) = a}. Finally (7.2.7) implies that for each x, lim n (X) = lim P N (X) . 0 N+= N N+=
7.2.8 COROLLARY. IF 0 :::; an < 1, n E N, THEN
N
PROOF. If P N �f II (1  an ) , for some p , PN ..l p. If n= 1
=
=
L an < 00 ,
7.2.5
n= 1
implies P > O. On the other hand, if L an = 00 , for each N, n= 1
N
whence lim II (1  a n ) = o. N+=
o
n=1
[ 7.2.9 Remark. The last sentence provides another motivation for b) in 7.2.1.] If z E 1U the series
ex:>
n
 L =n converges, say to l(z) . If g �f exp(l), then
( )
n= 1
g(Z) ' g'(z) (1  z)' . , whence = o. Smce l(O) = 0, g(z) = 1  z, I.e., 1z 1z g (z) (1  z) exp[l(z)] = 1 and so l(z) is a determination of In(l  z). It follows that for z fixed in 1U and K in N, for some N(K, z) in N, .

(
� 1  (1  z) exp t;Z �k =
(
)
N(K, )
II (1  z) exp t;Z znn =
According to 7.2.3,
N (K' )
:::;
)
=
� 2  K < 2. converges.
Section 7.2. Infinite Products
317
{an } nEN C C and 0 < I an I :::; I an+1 1 t oo, for any {N(K, n, z)} KEN such that k ( ) K, n ,z ( Z ) N ( ) :kn II 1  an exp L n= 1 k= 1 converges to some number J(z). If N(K, n, z) can be chosen to depend only on n, i.e. , if there is a sequence {Nn } nEN such that More generally, if fixed there is a sequence
Z
00

(
)
ft
J : C 3 z J(z) is entire and 7.2.8 implies J(b) = 0 iff for some z The expression E N (Z) �f (1  z) exp (� : , the product of 1  z and the exponential function of the sum of the first N terms of the Maclau rin series for a particular determination of In ( 1  z), is approximately (1  z) l 1 z 1. The next result serveS to estimate the error of the ap converges, =
n, b an .
.

proximation.
=
)

I z l :::; 1
N Z+ ,
THEN AND E 7.2.10 LEMMA. IF PROOF. There are nonnegative numbers kE
Ck ,
1 1  EN (Z) I :::; I z I N +1 . Z+ such that
En (O) = 1 and there is a sequence {ddkEN of nonnegative numbers such that E�(z) _ z N ( 1 + f dk Z k . If k= 1 1[o ,z] E� (u ) du 1 (Z) E N _ g N ( Z) �f Z N +1  ''�z';N:;+:1 , then g N (Z) = L e k z\ e k 2': 0, k E N. If I z l :::; 1, then I g N (Z) 1 :::; g N (I) 1. k= 1 o Furthermore,
=
00
)
=
Chapter 7. Meromorphic and Entire Functions
318
a sequence 0 l art l :::; l an+ l l t oo, then NNcontains n +l
7.2.11 Exercise. a) If <
{Nn}nEN such that for each positive R,
more,
� C:, I )
< 00. Further(7.2.12)
converges for all z in C. b) If, for some k in N,
� n�l for all positive R, then
�
(R) �
k+ l
k+1
< 00
(7.2.13)
(7.2.14)
converges for all z in C. c) The infinite products in (7.2.12) resp. (7.2. 14) represent entire func(v. 7.2.5). tions resp. G such that = = 7.2.15 Remark. If (7.2.13 ) obtains for some k in Z+ , there is a least such k, say h, the of the function When k = h, the representation (7.2. 14) is the for
[
F
Z(F) Z(G) {an} nEN G. genus canonical product representation
G.
p
G
More generally, when is a polynomial function, and h are as described above, and m N+ , then zTn G(z) is a
product.]
eP( z )
E
canonical
E M (C) there are entire functions f and g such {Pn}" EN ' m = �. [Hint: It may be assumed that the poles of m are listed according to their orders, i.e., if p is a pole of order k, then p appears k times in the listing of P(m). There is an entire function g such that Z(g) = P (m ) . ] 7.2.17 Exercise. If f E E and Z(f) = 0, for some entire function w, f = exp(w). r f'f ((tt )) dt [Hint: The functions f'f and w C 3 z r+ w (z) �f J[o.z] ( exp)w) ) ' = 0.] are entire and
7.2.16 Exercise. t hat off P(m ) �f
If m
:
Section 7.2. Infinite Products
319
7.2.18 Exercise. If 1 E E , for some in N, and some entire function w , I(z) = z k exp[w (z)] II ENn(a) a E Z (f) The preceding discussion shows that if 0 < l an l t oo , although an infi
) may fail to converge, nevertheless, with the aid zk + l of the convergence inducing factors, exp ( t E N, the k l) k=O ( k + l)an+ zk+ l infinite product IT (  ) exp ( t k l ) does converge. The an k =O ( k + l)an+ n=1
nite product
11. (
(�) .
k
1
:
n
, n
�
1
next paragraphs motivate and introduce Blaschke products that deal with the problem of finding function 1 such that 1 E H (1U) and Z (I) IS a preassigned subset of 1U. The Identity Theorem 5.3.52 implies that if 1 =j=. 0, then a
Z (ft
n 1U
=
0.
If Z(f) is infinite, it is countable, i.e., Z(f) f {a" } n EN ' and thus l an l + 1 , whence the assumption lan l t 1 is admissible. A natural conjecture for the form of 1 (z) is the infinite product
�
DC
(7.2.19)
n=l
If (7.2.19) converges, lim ( a  z) 1 . The preceding experience with n+ = n infinite products suggests that some convergence inducing factors {Fn } n EN can force the relation =
lim (a  z) F,, (z) n+ oo n
=
(7.2.20)
1
and, more to the point, the convergence of 00
(7.2.21 )
II (an  z) Fn (z ) .
n=1
7.2.22 Exercise. If 0 < l an l t 1, and l an l Fn (z) clef = an ( 1  a n z ) _
( 7.2.20 ) holds.
(
1
= sgn (a n ) 1  an z _
),
( 7.2.23 )
Chapter 7. Meromorphic and Entire Functions
320
7.2.24 THEOREM. FOR
Fn AS IN (7.2.23 ) , IF
L ( 1  l anl ) < 00 n= l 00
AND 0 < Z
(7.2.25
l anl t 1, THE BLASCHKE PRODUCT
00 (an  z) Fn (z) ( k II00 sgn (an ) an ) 1  an z n= l n= l
k II
_
Z
=Z
)
(7.2.26)
CONVERGES UNIFORMLY ON COMPACT SUBSETS OF 1U ( THUS REPRESENTS A FUNCTION B IN H (1U) ) AND Z(B) [ 7.2.27 Remark. Since =
l(z)
{ an} nEN l:J {O} .
F"
in 7.2.9 ) , the have forms reminiscent (v. the discussion of of the convergence inducing factors encountered in the Weierstrafi Product Theorem. In the latter the arguments of exp are finite sums, whereas the corresponding arguments in the are infinite series.]
Fn
I z i :s; < 1, and bn (z) clef= 1  sgn (an ) 1an anZz I bn(z ) 1 I ( 1  Iaannl()1 (ana+n z)l anl z) I 1 1 sgna (a)zz I :s; ( 1 l anl ) 11 += ( 1  l anl) 1 n whence 7.2.5 applies to the sequence {l bn (z) +l} nEN " o 7.2.28 Exercise. If E c 1I', then 1U contains a sequence {an } nEN such that EC = ({an} nEN r and the Blaschke product ( 7.2.26 ) converges. ( In
PROOF. If ( 7.2.25 ) holds, then
_
r
=
_
r
r
,
'
particular 1I' can be a natural boundary for the function B represented by (7.2.26 ) . )
321
Section 7.2. Infinite Products
[Hint: When k
Gkrn =
clef
{
re
iO
=
2, 3, . . .
1
and ° :::; m :::;: k
 T k :::;
r
:::; 1 ,
k
2m71"
 1, if 2( m + 1 ) 71" k
:::; () <
},
Gkrn i (/)
if E e n otherwise
{an L" EN {bkrn } 2"Sk<=, o "S rn
of for some enumeration a) n E N; b) E e = c) (7.2.26) converges.] The following paragraphs offer a development that leads to an elemen tary proof that (7.2.25) is not only sufficient but is also necessary for the convergence of the Blaschke product. z . clef = cPa" ( Z ) 0f t he facThe argument centers on the negatIves 1 tors appearing in the Blaschke product. These are of independent interest, e g., in the theory of conformal mapping, v. Chapter 8. 7.2.29 Exercise. If < 1: a) cPa E H (1U) ; b)
l anl :::; l an+l l , ({an} nEN r;
 an  an z
_
c) cPa (1U)
=
l al { I z I :::; I }
1U; d) cPa (1I')
=
'*
{ cPa 0 cP a ( z) =
z} ;
1I' .
l al
7.2.30 Exercise. If < R , then
3
7.2.31 LEMMA. (Jensen) IF 1 E H (1U) , Z(f) n 1U ::) = M < 00, THEN SUp II (z) I clef
I zl < l
z ft RcP ]';
(�) IS a
{an } l "Sn"SN
PROOF. If g clef =
N i . II cPa" (z), t hen  E H (1U) . For z fixed
n= l
g
m
1U and
Chapter 7. Meromorphic and Entire Functions
322
if f.l < <
r 1, the Maximum Modulus Theorem implies I z l = 1, then I g (z) 1 = 1 and lim min I g (w) 1 = 1. Ttl I wl=T
Furthermore 7.2.29d) implies that if
7.2.33 COROLLARY. ( Schwarz's Lemma) IF
J E H (1U) ,
I J (z)1 :::; 1,
sup
Iz l < l
AND
o
J(O) = 0,
THEN:
IJ(z)l :::; I z l if I z l < 1; 11'(0) 1 :::; 1.
( 7 . 2 . 34) (7.2.35)
z, J(z) e iO z. 1, 0,
IF EQUALITY OBTAINS IN ( 7 . 2 . 34) FOR SOME NONZERO OR IF EQUALITY OBTAINS IN ( 7 . 2 . 34) , FOR SOME REAL (J , == PROOF. The inequalities follow from 7.2.31 ( n = a l = and = . If  def = then h E H (1U) . The MaXImum Modulus Theorem implies in the current context that if < and = then h is a == constant function and If = then = The Maximum Modulus Theorem applies once more to show that h is a constant function. 0
J(z) h(z), z
I zl 1
I h(z) 1 1. h(O) 1.
11'(0) 1 1,
M 1).
I h(z) 1 1,
7.2.36 COROLLARY. IF , IN (7.2.34) , EQUALITY OBTAINS FOR SOME
1U, FOR SOME (J IN JR.,
J(z) MeiO nII cPan (z) . N
==
z IN
=l
PROOF. Schwarz's Lemma (7.2.33) applies to the function
N
Hence, if
Izl < 1 and J(z) = M nII cPan (z), for some 1] in 1I', k (z) = 1]Z. =l
r 0, J E H [D(O, rt] ' J(O) i 0, and Z( f ) n D(O, rt = { a l , . . . , an } ,
7.2.37 Exercise. If >
o
Section 7.3. Entire Functions
323
then (7.2.38)
7.2.39 THEOREM. IF 00
II sgn (an )
n=1
an 1 � an z
0 < l anl < 1,
n
E N, AND THE BLASCHKE PRODUCT
CONVERGES ( THUS REPRESENTING A FUNCTION
NOT IDENTICALLY ZERO AND IN H (1U) ) , THEN PROOF. Since
00
L (I  I anl ) < 00
n= 1
1 (0) i 0, 7.2.31 implies that for each N,
1
.
N
0 < 11 (0) 1 < II l anl n= 1 and 7.2.8 applies. 7.2.40 Exercise.
1 ' If 1 0' 1 < 1, then 4>�(0) 1  10'1 2 ; 4>;, (0') = 1  1 12 =
o
a
7.3. Entire Functions
1
Since a nonconstant entire function cannot be bounded, a study of the behavior of max M'\ R ; particularly for large R, is in order. Izl= R R. Nevanlinna [NevI] developed this subject in a very significant manner. Only the introductory aspects of the material are treated below.
I I(z) 1 �f
I),
1 an ao 0, M (R ; I) <2 a . Rn  1 0 I
7.3.1 LEMMA. AN ENTIRE FUNCTION IS A POLYNOMIAL OF DEGREE NOT EXCEEDING n IFF FOR SOME POSITIVE CONSTANT e, M(R; :s; eRn . PROOF. If + . . . + and i for large R, =
I(z) aozn
I) I zl, I I (z)1 1 C l z l n . series L Ck z k representing 1 implies that for all large R, if k=O
I)
Conversely, if, for some positive constant e , M(R; :s; eR" , for large The Cauchy formula for the coefficients in the power :s; + 00
k
>
n,
then (7.3.2)
Chapter 7. Meromorphic and Entire Functions
324
As R t 00 (7.3.2) implies
Ck
=
0,
z L Ck Zk . k=O M(R; f)
whence J ( ) =
n
n
Thus only those J such that for each . N, sup R>o attention. A convenient classification of the tions is achieved as follows. E,
7.3.3 DEFINITION. WHEN J E
m
Rn
o
= 00 , merit
orders oj growth of entire func
= lim p(f) clef
 In ln M(R' f) nR R+=
1 In M(R; f)
IS THE
order
IS THE type oj p(f) . oj J. IF p(f) E (0, 00 ) , THEN T(f) clef= R+= lim Rp 7.3.4 Exercise. If M(R; f) exp (RT ) , then p(f) = r. 7.3.5 Exercise. If M(R; f) = exp ( tRa ) , then T(f) = t. (1)
=
An entire function J is representable as a power series, i.e., =
z L cn zn ,
J( )
n=O
iO
1· I C I r if n � vf(r), , if n > vf ( r ) vf(r) is the central index and f.l, f(r) = I CVf (T) rVf(T) I · Thus CVf (T) ZVf (T) may be regarded as the last maximal term of the series. 7.3.6 Exercise. a) If p > 0, then M(r pf ) pM( r ; f) and f.l,pf(r) = pf.l,f (r). b) As a function of r, vf(r) is a rightcontinuous Z+valued stepfunction and if J is not constant, vf(r) t oo as r t 00. f 7.3.7 LEMMA. a) p(f) = inf { ). : I J( z ) 1 < exp ( I z l ), ) } � w(f) ; b) WHEN convergent for all
=
z in C. If z clef re
;
p(f)
< 00
, t hen n+= 1m n n = 0, and for some
=
AND
{ /'£ : /'£ > ° and for large r, M(r; f) < exp (/'£rP) } i 0, f THEN T(f) = inf K(f) � /'£(f). In ln M(r; f) PROOF. a) If E > 0, for large r, 1n r < p(f) + E , whence K(f) �
325
Section 7.3. Entire Functions
The Maximum Modulus Theorem implies that for all
z,
Hence w(f) :::; p(f). On the other hand, if < p(f), for some
{rn }nEN ' rn t oo and M (rn ; f) > exp (r�) , whence < w (f) : w (f) = p(f). b) For large r, InM(r; f) < IWP (f ) , whence T(f) :::; /'£(f) . If for some {rn } nEN ' rn t oo and M (rn ; f) > exp ((J"r� (f ) ) , whence (J"
(J"
(J"
/'£ (f) , T(f) : T(f) /'£(f). 0 A sense of the behavior of p and T is provided by the functions in =
r,
7.3.8 Exercise. a) If t > 0 and
J(z) �f
b) If then p(p)
=
eCz,
0;
J(z) cle=f
e
t
(J"
< <
Z , then r
p(f) = r and T(f) = t;
z
eP,
then p(f) = 00; c) If p E C[ ] , i.e., if p is a polynomial, d) If p E C[ ] and J �f then p(f) = degp; e) If 0 < <
z
00
Iql 1
J(z) �f L qn2 zn , then J is not a polynomial although p(f) = O. n=O 7.3.9 Exercise. a) If a > 0 and g(z) �f J(az), then p(g) = p(f). b) If and
0<
p(f), then p[exp(f)] = 00. c) p (f') = p(f).
7.3.10 THEOREM. IF J E
E,
THEN p(f)
=
ln ln (r ) . lim
T+oo
/lj
In r
PROOF. The Gutzmer coefficient estimate implies ln ln /l j (r) ((f) �f lim :::; p(f). In If ((f) = 00, then p(f) :::; ((f) . If ((f) < < 00 , for large < exp (rf3 ) :::; /l j
r
and if = then
( �)
T+oo r I en I rn ( r) is large, then lenl rn <
(3
� and
n
r,
( e(3:f3 ) 2e(3 �f �.
If
l8T� J M(r; f) :::; L lenl rn L lenl rn n=O n=r8T�1 1 ) (r ) L � �f I(r) II(r) < (r5r n=L8T(3J +
f3
+
/l j
00
+
00
T
+
15,
Chapter 7. Meromorphic and Entire Functions
326
whence . In ln M(r; I) < In ln[I(r) + II(r)] , hm In r In r lim II(r) = o.
T HJO
T + =
If 0 < x :s; y and r is large, the inequality x + y :s; 2y yields In[I(r) + II(r)] :s; In 2I(r), In I(r) :s; In ( 28ri3 ) + In llf (r) = In 28 + ,B ln r + In llf(r).
If some Cm ¥= O and r is large, 7.3.7b) implies Il(r) > I cm l r m ; 7.3.7a) implies p(pl ) = p( l ) and ((pI) = ((I). Hence it may be assumed that I Cm I > 1 , and so In Il f (r) > m In r. If m > ,B, the estimate applied earlier yields In 28 + ,B In r + In Il f (r) :s; 2 1n Il f (r) and ( r ) < In 2 + In In Il f ( r ) In In I'', In r
In r
whence ((I) ? p( l ) . 7.3.11 THEOREM. IF a > 0 AND Fa (z)
=
clef
n ) ;.; zn , then ( � n= l ae �
p (F,, ) = a and T (Fa) =
o

1.
PROOF. The maximization techniques of the calculus show
Because the coefficients in the power series representation of Fa are positive, M (r ; Fa )
=
=
L (:J rn , n= l 
n
�
a monotonely increasing function of r, whence · In ln M (r; Fa) . p (F", )  11m In r T + =
From 7.3.10 it follows that p (Fa) = a. Furthermore, if E > 0, for large r, dn r < In ln M (r; F", )  a ln r < dn r and r
_(
<
In M (r; Fa ) < r( . ra
o
327
Section 7.3. Entire Functions
The behavior of an entire function f is related to its Weierstrafi prod uct representation, in particular by the way in which Z(f) is distributed throughout C. Here is a list of some entire functions correlated with their orders and the cardinalities of their sets of zerOs:
f
p(f)
#[Z(f)]
eez
00
o o n
1 1
z eZ
sm
n k L ak z k=O sm z
No
0
1
The picture that emerges is far from clear because in the list no atten tion is paid to the relative density of the set of zeros, e.g.,
#[Z(f)
n D(O, R)] R
Two useful measures of the frequency of occurrence of the zeros of f clef are, when a clef = Z (f ) , =
{ an }nEN
. = mf v(a) clef clef
{
a
: a
E
JR.,
" I
� la l < 00 un EZ( f ) \ {O} n "
}
= the exponent of convergence of a,
8(a)
}
{
�f sup m : m E Z+ , L la n11 Tn = 00 an E Z ( f ) \{O} �f the exponent of divergence of a.
7.3.12 Exercise. If F(z)
1
�f IT Eh ( :n ) is a canonical product (of genus
h), then h :s; v(a) :s; h + and h :S; p(F). 7.3.13 Exercise. If p (f) p (g) , then p(fg) = p(g). If p(f) = p(g), then p(fg) = p(f) [= p(g)]. 7.3.14 Exercise. a) For some sequence a, t oo and v(a) 8(a) = 00
<
lan l
=
b) The numbers v(a) and 8(a) are both finite or both infinite. 7.3.15 LEMMA. IF f E E AND Z(f) =
f
{an }nEN � a, THEN v(a) :s; p(f) .
.
328
Chapter 7. Meromorphic and Entire Functions
p(f) = #[Z(f)]
p(f)
PROOF. If < 00, it may be 00, the result is automatic. If assumed that l an l :::; l a n + l l and that J (O) = 1 (whence l al l > 0). Since = No, J is not a polynomial function. Thus, for each M (r; is large. (For a polynomial function , in N, if r is large, then J rn is finite and neither v ( a) nor J ( a) is defined. If the definition of v ( a) is extended in a natural way to apply to finite sequences, for any finite sequence a, v ( a) = 00 < 0 = i.e., the result is automatic.) If is large, large r, and positive n < 1 . Thus, for large
n
#[Z(f)]
f)
p(f),
�
n
la l
E,
n,
f) < exp I�I exp (e[p(f)n E] ) P(f)+' , achieved when The minimum of is <
a� 
[rp(f) +<j . rn
M(r; rn
+
[rp(f) +<j rn
n is large, p(f� E] ) P(t\+, , and if J E, then (e[ < Hence I I L 1 I < 00. Furthermore, E and J are arbitrary positive numbers. The conclusions above are encapsulated in
which is large, since
+
1 an
>
1 P (f ) +,s n = l an 00
o
7.3.16 THEOREM. (Hadamard) IF
FOR SOME POLYNOMIAL P SUCH THAT deg(p) SOME P(z) (cf. 7.2.15) ,
canonical product
J (z) PROOF. Since
p(f) �f
P
E,s(a)
=
:::; p(f), SOME
exp[p(z)]z k P(z).
< 00, v ( a) + J ( a) < 00. In
( z ) clef ( an
=
z
1an
) exp [P (z)] , n
k
IN Z+ , AND
Section 7.3. Entire Functions
329
Pn is a polynomial and deg (Pn ) = J(a). Then 7.2.10 implies whence
IT E8(a) ( :n ) converges and defines an entire function F. For
I(z) some ), in N, A Z F( z) 1 (z)
=
ZA
n (R)
g ( 1  :n )
x
[ [ � P ] JL, (1  :. ) cPO" '] x
(z)
qR(Z).
R)O, In ( 1  :n ) may be determined so that
and if hR(Z) � p(z) + qR(Z) = e h R ( z) . If Z E D(O, then
,
exp P (z) +
�f PR(Z) In D(O,
. functIOn . p, i ° and hence, for some entIre
n ( R)
� Pn ( z) + n= �) l [In ( 1  �J + Pn ( Z )] , then r +
Rt and
ex:>
n
>
# [Z(f) n D(O,
2Rr l �f n(R) ,
I �, I < �. The power series representing In ( 1 :n ) implies I In (1  aZn ) + Pn (z) I :::; l2Ranl88((aa)+1) +1 · 
Hence the definition of J (a) implies that for large n in the infinite series representing hR, the terms are dominated in absolute value by the terms
2R8 (a) +1 . � l anl 8 (a) +1 If R > 1 and Izi = 2R, then IpR(z) 1 2': 1 , whence M(2R; f ) 2': M (2R; qR) . 00
of '""'
330
Chapter 7. Meromorphic and Entire Functions
Thus in D(O, R) , � [hR(Z)]
I M(2R; I) ,
hR(Z) �f then I bm I
<
2
I
1
,
n.
, < 2
p (n) (o) n.
<
00
L brr/n , m=O
In M(2R; I)  � (bo ) ,m E Rrn n p ( ) (0) 1
n > p, then bn =

and according to 5.3.34, if
. . shows that If. N . DIrect calculatIOn
CX) n and L k=n R +l nak ( )
In M(2R ; 1)  � (bo )
Rn
+
<
�
1
__ . n � k=n ( R) +l I a n I
(7.3 . 1 7)
If p + E n, then In M(2R; I) (2R)P+€ and since n > v(a), R + 00, both terms in the right member of (7.3.17) approach zero. It follows that if n > p, p (n l (O) = 0, i.e., p is a polynomial deg(p) :::; [pl . 0 7.3.18 COROLLARY. IF f E E, p(f)
as
<
00 ,
AND p(f) tJ. N+ , i.e., IF
p( f ) E [0, (0) \ N+ , FOR EACH a IN C, #[Z(f  a)] 2': No ·
<
( )
a PROOF. If #[Z(f  a)] No , for some polynomial g , Z f g � = (/) and 7.3.16 implies for some polynomial p, f a = eP • Hence p(

f  a ) [= p(f)]
=
p.
o
[ 7.3.19 Remark. In 7.3.18 there are resonances with the Weier strafiCasorati Theorem (5.4.3c)) and the Little Picard Theorem (v. Chapter 9) and the general phenomenon of defective func tions . ]
The development above is rounded out by the following items. The first is an application of contour integration. The others are formulre that relate in a direct way the density of the zeros of an entire function f to
p(f ) .
7.3.20 Example. The existence and evaluation of
Section 7.3. Entire Functions
331
turn on the existence, for E in (0, 1) , of the (improper) integral,
For small positive t,
1E
l In 1 1  e it I I = � l In 12  2 cos ti l l 2 � � ( In 2 + 2 1 ln sin � I � I I ) � + l In I � I I '
Iln xl dx =  limo (x ln x  x) I �= E  dn E. Hence J exists. 8t For z fixed, z  w a function of w is in H (1U) and is never Zero in 1U. Hence In Iz  w i E Ha(1U) MVP(1U) , and so
and
°
as
1
=
1271"
271" 0
l
In I z  e i t dt = In I z i .

Furthermore, In Iw  1 1 and In Iw  zl are negative, but Iw  1 1 < Iw  zl, whence l In I w 1 1 1 > l In Iw  zl l · In Figure 7.3.1, 0 < r < 0.5, z 1 r , and w lies on the arc ABC.
=
+
0
Figure 7.3.1.
l +r=z
Chapter 7. Meromorphic and Entire Functions
332
w z [1, 1.5]' I w  zl l I w  zl l r 0 z 1) I zl 0, 1 271" . 1 271" . lim  1 ln lz  e,t l dt =  1 In I1  e,t l dt = 0. 27r 0 27r 0
In '][' \ {AlB}, l In is bounded. In sum, for in '][' and in is dominated by an integrable function. Hence a passage to the l In limit as ..l (and + is justified. Because liml In = it follows that z+
rto
7.3.21 THEOREM. (Jensen ' s formula) IF
f E H (D(O, Rn , 0 < r < R, f(O) i 0, AND (f) n D(O, r) = { } (A ZERO OF ORDER k OCCURS k TIMES) , THEN r 1 271" In I f (O)1 + 2:)n  =  1 In I f (reiO) I dO. I I 27r 0 [ 7.3.22 Remark. Jensen ' s formula is an analog of the Poisson Jensen formula (6.2.11). ] al , "
N
n= l
"
aN
an
PROOF. There are three cases. The first provides the context for the other two. Case 1. If i in a, Rt , Jensen's formula is valid in the cir cumstances because In E HaIR a, Rt] ' and thus Jensen's formula is an expression of the MVP. Case 2. If c D(O, rt , for any function for which = n n Case 1 applies to L
f 0 D( I f I [D( Z(f) n D(O,r) Z(f) D(O, r) Z(g) D(O, r),
g
g �f h: In lh(O)1 = 2� (1271" In If (reiO)I dO)  2� (1271" ln l g (eiO)1 dO) . (7.3.23) The choice of g is rather unrestricted. Owing to the special and useful properties of the functions cPa (v. 7.2.29), g(z) �f II [cP�n (z)] may be used. Then (7. 3 . 23) is In l h (0)1 = 27r1 (10 271" In lf (reiO)l dO)  27r1 ( 10 271" � ln l cP arn (eiO) l dO) . (7.3.24) Since I cP"rn (eiO) I = 1, at most the first term in the right member of (7. 3 . 24) N
n= l N
is not Zero.
Section 7.3. Entire Functions
333
Furthermore, In I h(O) 1 = In 1 1 (0) 1 
N
2:)n I
n= l
In I h(O) 1 = In 1 1 (0) 1 +
N
� In I L l ·
Case 3. If Z(f) n 8[D(a, r)l i 0, Z(f) may be enumerated so that
< r and l am +l l . . . l aN I r. r 1 :::; :::; N. The g in Case 2 is reIn Jensen's formula, In 0, l an l 10' 1 1 :::; · · · :::; l am l
=
=
=
n
m+
=
placed by ?i defined by the formula
(7.3.25) I . (r, R) , and h by h clef = :::; . For some s m
g
s
h E H [D(O, t l and Z Thus
(h) n D(O,
s
t = 0.
< < N , then
clef However, 1· f a n = re iOn , m _
In
n
_
I h (re iO ) I = In I I (re iO ) 1  n=m+ L In 1 1  e i(OOn ) I · N
From 7.3.20 it follows that for each n ,
1271" 1 1 o
Consequently
In

l
e i(OOn)
I dO = O.
(7.3.27 )
(7.3.28)
(7.3.29)
Chapter 7. Meromorphic and Entire Functions
334
The contents of (7.3.26)(7.3.29) imply the required conclusion. 0 [ 7.3.30 Note. Jensen ' s formula is a valuable tool in the extended study of E, v. [NevI], and of the Hardy
HP (lJ) , 0 �
p
spaces
� 00 ,
as defined below, v. [H i l , Rud]. When f E
H (lJ), and
{
In + t �f In t if t 2: 1 ' o if t < l 27r dO exp In + f 2 27r dO P f 2 sUPO S097r f
{ � 1 I (re iO ) I } { � 1 I (reiO ) I P } 1
if p = 0 if 0 < P < 00 ' p =
if 00 I (re iO ) I then Mp ( f r) a function of r increases monotonically on [0, 1), ;
as
v. [Ge3, PS, Rud] . Moreover,
HP (lJ) �f { f : f E
7.3.31 Exercise. If
H(lJ), Il f llp �f lim ..... l Mp (f; r) < 00 } .] r
n(r; I) �f #[Z(f) n D(O, r ) ] and f(O) n(r; I) � In M(2r; I) ;
=
1: a)
Inn(r; 1 ) � p( f ) . ..... = In r [Hint: Jensen's formula applies.]
b ) lim r
7.4. Miscellaneous Exercises
lJe
7.4.1 Exercise. If C fl, f E M ( fl) , and f (1I') c 1I', then f is a rational function. 7.4.2 Exercise. If f E M (([= ) , f is a rational function. =
7.4.3 Exercise. If
=
L I an  bn l < 00 , then II : = �: converges in 71=1 n= l
Section 7.4. Miscellaneous Exercises
335
and represents a function in M (Q) . =
7.4.4 Exercise. For
I(z) �f � zn and, when
what are vf (r) and vg (r)?
a >
0,
= n g(z) �f � (:! ) O! '
r = II(z ) l l , z , = r .:s: e1 z1a } . b) If I(z) �f L Cn z n , by abuse of notation when Cn = 0, n=O 7.4.5 Exercise. a) p (f) = inf =
{a : a 
2:
.....
0, lim
n ln n
n+ CXJ _ n I en I . 1 c) If p (f) < 00, then T(f) =  lim n I cnl ep (I) n = p(f) = lim
1
.....
£ill. n •
7.4.6 Exercise. (The Open Mapping Theorem for meromorphic func tions) If 1 is meromorphic in a region Q, then ( Q ) is an open subset of C= .
I
=
7.4.7 Exercise.
nEN I cnl < 00.
If
then sup
[Hint: # (S n 1UC )
I(z) = L Cn z n and S(f) n 1Uc = S(f) n 1I' = P( f ), n=O
< 00.]
II(z) l l lI'=
7.4.8 Exercise. If 1 E [H (1U)] n [C (1UC , C)] and K, 1 is a ratio nal function. [Hint: The Schwarz Reflection Principle and the CauchyRiemann equations apply.] 7.4.9 Exercise. a) For the meromorphic function
T : C \ {i} 3 z ft
zz z + z. ,

T (n + ) = 1U. b) Is there an entire function 1 such that 1 (n + ) = 1U?
8
Conformal Mapping
8.1. Riemann ' s Mapping Theorem
In each of 5.5.85.5.11, 5.5.175.5.19, 7.1.21, and 7.1.22 a simply con nected region Q plays a central role. Combined with 8.1.1 below, the contents of the cited results pro vide a useful edifice of logically equivalent characterizations (v. 8.1.8) of simply connected regions in C. 8.1.1 THEOREM. (Riemann) IF Q IS SIMPLY CONNECTED AND Q i C, FOR SOME UNIVALENT f IN H(Q) , f(Q) = 1U. [ 8.1.2 Remark. The result 8.1.1 was stated by Riemann. It was first proved by Koebe who created an algorithm for constructing a sequence {fn } nEN of univalent functions in H (Q) . He showed that for some univalent f in H (Q), fn � f on each compact subset of Q and f(Q) = 1U. Riemann's Mapping Theorem is frequently called the Conformal Mapping Theorem. The term conformal refers to the fact that the mapping f preserves angles (v. 8.1.6, 8.1.7). The PROOF below consists of the crucial 8.1.3 LEMMA followed by the main argument. The line of proof is nonconstructive (ex istential) and is based on the ArzelaAscoli theme as expressed by Vitali's Theorem (5.3.60). Other tools in the argument are Schwarz's Lemma (7.2.33) and the functions cPa used in the study of Blaschke products (v. 7.2.297.2.31 and 7.2.36) .] PROOF. 8.1.3 LEMMA. IF Q IS SIMPLY CONNECTED AND Q i C, FOR SOME UNI VALENT g IN H (Q) , g(Q) c 1U. 336
337
Section 8.1. Riemann's Mapping Theorem
PROOF of 8.1.3. If (C \ flt i 0, even if fl is not simply connected, for some b and some positive r, D(b, rt c C \ fl and 9 : fl 3 z ft � meets zb the requirements. On the other hand,
fl �f C \ (

00 , 0] =
{ z : z = Re iO , R > 0,
Jr
< (J <
Jr
}
is also simply connected but (C \ flt = 0 . The function 9 above is use less. For this fl and in general, the importance of simple connectedness is revealed. If a tJ. fl, for some h in H (fl), [h(z W = z  a (v. 5.5.19) . If then Z 1 a = Z2 a, whence Z 1 = Z2 : h is univalent. The Open Mapping Theorem (5.3.39) implies h(fl ) contains some a other than 0, hence, for some small r in (0, 10'1 ) , 0 tJ. D(a, r) Co h(fl ) If Z2 E fl and h ( Z2 ) E D( a, r), then h ( Z2 ) E D(a, r) and so for some Z 1 in fl, h (zd =  h ( Z2 ) . But, shown above, Z 1 = Z2 , 

.
as
h (zd = h (zd = 0 E D(a, rt, a contradiction: D(a, rt � h ( fl ) . Hence, if z E fl, then I h(z ) + 0'1 > r and cle r 9 =f
h+a
meets the requirements. 0 PROOF of 8.1.1. Vitali's Theorem (5.3.60) implies that the nonempty family F of univalent maps of fl into V is precompact in the I ll aoinduced topology of Hb(fl), the set of functions bounded and holomorphic on fl. For b fixed in fl, if k E F, then h �f [k  k(b)] E F and h(b) = O. If
�
h'(b) = I h'(b) 1 e i O and 9 �f e io h, then 9 E F and g'(b) > O. Hence atten
tion is focused on the nonempty set 9 of functions 9 such that: 9 is univalent in fl , g( fl ) c V, g(b) = 0, and g ' (b) > O. If M = sup { g' (b) : 9 E 9 }, for some sequence {gn } nEN contained in 9, g� (b) t M. Hence, for some subsequence, again denoted {gn } nEN' and some 9 in Hb(fl), g n � 9 on each compact subset of fl and g� (b) t M. Furthermore (v. 5.3.35) , nlim g�(b) = g'(b) = M < 00 The next arguments show that: a) 9 is univalent, whence 9 E 9; b ) g(fl) = V. a) . Since each gn is univalent, M > O. If Zo E fl, for each n, ..... =
.
Chapter 8. Conformal Mapping
338
Hurwitz's Theorem (5.4.39) implies
g(z)  g (zo ) == or [g (z )  g (zo ) ] l n\{zo} i 0.
°
If g(z) == g (zo ), then g ' (b) = 0 < M, a contradiction. Hence, for any Zo in Q, [g(z)  g (zo ) ] l n\{zo} i 0: g is univalent. b) If lei < 1 and e tJ. g(Q), the simple connectedness of Q enters the
g(z)  e is 1  cg (z ) welldefined and in H (Q) . Direct calculation shows G is univalent on Q and G ( Q ) C 1U. G(z)  G(b) , then H E 9 . Since Finally, if H : Q 3 z ft cPO ( b) (Z) = 1  G ( b )G (z ) Ii:::I 2 Ii:::I , lei (1  y l e i ) = 1  2 y lei + lei > 0, H (b) = G'(b) 2 = 1 +fi:I ' M > M, 1  I G (b) 1 2 y lei
argument again: 5.5.19 implies G : Q 3 z ft JcPc ( g (z ) =
a contradiction.
0
8.1.4 DEFINITION. WHEN (J E [0 , 27r) , THE straight line through a at incli nation (J I S L( a, (J) cle=f { z : z = a + te t° O , t E }
ffi. .
8.1.5 Exercise. If L is a straight line in C and a E L, for a unique (J in [0 , 27r) , L = L ( a, (J). 8.1.6 Exercise. (Conformality, first version) If
f E H (Q) , a E Q, J'(a) i 0, and L �f L (a, (J) : a) For t in [0, 1] , the equation /' (t) = f (a + te i O ) defines a curVe through f(a) . b) For the line L (f(a), cP) , tangent to /' * at f(a),
cP  (J E Arg [J'(a)] . c) For two differentiable curveimages intersecting at a (whence their f images intersect at f (a)) , the size of the angle between their tangents at a and the size of the angle between the tangents to their f images at f (a) are the same. [Hint: b) The chain rule for derivatives applies to the calculation of /,' . ]
339
Section S.l. Riemann's Mapping Theorem
8.1.7 Exercise. (Conformality, second version) If
f �f u + iv E H (Q) , J'(a) i 0, and "/ �f x + iy is a differentiable curve such that ,,/(0) = a: a) x'(O)e l + y'(0)e 2 is a vector parallel to the tangent line at ,,/(0); cle b) for U cle =f u 0 "/ and V =f v 0 ,,/,
U'(O) = u x (a)x' (O) + u y (a)y' (O) , V'(O) = v x (a)x'(O) + vy (a)y'(O);
)
c) U'(O)e l + V'(0)e 2 is a vector parallel to the tangent line at f 0 ,,/(0); u (a (a d) for some ¢ in [0, 27r) , the matrix V x a ) u y a ) is a multiple of the (orthogonal) matrix
( coSlll� ¢,A/,
( x( ) Slll � ) ; cos ,/,
Vy ( )
e) the vector U'(O)e l + V'(0)e 2 is the vector x'(O)e l + y'(0)e2 rotated through an angle of size ¢.
[Hint: d) The CauchyRiemann equations apply.] The two versions of conformality may be reworded as follows. If f E H (Q) and f is invertible at a, f preserves angles at a and the sense of rotation at a. Central to the phenomenon are the CauchyRiemann equations, i.e., the differentia bility of f. The material in 10.2.46 is related to the current discussion. 8.1.8 THEOREM. FOR A REGION Q, THE FOLLOWING STATEMENTS ARE LOGICALLY EQUIVALENT: a) Q AND 1U ARE HOMEOMORPHIC; b) Q IS SIMPLY CONNECTED; c) IF a E Coo \ Q AND "/ IS A LOOP FOR WHICH "/* C Q, Ind y(a) = 0; d) Coo \ Q IS CONNECTED ; e) IF f E H (Q) , FOR SOME SEQUENCE {Pn } nEN OF POLYNOMIAL FUNC TIONS, Pn � f ON EACH COMPACT SUBSET OF Q; f) IF f E H (Q) , "/ IS A RECTIFIABLE LOOP, AND "/* C Q, THEN
i f dz
= 0,
i.e. , IF "/ IS A RECTIFIABLE CURVE AND "/* C Q, THEN
i f dz DE
PENDS ONLY ON ,,/(0) AND ,,/(1) AND NOT ON THE PARTICULAR CURVE "/;
Chapter 8. Conformal Mapping
340
g) IF 1 E H(Q) AND 0 tic I(Q) , FOR SOME F I N H (Q) , F' = I; h) IF 1 E H(Q) A N D 0 tic I(Q) , FOR SOME G I N H (Q) , 1 = exp(G) ; i) IF 1 E H (Q) , 0 tic I(Q) , AN D n E N, FOR SOME H I N H (Q) ,
I = Hn .
[ 8.1.9 Remark. Listed below are implications already estab lished and their provenances. These implications and their deriva tions are the root of the subsequent argument. The entire set is intimately related to 8.1.1.] Implication b) '* d) b) '* e) b) '* f) b) '* g) b) '* h) b) '* i) PROOF. a)
'*
Provenance 7.1.21 7.1.19 5.5.16c) 5.5.17 5.5.18 5.5.19
b) : If "( is a loop, "(* C 1U, and J(t)
F(t, ) clef = S"((t)
==
0, then
s
is a homotopy such that "( '" F,1U J. If \11 : 1U r+ Q is a homeomorphism and f r is a loop such that r* c Q, then \11  1 0 r � "( is a loop such that "(* C 1U and \11 0 F 0 \11  1 �f
Ll.
b) '* c) : v. 5.5.16c ) . c) '* d): v. the PROOF of 7.1.21. d) '* e): v. 7.1.14. e) '* f): Every polynomial is a derivative and f) obtains for derivatives. f) '* g): If { a, z} C Q, there is a polygon n connecting a to z and a corresponding "( such that "(* = n . For the map "' ''' , n
f
F : Q '3 z r+ F(z) �
11 I b (t)] d"((t) ,
F'(z) = I(z). Owing to f), F(z) is welldefined, Le., is independent of the choice of "( so long as "((0) = a. f' f' g) '* h): For some F, F' = 7 ' If
(
 c)
clef
= exp(G) .
Section 8.1. Riemann's Mapping Theorem
341
( �).
h) '* i): H exp i) '* a): If Q j. C, since the heart of the PROOF of 8.1.1 is the existence z of H when n = 2, a) follows. If Q = C, the map C '3 z r+  E 1U is a 1 + 1 z1 homeomorphism. D
=
8.1.10 THEOREM. IF f IS A conformal selfmap (holomorphic autojection) OF 1U, FOR A (J IN [O, 27r ) AND AN a IN 1U, f = ei O ¢a (WHENCE f  1 E H (1U) , i.e. , f IS biholomorphic) . PROOF. If f is a holomorphic autojection of 1U, for some (unique!) b in 1U, f(b) 0.5. Hence g ¢0 . 5 0 f is a holomorphic autojection of 1U, and g(0.5) = o. Since ¢  b 0 g(b) = b, Schwarz's Lemma (7.2.33) applies. D 8.1.11 Exercise. For f as in 8.1. 10, what is f  1 ?
�f
=
8.1.12 Exercise. If Q is a simply connected proper subregion of C and a E Q there is no unique conformal map f of Q onto 1U and such that f(a) = o. There is a unique conformal map f such that f(a) = 0 and

f'(a) > o. [Hint: If f and g are two maps as in 8.1.10 applies to f o g I . ]
8.1.13 Exercise. In 8.1.12, the f such that f(a) = 0 and f ' (a) the unique solution of the problem of finding in F an f such that
f' (a)
= sup { h' (a)
>
0 is
: h E F} .
In effect, 8.1.13 restates the Riemann Mapping Theorem as a result in the calculus of variations. Lemma 8.1.3 provides a motivation for Rie mann's original attempt to prove 8.1.1. According to 4.7.18, if f : Q r+ 1U is univalent and holomorphic, the CauchyRiemann equations imply that the area of f(Q) is
A[J(Q)] =
�f in 1 f' (z) 1 2 dx dy
in (u; + u�) dx dy = in (v; + v�) dx dy ::::: 1.
Hence, if A[J(Q)] is maximal, e.g., if A[J(Q)] = I, f is a good candidate for the biholomorphic map of Q on 1U. Euler's equations for the stated vari ational problem take the form t:m = v = 0 and the boundary conditions for and v are simply that for all a in 8(Q), lim l (z) 1 lim I v(z) 1 ::::: 1. The z=a z = aregion, for some discussion in 6.2.176.2.22 implies that if Q is a Dirichlet 0, lim l u(z) I , i.e., Dirichlet ' s problem has a solution. z=a
u u, l1u ln= u la( n) =
l1
u V
Chapter 8. Conformal Mapping
342
Riemann's proof of 8.1.1 involved the implicit assumption that there is a solution to the problem of maximizing (u ; + u�) dx dy subject to
In
In
the normalizing condition, (u; + u�) dx dy � 1. The assertion that this kind of variational problem has a solution became known as Dirichlet 's
Principle.
However there appeared 8.1.14 Example. (Weierstrafi) Among all 1 in C OO (lR, lR) such that I(x) = 0 if I x l ::=: 1 and 1'(0.9) = 1 '(0, 9) = 1 there is none for which
11 { t2 + [J' (t)] } 2
�
dt is least. (The problem is to minimize the length of a curve "( : [0, 1] '3 t r+ t + i l (t) such that "((0) "((I) = 1, "( t o. The =
geometry of the situation shows that the infimum of all such lengths is 2 but that the length of each such curve exceeds 2.) Thus Dirichlet's Principle, as a statement about the solvability of a variational problem, was suspect. In the hands of (alphabetically) Hilbert, Koebe, Konig, Neumann, Poincare, Schwarz, Weyl, and Zaremba, the validity of Dirichlet's Prin ciple for a simply connected fl achieved a semblance of validity. In mod ified form, Dirichlet's Principle is central to one of the derivations of the Uniformization Theorem, v. 10.3.20. Given the validity of Dirichlet's Principle for a si�ply connected fl, Riemann ' s approach leads to a u and a harmonic conjugate v such that 1 u + iv is the required conformal map of fl onto 1U. The intimate connection of Dirichlet's Principle to the solvability of Dirichlet's prob lem seems to bind the two to questions about Dirichlet regions, barriers, simple connectedness, etc. The PROOF of 8.1.1 resolves these questions. The discussion in Section 8.5 is also germane to the considerations above. For the harmonic function g corresponding to a Green's function G(·, a), v. 8.5.1 8.5.9, some harmonic conjugate, say h, of g leads to a function ¢ h + i g E H (fl) . If, for z in fl, I(z) (z a) exp[¢(z)] , then is a conformal map of fl onto 1U and I ( a ) o. 1
�f
�f
�f

=
8.2. Mobius Transformations
If Z; , 1
:::; i :::;
4 , are four elements of C, the number
is their cross ratio or anharmonic ratio. For Z2 , Z3 , Z4 fixed and pairwise different, X (z, Z2 , Z3 , Z4 ) M(z) is a function on Coo \ { Z3 } and may be
�f
Section 8.2. Mobius Transformations
00 There are constants a,
extended to Coo by defining M (Z3) to be a z + b clef such that M(z) Tabed(Z) and d
= cz =
.
+
343 b, c, d
More generally, when l1 j. 0, the map
Tabed : C '3 z r+
cz az + b d +
is a Mobius transformation. By definition,
(  � ) 00 and Tabed(oo)
Tabed Correspondingly, for
e
=
as
=
�.
in Section 5.6, there is
� clef e  1 T"b(.d8 : L: 2 \ { (0, 0, 1) } r+ L: 2 , Tabed =
L: 2 •
which may be extended by continuity to a selfmap of When ambiguity is unlikely, the subscript abed is dropped. Note that if a j. 0, then
=
1 whence, if a V75. ' then (a a) (ad)  (ab) (ac) = 1, and as the need arises the value of l1 may be taken to be 1.
8.2.1 Exercise. Each Mobius transformation T is invertible and
:

1 C '3 z r+ dz + b = T( d) be ( a ) . Tabed cz  a (Thus each T is oneone: {T(z) T (z') ) {} {z = z' } . )
=
8.2.2 Exercise. a) The set of all Mobius transformations Tabed is a group with respect to composition 0 as a binary operation. b) Those for which l1 = a d  bc 1 is a normal subgroup M 1 contain ing the normal subsubgroup E �f { TW01 ' T( l )OO( l ) }. c ) The quotient group M d E, denoted is isomorphic to SL(2, q , the multiplicative group of all 2 x 2 matrices M with entries from C and for which det (M) 1.
M
=
=
Mo,
344
Chapter 8. Conformal Mapping
zP
I w  Z l 2 = I Z  a l · I ZP  Z l = I Z  a l · I ZP  a l  I Z  a l 2 = I Z  a l · I ZP  a l  I W  ZI 2 ) , I Z  a l · I ZP  a l = r 2 . 
(r2
Figure 8.2.1.
When r > 0, z j. a, and k clef =
1
2
r za
1 2 ' the pomt .
zP clef = a + k(z  a) is the reflection or inversion of z in Ca(r) �f a [D( a, r rl and z is the reflec tion of z P in Ca(r): z = (zP) P . By abuse of notation, aP = 00 and ooP = a. Figure 8.2.1 above illustrates the geometry of reflection or inversion in the circle Ca(r). For a line L( a, 0) the reflection of z � a + z  a l in L( a, 0) is
1 Z P clef = a + 1 z  a 1 e  i ( q,  2 0 )
eiq,
and z is the reflection of z P in L( a, 0) . 8.2.3 Exercise. The reflection of z in the line L (O, O) IS z . For L(a, O) regarded as a mirror, zP is the mirror image of z.
345
Section 8.2. Mobius Transformations
The superscript P serves as a generic notation for a reflection z r+ zP performed with respect to some circle or line. 8.2.4 THEOREM. IF z E C AND ad be j. 0, THEN Tabcd(Z) ARISES FROM THE PERFORMANCE OF AN EVEN NUMBER OF REFLECTIONS (IN LINES OR CIRCLES) . 
PROOF. The argument can be followed by reference to Figure 8.2.2. For any z, TOl lO (z) arises by reflecting z in '][' and reflecting the result in JR. If b j. 0 there are (infinitely many) pairs of parallel lines L 1 , L 2 , sepa and perpendicular to the line through b and o. Direct calcu rated by lation reveals that T1 b0 1 (Z) arises by reflecting z in L 1 and reflecting the result in L2 •
I�I
Z1: the reflection of z in L1
z12: the reflection of z1 in L2
o
Figure 8.2.2.
346
Chapter 8. Conformal Mapping
If ° :::; (J < 27r, then TeiO OOl (z) arises by reflecting z in L3 and reflecting the result in L4• If ° < A E lR, then TA00 1 ( Z ) arises by reflecting z in '][' [= Co (I)] and reflecting the result in Co Finally,
(VA).
T.abed (z) 
{
r::.
be  ad + ( z d) if e j. ° e ee + b a if e = O z + d d
D
8.2.5 Exercise. The elements (z, z') in C� are a pair of mutual re flections in a circle C resp. a line L iff X (z', Z I , Z2 , Z3 ) = X (z, Z I , z2 , Z3 ) is meaningful, , i.e., # (z', Z I , Z2 , Z3 ) # (z, Z I , Z2 , Z3 ) = 4, and true. =
8.2.6 Exercise. The validity of X (z', Z I , Z2 , Z3 ) = X (z, Z I , Z2 , Z3 ) is inde pendent of the choice of an acceptable triple Z I , Z2 , Z3 . 8.2.7 Exercise. If T E or 2.
M \ {id }, the number of fixed points of T is 0, 1 ,
1 . sgn a 8.2.9 Exercise. If {Z I ' Z2 , Z3 } resp. { W I , W 2 , W3 } are two sets of three points in Coo , for precisely one T in T (Zi ) = Wi , 1 :::; i :::; 3. 8.2.10 Exercise. (Extended Schwarz Reflection Principle) If: a) r > 0; b) A dcl = { a + ret 0 :::; (J l < (J < (J2 :::; 27r } ; c) 8.2.8 Exercise. If ° < 10'1 < 1 the fixed points of 4;" are ±

Mo,
e
:
1 E H [D(a, rt] n C [D(O, atl:.JA, q ;
d) 1 (A) C Cb ( R) ; e) z r+ zPa resp. z r+ ZP b is the map z r+ zP performed with respect to Ca(r) resp. Cb ( R) ; f)
F(z) then Q
�f
{
if I z  a l < r I( z) [J (zPa Wb if I zPa  a l < r , if z E A; I(z)
�f D(a , rtl:.JAl:.J [D(a, rtJ Pa is a region and F E H (Q) .
8.2.11 Exercise. If K is a circle lying on L 2 , then 8 (K \ { (O, O, I)}) is a circle or a straight line. 8.2.12 Exercise. The group is generated by the subset
M
1
To : C \ {a} '3 z r+ , Tab : C '3 z r+ az + b, a, b E C. z 
Section 8.2. Mobius Transformations
347
8.2.13 Exercise. a) If .c is the set of all circles and straight lines and then T(.c) = .c. b) If D is the set of all open discs and the comple ments of all closed discs and T E then T ( D) = D. 8.2.14 Exercise. If a, b , c, d E lR and ad  bc > 0, then Tabed leaves n + invariant: Tabed (n + ) = n + .
TE
M,
M,
8.2.15 DEFINITION. FOR A REGION Q, Aut (Q) IS THE SET OF CONFOR MAL AUTOJECTIONS OF Q. 8.2.16 Exercise. If Q is a region, with respect to composition 0 as a binary operation, Aut (Q) is a group. 8.2.17 Example. According to 8.1.10, Aut (V)
{ T : T(z) = e iO ¢,,(z ) ,
=
a
E V, 0
M.
::;
(J <
27r
},
I E Aut (C) iff for some constants a and b, I(z) az + b. b) I E Aut (Coo ) iff I E M. c) If I E Aut (V) , then 1 1(0) 1 = 1 1  1 (0) 1 . a proper subgroup of 8.2.18 Exercise. a)
==
8.2.19 Exercise. For Z 1 and Z2 in a region Q and a subgroup G of Aut (Q), the relation rvG defined by {Z 1 rvG Z2 } {} {Z 1 E G (Z2 ) } is an equivalence relation. The set QIG (the quotient space ) consists of the rvGequivalence classes of Q. 8.2.20 Exercise. The set VIAut (V) is a single point. If k E N and = 1, then G �f { n : n E N } is a finite subgroup that may be identified with a subgroup of Aut (V). The set VIG may be identified with the open sector
{ reiO : 0
w
::;
r<
1,
0 < (J <
The operator Sch : H (Q ) '3
wk
2: } .
I ( 7'II ) '  "2 ( 7'II ) 2 �f {f, } 1
r+
is the Schwarzian derivative. The operator
I"
Lg : H(Q ) '3
I'
I
r+
(ln l' )
' cle=f L
[I'
f is the logoid derivative. If ( z ) j. 0, for any determination of In ( z ) ] ' 7' {In ( z ) ] } is unambiguously defined. Hence both {f, z } and L f ( z ) are welldefined if ( z ) j. o. =
[I' '
I'
Chapter 8. Conformal Mapping
348
8.2.21 Exercise. a) {J 0 g, z} = {J, g} [g ' (z)] 2 + {g, z}; b) for a Mobius transformation T, {T, z} = 0 and {T 0 g, z} = {g, z}; c) Laf + b = Lf; Items b) and c) above serve as motivations for introducing Sch and Lg : Sch is Mobiusinvariant while Lg is invariant with respect to an important The next result is the converse of b) above. subgroup of
M.
8.2.22 LEMMA. IF {J, z} = 0, THEN f E
M.
PROOF. If F � Lg (I) , because {J, z} = 0, it follows that 2F' = F 2 . Hence
1 2' 2 i.e., for some constant a, Lg (I)(z) = F(z) = (ln !') ' = _ __ . Hence, for za any determination of In(z  a) , [  2 In (z  a)l ' = (In ! ' ) ' . Successive inte A grations imply that for constants A and B , f =  + B. D za There follows an interesting link between simple connectedness and
M.
8.2.23 THEOREM. IF Q IS SIMPLY CONNECTED SUBREGION OF COO AND Aut (Q) c FOR SOME T IN T(Q) IS C, Coo OR, 1U. PROOF. Since Coo with one point removed is equivalent to C, via some T in only the possibility that # [800 (Q)] 2': 2 needs treatment. If 00 E Q, for some T in T(Q) C C (v. 8.2.9) . Hence the assumption Q c C is admissible. There is a biholomorphic bijection h : 1U r+ Q such that h' (0) = 1. If h E M, then h(1U) ( Q) is an open disc or an open halfplane, i.e., Q is conformally equivalent to an open disc. The next argument shows that
M,
M,
M,
M,
=
hE
M.
If g E Aut (1U) and f
�f h o g 0 h  1 , then f E Aut (Q) c M,
f o h = h o g, and {J o h, z} = { h o g, z}. Moreover 8.2.21a) and 8.2.21b) imply
{h, g} [g'(z)] 2 = {h, z}. If w E 1U, g
=
( 8.2.24 )
¢ w and z = 0, then (8.2.24 ) reduces to
{h, w} (1  lwI 2 ) = {h, O}.
iO
(8.2.25)
i.e., by abuse of Hence, if {h, O} � Rei O , for w in 1U, { h, w} = 1 wI2 ' notation, {h, 1U} c L(O, O). Off Z (h'), {h, w} is holomorphic in w. The
�l
Section 8.3. Bergman's Kernel Functions
349
{h, w} is a constant {h, O}. As I wl t 1,
Open Mapping Theorem implies that the map 1U '3 w r+ map: == C. Thus, for w in 1U, C (1  lw I 2 ) = = C + By virtue of 8.2.22,
{h, O}
{h, w}
all
O.
h E M.
D
8.3. Bergman's Kernel Functions
The existence of a conformal map 1 of a simply connected proper subregion
fl of C onto 1U leads, via Bergman's kernels described below, to an explicit
formula for the mapping function I . The starting point for the development is the study, for any region fl, of SJ(fl) �f L 2 (fl, ), 2 ) n H (fl). As a subspace of L 2 ( fl , ), 2 ) , SJ(fl) is naturally endowed with an inner product ( , ) and an associated norm I I 11 2 , the latter providing a metric.
u iv E H (fl): a)
8.3.1 Exercise. If 1 �f +
)
(u, v) E
]R2 , the derivative F ' (v. Section b for the map F : ]R2 '3 (x, y) r+ 2 4.7) exists and p (F') = 1 1' 1 . + The next result, despite its negative character, reveals something useful about SJ(fl). =
u;, u�
a E fl, THEN SJ(fl) = SJ(fl \ {a}) �f SJ (fla ) .
8.3.2 LEMMA. IF fl IS A REGION AND
PROOF. If 1
SJ(fl)
C
E SJ( fl) , then 1 E H (fla ) and
SJ (fl a ) .
E
1
n
111 2 d),2 =
r
ina
11 1 2 d),2 then
z z 1 a
On the other hand, if 1 H (fl a ) \ H (fl), e.g., if I ( ) = , for some positive C fla and I IA a;O,R O is represented by a Laurent 0, ( ) series:
R, A(a; Rr
za

00
n=  (X)
( 8.3.3)
{ R },
Just as in 5.3.28, if �f reiO , 0 :::; (J < 27r, and 0 :::; r < < min l, 2 C L n crn rn+rn ei(n  rn) O and then II ( ) 1
z
=
s
n,rnEZ
(8.3.4)
350
Chapter 8. Conformal Mapping
The only nonzero inner integrals in the right member of (8.3.4) are those for which n  m = o. Hence
(8.3.5) For at least one term of the series in the right member of (8.3.3) , n < 0 since otherwise 1 ( Q ) . The corresponding integral in ( 8.3.5) is divergent and thus r I I ( z d)..2 = 1 tic SJ (Qa ). D ina
EH W
00:
8.3.6 COROLLARY. IF 0 j. 1
E E, THEN [ 1 1 1 2 d)..2
= 00 :
SJ (C) =
{O}.
PROOF. If 0 j. I, then 1 is represented by a power series 00
I ( z ) = L cn zn
n=O
convergent in C. If R > 0, as in the preceding argument,
For some positive
n,
Cn j. o.
D
8.3.7 LEMMA. IF Q IS A SIMPLY CONNECTED PROPER SUBREGION OF C, dimSJ(Q) = No . PROOF. For any bounded region Q l , SJ (Q I ) contains the set of all poly nomial functions. Thus dim SJ (Q I ) No. If g Q r+ is a biholomorphic bijection and 1 SJ (Q) . Furthermore, for the trans then l o g formation g Q r+ the CauchyRiemann equations, 4.7.22, and 4.7.23 imply: a) Idet [J(g)l l 1 9' 1 2 ; b) if 1 SJ then l o g · 1 g' 1 SJ(Q) ; c) if n nEN is an orthogonal setn in SJ then n 0 g . I g' I nEN is an orthog onal set in SJ(Q) . d) z r+ z nEN is an orthogonal set in D The next result is the basis for many of the arguments that follow. :
U}
=
E (1U), 1U, =
{
}
EH E (1U), (1U), U
:
1U
}
E H(1U).
8.3.8 LEMMA. IF
1 E SJ(Q) , THEN II ( a ) 1
:::; ��.
a
E Q,
AND r5
�f inf { I z  a l z E 8(Q) } ,
Section 8.3. Bergman s Kernel Functions
351
'
PROOF.
8.3.9 LEMMA. IF Q IS A REGION, THEN SJ (Q) IS II 11 2 COMP LETE. PROOF. If # [8(Q)] :::; 1, 8.3.2 and 8.3.6 imply SJ(Q) = {O}. If # [8(Q )] > 1 and {fn } nEN is a II 11 2 Cauchy sequence in SJ(Q), since limCXJ Illn  F I1 2 = o. L 2 ( Q, ), 2 ) is complete, for some F in L 2 (Q, ), 2 ), n+ If a Q, for some positive
J,
E
a E 58 �f Q \
[ U D(Z, Jtj , 58 E Sp, z E8(fl)
and 58 ¥ 0.
7r Moreover, 8.3.8 ImplIes Il fm  In l1 22 ? (a)  In (a) 1 2 . Hence on Q, m 2 Ilrn nlim +CXJ In (z) �f I(z) exists, and on each 58 , In (z ) � I(z). Thus I E H(Q ) lim= Illn  1 11 2 = o. D and on Q, 1 == F: I E SJ ( Q ) n L 2 (Q, ), 2 ) and n+ •
•
8.3.10 LEMMA. IF {¢" } n EN IS A CON IN SJ ( Q ) AND I
E SJ(O) , THEN
=
(8.3. 11)
n= 1 CONVERGES TO I THROUGHOUT Q AND UNIFORMLY ON EACH PROOF. From 8.3.8 it follows that if z then
E 58 ,
8.3.12 THEOREM. I F z ex:>
L l¢n (z) 1 2 :::; 1J2 n=1 7r
n
58 .
E 58 AND {¢n } nEN IS A CON IN H (Q), THEN
E N, no further argument is needed.NThat possibil ity aside, for some N, the map eN '3 � (a l , . . . , a N ) r+ L an¢n (z) is a PROOF. If ¢n(z) = 0,
a
n= 1
Chapter 8. Conformal Mapping
352
continuous open map of the Banach space eN onto C. Hence, for some a, N L an q)n (z) = 1, and A �f { f : f E span ( q)1 , . . . , q)N ) , f (z) = 1 } j. (/). If
n= l N N N f E A, then f L (I, q)n ) q)n �f L cnq)n and so 1 = L cnq)n( z). n= l n= l n= l Schwarz's inequality in the current context takes the form =
1�
N
N
L I cnl 2 L lq)n (z) 1 2 n= l n= l •
•
( 8.3.13 )
Since ( 8.3.13 ) is valid for all large N, direct calculation yields the conclusion.
D
8.3.14 COROLLARY. THE SERIES ( 8.3. 1 1 ) CONVERGES ABSOLUTELY IN
Q.
a
PROOF. If E Q there are sequences
bers such that: •
•
•
•
g
=
{an } nEN ' { ;3n } nEN of complex num
lanl = 1 ;3,,1 1, n E N; {anq)n } nEN is a CON in SJ (Q) ; anq),, (a) = l q)n (a) l , n E N; (;3nf, anq)n ) 1 ( 1, q)n ) l· =
=
The FischerRiesz Theorem (3.7.14) implies that for some g in SJ ( Q ) , 00 L (;3n f, anq)n) anq)n and dn �f (g , anq)n) 1 (I, q)n ) l , n E N.
n= l
=
S,h=,,', inequality implie, 8.3.15 Exercise. The series
00
n= l
(� I (t, ¢,,) I ' 1 ¢,, (a) I ) '
<:
1�1�l .
D
( 8.3.16 )
Section 8.3. Bergman' s Kernel Functions
353
converges throughout Q2 and if r5 is small and positive, uniformly through out sl. [Hint: Schwarz's inequality applies. ] The series ( 8.3.16) defines a function 00
K : Q2 '3 (z, w) r+ n=L cPn (z) cPn (W), l Bergman's kernel. As it stands, K appears to depend on the choice of the CON { cPn } nEW The developments below reveal that K depends only on Q. 8.3.17 Exercise. I K(z, w) 1 2 :::; K(z, z)K(w, w). 8.3.18 THEOREM. IF J E S)(Q) AND z E Q, THEN J(z) = In K(z, w)J(w) dA2 (w). [ 8.3.19 Remark. Because of the preceding equation, K is a reproducing kernel.]
z
K(z, · ) E S)(Q) , whence the integral above exists. Fur denoting L (I, cPn ) cPn , n=l
PROOF. For fixed,
thermore, S N
N
and 8.3.17 implies
[ 8.3.20 Remark. More generally, if (X, 5, f.l) is a measure space, S) is a closed ( Hilbert ) subspace of L 2 (X, f.l) and for each x in X, the evaluation map 1]x : S) '3 J r+ J(x) is in S)' ( v. 8.3.8) , Riesz's Theorem (3.6.1) implies there is in S) a function Kx such that 1]x (l ) = (I, Kx ). The function K : X2 '3 (x, y) r+ (Ky, Kx ) dcl= K(x, y)
is a reproducing kernel, i.e., J(x) = [J( . ), K( . , x) ] . ]
In 8.3.218.3.23 the underlying context is that of 8.3.20.
354
Chapter 8. Conformal Mapping
8.3.21 Exercise. The reproducing kernel K is unique, i.e., if K is such
f(x) [J( . ), K( . , x)], then K K. 8.3.22 Example. If SJ SJ(1U) , then f E SJ iff for the power series repren=O n=O n=O (I, g) L cn dn . The reproducing kernel K in this case is Szego 's kernel: n=O 1 K(z, ) 1  zw [ 8.3.23 Note. From 8.3.18 it follows that for each SJ(fl) there is
that for all f in SJ, 00
=
w
=
=
= 00
00
00
=  . _
a reproducing kernel K; 8.3.21 implies that K is unique. Hence K is independent of the choice of the Kgenerating CON and depends only on fl: K = Kn . ]
8.3.24 Exercise. If fl is simply connected, a E fl, and #[8(fl)] > 1 , for some f in SJ( fl), f(a) j. o. [Hint: For some b in fl and some g in SJ ( fl), g (b) 1. For some h in Aut (fl), h(b) = a . ] =
fl, K ( , ) j. O. IF g ( z, ) �f KK(( z,, THEN g E B clef { f f E SJ(fl), f( ) = I } AND I I g l1 2 min { l fl 1 2 : f E B } . 8.3.25 THEOREM. FOR
w
IN
w w
w

w
) )'
w w
w
=
=
FURTHERMORE, g IS THE ONLY ELEMENT IN B FOR WHICH THE MINIMUM IS ACHIEVED. 8.3.26 Remark. The independence of K from the choice of the CON by which it is generated is reaffirmed by 8.3.25.]
[
f B, for some sequence {cn} nEN '
PROOF. If E
00
n= l
00
n= l
355
Section 8.3. Bergman's Kernel Functions
(g, cPn ) = :t��2) and I l g l � = K(�, w) If k E B and k  g clef h, then h(w, w) = ° and, since h E SJ ( Q), for some sequence {dn} nE J'i ' Since w is fixed,
=
00
00
00
n= l n= l However, k = g + h, whence
n= l
D
8.3.27 Exercise. The last paragraph in the PROOF of 8.3.25 is valid. 8.3.28 Exercise. The sequence
a CON for SJ (D(O, Rr).
{ '1/Jn :
D(O, 00
n Rr '3 z r+ VE; zR:l } nEJ'i is
I : D(O, Rr '3 z �f L cn zn is in SJ [D(O, Rr] n=O (f, '1/Jn ) R2 n an  I · ] 8.3.29 LEMMA. IF I : U r+ V IS A BIHOLOMORPHIC ( HENCE CONFOR MAL ) MAP BETWEEN TWO OPEN SUBSETS OF C AND SJ ( U) j. {O}, THEN SJ(V) j. { O } AND: a) BOTH F : SJ(V) '3 g r+ 1' . (g o f) AND G : SJ ( U) '3 h r+ (I  I ) ' . (h o 1 1 ) ARE UNITARY, ( v. 3.6.16) ; b) EACH OF F AND G IS THE ADJOINT OF THE OTHER , i.e. , [F (g), h] == [g , G(h)]. PROOF. a) The formula in 4.7.18 for changing variables implies that if g E SJ(V), then I g l 2 E L I (V, ),2 ) and I g o 11 2 11' 1 2 E L l (U, ),2 ). Hence I' . (g 0 f) E SJ(U). [Hint: The map 7r and =n
Direct calculations lead to the remaining conclusions.
D
356
Chapter 8. Conformal Mapping
8.3.30 Exercise. For F and G as in 8.3.29: a) {h E S:J (U) } '* {F o G(h) = h} , {g E S:J(V) } '* {G 0 F(g) = g} . b) F [S:J(V)] S:J(U) and G [S:J(U)] S:J(V) . c) If <1> is a CON in S:J(U), then G (<1» is a CON in S:J(V). 1 . a) If I(z) = w, then ( 1  1 ) ' (w) [Hmt; I' (z) . b) The result a) applies. c) The result 8.3.29a) applies.] =
=
=
8.3.31 LEMMA. IF 1( · , w) IS A CONFORMAL MAP OF A REGION Q ONTO
D(O, R)O AND I(w, w) = 0, !'(w, w) = 1 : a) !' E S:J(Q) ; b)
{ ¢n (. , w) : Q '3 z
r+
}
(i [J(z ';2] n  l f'(Z, w) n EN V :;
IS A CON FOR S:J(Q) . PROOF. a) The formula for change of variables (v. 4.7.18) shows
b) The results in 8.3.29 and 8.3.30 apply.
D
8.3.32 THEOREM. (Bergman) IF Q IS A SIMPLY CONNECTED PROPER SUB1 , THE REGION OF C, w IS A FIXED ELEMENT OF Q, AND R clef V7rK(w, w) z K(s, w) ds = ",,,w DEFINES A CONFORMAL MAP OF Q FORMULA <1>(z, w) clef K (w , w ) a<1> ONTO D(O, Rt . FURTHERMORE, <1>( w, w) = ° and = 1. az I zw =
l
_
PROOF. Riemann's Mapping Theorem (8.1.1) implies that there is a con formal map 1 : Q r+ D(O, R) O , and 8.1.12 implies that there is one and only al one such I, say I( · , w) for whI. ch I(w, w) = 0, az (z, w) 1 zw = 1. Moreover, 8.3.7 implies that dim S:J(Q) = No. Since K is independent of the choice of a CON for S:J(Q), the CON {¢n } nEN of 8.3.31 may be used to define K. For that choice, _
1 , '1/J (W) 0, n > 1, J7rR n = 1 al K(z, w) = (z, w), K(w,w) = 7r R1 2 ' 7rR2 az <1>(z, w) I(z, w).
'1/J l (W)
=
=
D
Section S.4. Groups and Holomorphy
357
K(w,w) K w w 1 V . 1z I
8.3.33 ExercIse. ( s, ) ds :::; 7r . [ 8.3.34 Note. By virtue of 8.3.2, if Q is a region and a E Q, Bergman's kernel functions for Q and for Qa �f Q \ { a } can be identical. If Q is simply connected, provides a confor mal map of Q onto 1U, but does not (nor can it) do the same for Qa. Hence, absent the existence of a conformal map f Q r+ 1U, the kernel does not serve as the basis for the existence of su�h an f.
Kn Kna
Kn
On the other hand, if dim SJ ( Q )
Kna Kn
=
00
:
No, the formula
K(z, W) clef n=,",l ¢n(Z)¢n(W) = �

w
is meaningful. However, in the current context, for fixed, <1> : Q '3
Z r+ LZ K(s, w) ds
w
is not necessarily in H ( Q ) . Nevertheless, for a maximal simply connected subregion (cf. 5.9.5 ) Q 1 of Q, if E Q l , as a function of z, <1>(z, E H ( Qt ) .
w)
An extensive treatment of the material in this Section can be found in [Berg] . The applications of Bergman's kernel functions are not confined to the subject of conformal mapping.] 8.4. Groups and Holomorphy
The appearance of M, e.g., via ¢c , in the treatment of 8.1.1, in 8.1.10, and in 8.2.18b), where the isomorphism of Aut ((Coo ) and M is implicit, suggests that M, regarded as a group with respect to the binary opera tion M2 '3 ( Tl ' T2 ) r+ Tl T2 E M, deserves examination. Among the subgroups of M are the following: a) when {W l .W2 } c (C \ {O}) and W2 IS. not real, the set 0 :
0

WI
b) the modular group: Mod �f { Tabed : {a , b, c, d} c z } n Mo.
358
Chapter 8. Conformal Mapping
8.4. 1 DEFINITION. A SUBGROUP G OF Mo IS properly discontinuous IFF FOR SOME P IN C, SOME OPEN NEIGHBORHOOD N(p) OF {p } , AND EACH T IN G \ {id } , T(p) tic N(p) . 8.4.2 Exercise. The groups GW1 ,W2 and Mod are properly discontinuous and GW1 ,W2 is a normal subgroup of Mod. [Hint: If T E (GW1,W2 \ {id }), p = 0, and
I
N(p) �f { z : z i < min
{ I wl ± w2 1} } ,
then T(p) tic N(p). If Tabed E Mod, Tabed j. id , p
{
=
2i,
and
I
N(p) = z : z  p i <
� },
then T(p) tic N(p).] 8.4.3 Exercise. A properly discontinuous subgroup G of Mo is finite or countable. [Hint: If #(G ) > No , then #[G(p)] > No, p E C and G (pt j. 0. If q E G(pt , then p E G ( q t . ] 8.4.4 Exercise. a) When G is a properly discontinuous subgroup of Mo and p and N(p) are the objects in 8.4.1, for some nonempty subset 5 of Mo, if S E 5, then S(p) = 00. b) If S E 5 the set r �f SGS  1 �f { STS 1 : T E G } �f { T
}
is a properly discontinuous subgroup of Mo and is isomorphic to G. c) If Tabed E (r \ {id }), then Tabcd( (0 ) j. 00 and c j. O. [Hint: c). If 00 T (oo) , then p E [Coo \ N(p)] .] =
Below, each properly discontinuous group, however denotedr or Gis assumed to conform to c) in 8.4.4. 8.4.5 LEMMA. IF p AND N(p) ARE THE OBJECTS IN 8.4.1, FOR SOME POSITIVE p, IF I z l > p AND T E (r \ {id } ) , THEN T( (0 ) j. z. PROOF. If Tn E r \ {id } , Tn (oo) �f Zn , and I Zn l > n, for some Rn in G, Tn = SRn S 1 , Zn = SRn S 1 (00) SRn (p) , S  1 (zn ) = Rn (P) E [Coo \ N(p)] , 1 S (zn ) = S  I (oo ) = P E [Coo \ N(pW = Coo \ N(p) , nlim + oo =
Section S.4. Groups and Holomorphy
359
a contradiction.
D
8.4.6 Exercise. If Tabed E Mo, then T�bed (z) =
1
(cz + dF
.
8.4.7 DEFINITION. FOR A GIVEN Tabed IN Mo, WHEN c j. 0 THE SET { z : I cz + d l = I } IS THE isometric circle Cabed OF Tabed:
( �, � )
THE ( CLOSED ) DISC D ed. WHEN r IS I I IS THE associate OF Tab A PROPERLY DISCONTINUOUS SUBGROUP OF Mo,
clef AND C =
{  d : Tabed E r } . �
8.4.8 Exercise. a) The isometric circle of T;;;'�d is the image under Tabed
of Cabed. b) If
Tabe d E (r \ {id }) and z E Coo , then Tabe d(Z) arises by inversion of z in Cabe d, a reflection in L, the perpendicular bisector of the line joining  dc and ac , and a (possibly trivial) rotation centered at � . c [Hint: For a), if z E Cabe d, for some 8, z=
d + e i O and Tabed(Z) c
a _ e i O c
= 
For b) , 8.2.4 applies.] 8.4.9 Exercise. a) 00 E R r ; b) If T E r \ {id } the radii of the isometric circles of T and T 1 are equal. 8.4.10 Exercise. If {S, T} c r, ST 1 j. id , {CS , CT, CST, CS �l , CT �l } is the set of centers, and {rs, rT, rsT, rs�l , rT�l } is the set of radii of the isometric circles of { S, T, ST, S 1 , T  1 } : a) rT2 rSTrT =  = rST C rSl rT c l e ST TI rs I CT� l  cs l .' I T�  cs l ' b)
=
l esl < p; c) rs < 2p.
360
Chapter 8. Conformal Mapping
The union of all the discs that are associates of elements of r is contained in a disc of finite radius, say D(O, A) . If I z l > A, then z E Rr. [Hint: b): If I cs l
>
p, then 5 1 (00)
8.4.11 THEOREM. a) THE UNION U
=
cs ; v. 8.4.5.]
�f Ur T (Rr) �f r ( Rr) IS DENSE IN TE
C. b) IF u E Rr AND T E r \ {id } , THEN T(u) tic Rr. c) I F u E 8 ( Rr), N(u) I S A NEIGHBORHOOD OF u, AND v E N(u), FOR SOME W I N Rr AND SOME T IN r \ {id } , T(w) = v . PROOF. a) If U is not dense in C, for some positive r and some u, D(u, rt
C
Coo \ u.
Furthermore, for each T in r, T [D(u, rn T [D(u, rt]
c
C
Coo \ U. In particular,
C \ Rr.
Since 00 E Rr, the center of each isometric circle is in r ( R r), whence u is not the center of any isometric circle. On the other hand, since u tic R r, u is in some D , the associate of some Tabe d. Furthermore, 8.4.8 I I implies that Tabed arises by a reflection in Cabed followed by a reflection in
( �, � )
a line and a possibly trivial rotation (a pair of reflections in lines) . If z E Cabed and the radius of Tabe d [D(z, r)] is R, direct calculation r shows R = 1  c 2 r2 '
11
(
)°
Since r < � < 2p, if z E D  �d , �1 , then 1
R
> ,2,.
r
r 14 2 p
cle=f kr
>
r.

Hence the radius Rm of T;bed [D ( z, r)] exceeds k Tn r, m E N. If m is large, T;be d [D(z, r)] meets Rr, a contradiction. b) If T E r \ {id } , since u is not in the associate of T, 8.4.8 implies T(u) is in the associate of T 1 , hence is not in Rr. i.e., c) By definition, N (u) meets some Coo \ D _
(
)
( � , I �I )
d 1 ° N(u) n D  � ' � = 0.
0,
361
Section S.4. Groups and Holomorphy
If D ( �e ' �) l e l is the associate of Tabed, 8.4.8 implies 
D
8.4.12 LEMMA. IF S AND T ARE TWO ELEMENTS OF r, THEN T (R r ) n S ( R r ) = 0.
PROOF. Since S  I T E r
\ { id } , 8.4.11b) implies S  I T (R r) n Rr = 0.
D
8.4.13 DEFINITION. FOR A PROPERLY DISCONTINUOUS GROUP r, A FUNC TION f IN M (UO ) IS rautomorphie IFF FOR EACH T IN r, f T = f.
) )
0
8.4.14 THEOREM. ( Poincare a IF Rl IS A RATIONAL FUNCTION, P (Rd n (et = 0, AND N '3 m > 2,
THEN ON EVERY COMPACT SET DISJOINT FROM P( R) u e· ,
CONVERGES UNIFORMLY AND DEFINES A FUNCTION (h IN
{ Coo \ [P (R I ) u e·l } . b ) IF Tabed E r, THEN (h [Tabed(Z)] = ( ez + d) 2rn (h ( z ) . c) IF (h CORRE SPONDS TO A RATIONAL R2 , RESTRICTED LIKE R1 , AND, FOR Z NOT IN (h ( z ) , THEN F IS rAUTOMORPHIC. Z (t'h ) u P (Rd u e· , F ( z ) cle=f (h ( z ) PROOF. a ) Since the set e of the centers of the discs associated to the H

elements of r is bounded, and since the set of radii of those discs is also bounded, there is a positive number p such that for each Sa(3'Yii in R r ,
The circle a
[D ( � ) ] �f Cabed is concentric with Ca(3'Yii ' 
,p
362
Chapter 8. Conformal Mapping
All the associated discs are contained in each D the complement of each D
( � , p) , whence
( �, p) is contained in R r .
If Tabed E Rr, then Tabed (Cabed ) results from an inversion in Cabed fol lowed by reflections in lines. The reflections in lines are isometric maps, i.e. , they do not alter distances between points. The inversions in circles multi ply distances between points by a constant dependent on I c l (v. PROOF of 8.4.11). Thus the radius of Tabed (Cabed) is p 2 ' l 1 The complement of Cabed consists of a bounded open disc and an un bounded component. The unbounded component is mapped by Tubed onto the bounded open disc determined by Tabed (Cubed) . The bounded open disc is a subset of Tabed (Rr). Hence, owing to 8.4.12, if SOi(3'Yii and Tabed are two maps in Mo, the interior of the intersection of the bounded open disc determined by S"(3'Yii (COi(3'Yii ) and that determined by Tabed (Cabed) is empty. It follows that
�
(8.4. 15) .
If K is a compact set disjoint from P(R) u e· , for some constant M,
The statements b) and c) follow by direct calculation.
D
R1 is not a constant, then F is not a constant. 8.4.16 Exercise. If _ R2 When G is a properly discontinuous group, unrestricted by the conditions in 8.4.4c), a fundamental set
isometric circle and the associate of T is meaningless.]
,
8.4.18 Exercise. If G is an arbitrary properly discontinuous group and, in the notations introduced above, r = S GS  1 then S  1 R r serves as a fundamental set
363
Section 8.5. Conformal Mapping and Green's Functions
{
�
�,
},
set
{
:
}
D �f (A n B) U C and its reflection DP in lR is a fundamental set
When G is unrestricted, there are the following options . One can work with r �f SGS  1 for some appropriate S and apply the machinery developed above. One can stick with G and cope with complications that can arise when the set of centers of meaningful isometric circles has a cluster point at 00 .
Sometimes, e.g., when G = GW1,W2 ' there are no isometric circles, and the resulting discussion is straightforward. The emerging theory is that of elliptic functions to which SOme of the most important contributions came from Weierstrafi [ Hil] , v. 8.6.1 1. For some properly discontinuous groups r , detailed elaboration of the reasoning behind 8.4.14 leads to the construction [Ford] of rautomorphic functions enjoying special properties, e.g., having in R r exactly one pole of order one and one zero of order one. Manipulation of such functions leads to others that are holomorphic in n + and map n + onto C \ {O, I}. Such functions can be used to prove the Little Picard Theorem (9.3.1) [Rud] . 8 . 5 . Co nformal Mapping and Green's Functions
As noted 6.5.1 if h E H(Q) and 0 tic h(Q) , then In I h l E HaJR(Q). When Q is simply connected and f Q r+ 1U is a conformal map, for some a in Q, f(a) = 0, whence G( · , a) �f  In If I is not defined at a but is defined in :
Q \ {a}.
Q , a, AND f ABOVE: a) G( · , a) E HaJR (Q \ {a}); b) IF b E 8oo(Q) , zlim G(z, a) = 0; c) FOR SOME POSITIVE r , + b g Q '3 z r+ G( z, a) + In I z  a l IS IN HaJR [D(a, rt l. 8.5.1 LEMMA. IN THE CONTEXT OF
:
PROOF. a) The Hint in 6.5.1 applies.
364
Chapter 8. Conformal Mapping
b) If fl '3 Z + b E 8oo (fl) and In I f(z) 1 1+ 0, via passage to subsequences as needed, for some 15 in (0, 1), and some {zn} nEN ' fl '3 Zn + b while f (zn ) converges to some d in 1U and If (zn )1 � 1  15. For the sequence
of compact sets, Km
C
K� + 1 and
U Km = mEN
1U.
phic nature of f, each f  1 ( Km ) is compact and
Owing to the biholomor
Furthermore, 1.3.7 implies that for some {wn} nEN , W n + C � f l (d) and f (wn ) == f (zn). For some large m and some large n, Wn E f  1 (Km ) and Zn E fl \ f  1 (Km ) , whence f is not bijective.  a is in H (fl). c) The function fl '3 Z r+ zf(z) D
a
a)
8.5.2 DEFINITION. FOR A REGION fl AND AN IN fl, A FUNCTION G(·, CONFORMING TO a) c) IN 8.5.1 IS (A) GREEN'S FUNCTION FOR fl. Riemann's Mapping Theorem (8.1.1) and 8.5.1 imply that for a sim ply connected proper subregion fl of C and a point in fl, (a) Green's
a
function G(·, a) exists. The following items delimit to some degree the kinds of regions for which there are and are not Green's functions. 8.5.3 Exercise. If fl is a bounded Dirichlet region and a E fl, there is a Green's function G(·, a) for fl. [Hint: The solution of the Dirichlet problem for the boundary condition u la ( n / z) = In I z  a l serves.] 8.5.4 Exercise. If fl
for fl.
=
1U \
{o} � 10 there is no Green's function G(·, O)
[Hint: The discussion in 6.3.11 applies.]
a)
8.5.5 Exercise. If G(·, is (a) Green's function for fl, then G > 0. [Hint: The Maximum Principle applies.] 8.5.6 LEMMA. THERE IS NO GREEN'S FUNCTION FOR C. PROOF. If G is a Green's function for C, E > 0, r > I Z2 
{I z  zl l < r5} ::::} {IG(z)  G(zI )1 < E} ,
zl l > 15 > 0, and
Section 8.6. Miscellaneous Exercises
365
G (Z I ) + E (ln z  z  ln r) I. S harmomc. m . C \ {zd · Furthen gr ( z ) cl=e f (ln ll J  ln r) l thermore, if A �f A (z l ; J, rt, on 8(A) �f r �f CZ 1 (J) u CZ 1 (r), G l r :S: gr l r . The Maximum Principle implies G I A :S: gr I A · As r t 00 there emerges i.e., G (Z I ) :s: G (Z2 ) : G is a constant and cannot be a Green's functi�n.
D
8.5.7 Exercise. a) If G ( · , a) is a Green's function for Q , then G ( · , a) is unique, and if O(z, a) + In I z  a l is harmonic near a, then 0(. , a) > G. [Hint: The Maximum Principle applies.] 8.5.8 Exercise. If f : Q 1 r+ Q2 is a biholomorphic bijection and G2 (·, a) is the Green's function for Q 2 , then G2 (·, a) o f is the Green's function G1 [ . , J  l (a)] for Q l . [Hint: The Open Mapping Theorem applies.] 8 . 6 . Miscellaneous Exercises
8.6.1 Exercise. If T E Mo and p, q , r, s are four complex numbers, then
X[T(p), T( q), T(r), T(s)] 8.6.2 Bxercise. a) If 5 E Mo, and 5  1
=
X(p, q, r, s).
{O, 00, 1} = {Z2 , Z3 , Z4 }, then
b) Four points p, q, r, s are cocircular or collinear iff X (p, q, r, s) is real. Pq . · · p, q , r are co llmear 8 . 6 . 3 ExerClse. Three pomts 1· ff  IS. rea1 .
qr Q, and 5· n Q = 0: a) Q \ 5 is a
8.6.4 Exercise. If Q is a region, 5 c region; b) SJ(Q) = SJ (Q \ 5). [Hint: For b) the argument for 8.3.2 applies.]
If Q is a region, S ,B '3 5 c Q, Q \ 5 IS a region, and )' 2 (5) = 0, then SJ(Q) = SJ (Q \ 5). [ 8.6.6 Note. If 5 = CO! (some Cantor set contained in [0, 1]), ). 2 (5) = ° and 1U \ 5 is a region.
8.6.5 Exercise.
If 5 �f ( 1 , 1 ) , ). 2 (5) = ° but 1U \ 5 is not a region.
366
Chapter 8. Conformal Mapping
=
If 5 �f { z : z p + iq, {p, q} tains no region.]
C
QI } , >' 2 (5)
=
0,
but 1U \ 5 con
8.6.7 Exercise. If Q { x + iy : I x l < 1, I y l < 1 } what is the corre sponding Bergman kernel K? [Hint: The GramSchmidt algorithm applies to the sequence
�
8.6.8 Exercise. If Q 1
Q2 cle=f
�f { z
{z : z
:
=
0 :::;
a < '25(z) < b :::; 27r } and
Re iO , 0 < R, a < (J < (3 } ,
for some real c, h Q 1 '3 z r+ eic z is a conformal map of Q 1 onto Q2 . 8.6.9 Exercise. If the Schwarzian derivative, :
{w , z }
(=
)
2w'(z)w "'(z)  3 [W " (Z)] 2 ' 2 [w ' (z)] 2
is regarded as a function of w and z, and T E Mo, then {w, z} = {T(w), z }
=
[T(z) '] 2 {w, T(z) } .
8.6.10 Exercise. When {5, T} C Mo: a) If 5 has only one fixed point while T has two, then 5T j. T5; b) If 5 and T have the same fixed point ( s) , then 5T T 5. c) If 5T T 5 and each of 5 and T has only one fixed point, they share it. 8.6.11 Exercise. a) If =
=
� P ( Z l �l , �2 ) �f z12 L +
{
�}
I then P , the _ 2 2 ' 3w,tO [Z W] automorphic. b) If f is G Weierstrafl elliptic function, is G automorphic and 5 � { a + + : 0 :::; s, < I } is the period paral lelogram vertexed at a: bl) the sum of the residues of the poles of f in 5 is 0; b2) 2 :::; #[P( f ) n 5] = #[Z (f) n 5] < 00.
and
P
Q
W 1 , W2
tWl SW2
t
W 1 , W2 
Section 8.6. Miscellaneous Exercises
8.6.12 Exercise. If _ef 60 g2 �
p' p
" and g3 cl=ef 140 " :� � [Z ] 3w,tO 3w,tO [Z  w] 1
Q
367
then = 4 3  g2 P  g3. 8.6.13 Exercise. If �f W
 W
4
1
Q
6 '
m l Wl + m2w2 j. 0, then
(
(�l )
;
)
�)
8.6.1 4 Exercise. If �f P , e2 �f P WI W2 , e3 �f P ( 2 , then the ei , 1 :::; i :::; 3, are the three zeros of the polynomial function Z r+ 4z 3  g 2 Z  g3 .
el
a la p ( )]
8.6.15 Exercise. If E Coo and multiplicities are taken into account, for = 2. P as in 8.4.19, # [p n The statements in 8.6. 16 below are steps leading to the sufficient con dition for the existence of a barrier at a point in the boundary 8(Q) of a region Q, 6.3.29. The argument has been deferred to this part of the text because 8.1.8 is used. 8.6. 16 Exercise. If E r �f 800 (Q) and no component of Coo \ Q consists of alone: a) The assumption = 00 is permissible. (Otherwise, for the map v.
a
a
a a
a 1
¢ : Coo '3 Z r+ z  , the argument may be conducted on ¢ (Coo ). ) b) If C is a component of Coo \ Q and 00 E C, then Q 1 �f Coo \ C is simply connected, d. 8.1.8d ) , and Q C Q . c) For some f in H (Q I ), exp[ J (z)] = z, d. 8.1 .8h ) (f is a determination of In) . d) If Q2 �f f ( Q ) and L �f + it : E lR,  00 < t < oo } , the line L meets Q2 in at most count ably many open line segments ( w � , w �) of total length not exceeding 271" so that 'S ( w �) > 'S ( w U . (Otherwise, for some finite K, in Q 2 there are K points Z l , . . . , ZK such that
{a
a
l
Kl L [In (Zk+I )  In (zk)] = 271"i, k=l
exp (In ZK  In z I )
=
exp(271"i) = 1,
Chapter 8. Conformal Mapping
368
a contradiction.) e) If w �f u + iv, U 2: a there is a holomorphic function ( h such that w�  w = exp [il'h(w)] w�  w
and 0 � fh (w ) f) If
�
7r.
e(w) cle=f
{ � 7' 7r
'" 1'h (W)
if �(w) 2: a if �(w) < a
1
and Q3 �f { w : � (w) > a }, then Furthermore,
 7r2 arctan
7r
� (w )  a
�
e(w) � o.
(The function f is a (holomorphic) determination of In. Hence the imaginary part of f is harmonic and so is (J. The calculation in e) and plane geometry provide the estimate . ) g) If an t 00 and en corresponds to an as e corresponds to a, then (3(z)
00
� nL 2  n en (ln z)
=O defines a function subharmonic in Q and tending to zero as z + 00 . h) Near any point in r, for all sufficiently large n, the functions en take on the value  l . i) In the sense of Ahlfors and Sario [AhS] , the function (3 is a barrier at
a.
[ 8.6.17 Note. The reason for g) is that e(ln z) can converge to as z converges to some point on r .]
o
8.6.18 Exercise. If Q is simply connected, then Q is a Dirichlet region. [Hint: If Q ¥ C, then Coo \ Q consists of precisely one component: 8.6.16 applies. If Q = C, 8(Q) = 0.]
9
Defective Functions
9.1. Intro duction
E
The set of entire functions is divided into two subsets, namely the set P of polynomial functions and T \ P the set of transcendental functions. If p E P, then p(C) = C is an abbreviated statement of the Funda mental Theorem of A lgebra (FTA). On the other hand, z + exp(z) is a transcendental function and exp(C) = C \ {O} . If f T, the WeierstrafiCasorati Theorem (5.4.3c)) implies that near 00, the values assumed by f are dense in C. Thus zero is an isolated essential singularity of g C \ {O} '3 z r+ f By abuse of language, 00 is an isolated essential singularity of f. The following result embraces all the phenomena just described.
E
�f E
(�).
:
E
9.1.1 THEOREM. (The Great Picard Theorem) IF a IS AN ISOLATED ES SENTIAL SINGULARITY OF A FUNCTION f, R > 0, AND f H [A (a; 0, Rt] '
THEN # {C \ J [A(a; 0, Rt]} � 1 .
I n every sufficiently small punctured neighborhood o f an iso lated essential singularity of a function f, the range of f omits at most one point.
The discussion is facilitated by the introduction of some special vocab ulary and notation.
E
�f
9.1.2 DEFINITION. WHEN f CIC , C \ f(C) D(f) IS THE SET OF de fections OF f AND f I S # [D(f)]defective. If and f is 2defective, say {a, b} C [C \ f(C)] , then
fEE
] �f � = � E E and {O, I} C C \ [ ](C) ] . Similarly, if f E M and f is 3defective, say {a, b, c}
C
[Coo \ f (Coo )] ,
369
Chapter 9. Defective Functions
370
then I cl�f cC  ab . I  ab E M and {a, 1, 00} c [Coo \ I (Coo )] . The study of 12defective entire resp. 3defective meromorphic functions may be confined to functions with the simple sets of {0, 1} resp. {0, 1, 00} of defections. Hence, absent any further comment, for an entire 2defective function I, D(f) = { o, I}; for a meromorphic 3defective function I, D( f) = {a, 1, 00}. The discussion below is devoted to showing first that 2defective entire functions and 3defective meromorphic functions are constants. Elabora tions of those results provide the contents of the Great Picard Theorem. 1 9.1.3 Exercise. If I E M (C) and D(f) = {a , b, c}, then h cle=f _ E E I a 1 _ , _1_ } . (Thus the study of 3defective meromorphic and D ( h ) = { _ b a bc functions is reduced to the study of 2defective entire functions.) The arguments that follow are an amalgam of the efforts of several writers: Ahlfors, Bloch, Bonk, Caratheodory, Estermann, Landau, Minda, Montel, and Schottky. Picard ' s original proof of his Little Theorem is of an entirely different character, v. [Hil, Rud] . Four steps are involved. The first seems irrelevant to the goal. A. (Bloch) If I E H ( U) and 11' (0)1 2: 1, for some Ifree positive constant B, and some b (dependent on I) , B 2: 11 and I (1U) :J D(b, Bt. 2 B. If h is a nonconstant entire function and r > 0, for some b, 

h (C)
:J
D(b, rt ·
The range of a nonconstant entire function contains open discs of arbitrary radius. C. If l E E, I is not a constant, and D(f) = {a, I}, associated to I is a nonconstant entire function F such that D(F)
:J
S cle=f
{
± In( Vm +
� V
n7ri
m  1) + 2
:
m E N,
n E ;Z } .
Since, for any b, D(b, It n S j. 0, F (C) fails to contain any open disc of radius 1, a contradiction of B: the Little Picard Theorem is valid. D. Refined extensions, due to Schottky, of A together with a special ap plication of the ideas behind the ArzelaAscoli Theorem, v. 1.6.9, lead to the Great Picard Theorem.
Section 9.2. Bloch's Theorem
371
9.2. Bloch's Theorem
The next rather general result provides an entry to the entire complex of Bloch/Landau/Schottky theorems. 9.2.1 Exercise. If X is a topological space, Y is a metric space, f : X r+ Y is open, a E V E O(X), and r5 �f inf { d[y, f(a)] : y E 8[J(V)] }, then r5 > 0 and B[J(a), W C f(V). [Hint: 1 .3.7 applies.]
IF r 0, f E H[D(a, r)], M �f 11 !' I D(a,r)o 11 00 :::; 2 1!, (a)l, AND R = (3  2v2)r 1!, (a) l , THEN D[f(a), R]O C f [D(a, rt]. PROOF. Consideration of the translate fr  a] (z) �f f(z  a) and the func tion fr  a]  f(a) shows that the assumption a = f(a) = 0 is admissible. If g(z) �f f(z)  !, (O)z and I z l < r, then g(z) = Jrro,z] [!, (w)  !, (O)] dw, 1 I g(z) l :::; I z l · 1 I!, (zt)  !, (O) I dt. Cauchy ' s formula implies that when I w l < r, !'(u) _ !'(u) ] du = � r f'(u) d , r [ !, (w)  !, (O) = � 27rl J1 l r (  w) 27rl J1 l r w U Iwl I w l M ' 27rr = I f, (w)  f, (0 I :::; r  1w 1 M, 271' r ( r  I w I ) 2 I g(z )I :::; l z I Jto M r _I z ll zt l t dt = r I_z l l z l · 21 · M. The triangle inequality implies If(z)  f' (O)z l 2': I!, (O) I ' I z l  If (z) l. Be cause � :::; I!, (O) I , (9.2.3) I f(z)l 2': I!, (O) I ' I z l  I g (z)1 2': I!, (O) I ( I z l  r �1 I �Z I ) . Since I z l < r, the maximum value of the first factor in the rightmost member of (9.2.3) is achieved when I z l = l1 � (1  �) r « r). The maximum value itself is (3  2 v2)r and 9.2.1 applies when X = Y = C, V = D(O, l1t, and r5 = ( 3  2v2)r I!, (O) I . D 9.2.2 LEMMA.
>
u
=
u
u
=
u u
u
Chapter 9. Defective Functions
372
f
f(O) = 0, 1' (0) = 1, and R �f I l f l � l I oo ,
9.2.4 Exercise.
If E H (U ) ,
9.2.5 Exercise.
a) If E H (U ) , then '1jJ : U '3 z r+
then f(1U)
:J
D(O, Rt · [Hint: 3  2v2 > 61 .]
continuous. b) If
f
I I'(z) 1 ( 1  I z l ) IS
'1jJ ( z ) cle=f M, and R cl=ef (32 In) a E , '1jJ ( a) = max 2 M, z E llJC then f(1U) D(f(a) , Rt . c) R > I f��) 1 � 112 , l ' I a l then [Hint: M = (1  l a l ) I I' (a) 1 and if I z  a l < 2 I I'(z) 1 :::; 2 11'(a) 1 ;  V
IT 1C \U
:J
9.2.1 applies.]
( 1 )
It is a short step from 9.2.5 to A: b = f(a) and f(1U) :J D b, 1 2 [ 9.2.6 Note. A heuristic (but invalid) argument for A is the following. If 11'(0) 1 � 1, for z near 0, I f(z)  f(O) 1 � 1 ; 1 . Thus, if r is small and positive, each point in the Jordan curve image Jr �f f[ Go(r)] (the image under of the perimeter of the open disc D(O, rt) is at a distance not less than r 2" from 0 [= f(O)]. Thus
f
:J
f(1U) J [D(O, r)O] = U J [Go(s)] O�s
O�s
Even if the display above is valid, it does not imply A, since there r is no lower bound on "2 ' i.e., there is no implication that
f(1U) D [1 (0), 112 r nz Indeed, if f(z) = e n 1 , then f(O) = 0, 1'(0) = 1; if n is large, 1 [ 121 ] 0  n tic f(q, which shows that if n is large, f(1U) 1; D f(O), :J
Section 9.2. Bloch's Theorem
( )
373
Thus, although 1 (1U) contains some D b, 112 b is not neces sarily 1(0) (rather, for a as found in 9.2.5, b = I(a).] 0,
9.2.7 Exercise. The assertion in B is valid. [Hint: If M > 0, for some a, I h'(a) 1 > M. Translations and rescal
ings apply.]
9.2.8 THEOREM. I F R > 0, 1 E H [D(O, R)] , AND
(9.2.9)
{0, 1} c C \ I [D(O, Rr] , THEN: a) FOR SOME FUNCTION F IN H [D(O, Rr] '
I(z) =  exp[7ri cosh 2F(z)]; b) A VALUE OF F(O) CAN BE DETERMINED BY REFERENCE TO 1(0) ; c) THE RANGE OF F CONTAINS NO OPEN DISC OF RADIUS ONE, i.e., {b E q
'*
{D(b, 1 r � F [D(O, Rt] } .
PROOF. a) The truth of each of the following statements is implied by 8.1.8 and the hypotheses: for some h in H [D(O, Rr] ' 1 = exp (27rih); for some u in H [D(O, Rr] ' h = u 2 ; for some v in H [D(O, Rt] ' h  1 = v 2 ; u 2  v 2 = 1 and u  v = 1 ¥= 0; u+v for some F in H [D(O, Rt] ' u  v = exp(F); I I D ( O ,R) O =  exp(7ri cosh 2F ). b) If •
• •

•
•
•
{k, m } C IZ, 2 ln 1 1(0) 1 + \2k  l)7ri 7rl
I�f n 1 (3 �I ±
F(O)
�f
(3
+ 2m7ri
2
'
then 1(0) =  exp [7ri cosh 2F(0)]. c) If (T E L
cle=f { ± In (Vm + v m  1) + n7ri �
T
m E N, n E IZ
}
Chapter 9. Defective Functions
374
and, for some z in D(O, Rt, F(z) = a, then I ( z ) = 1, a contradiction of (9.2.9). The remainder of the argument consists of showing that L meets every open disc of radius one. The estimates In(vm +
1 + vim)  In(vIm + vm  1) if m = 1 = In(h + 1) < 1 m+ 1 :::; In y'3 < 1 if m > l ' < In
{
J
m 1
imply that if a E C, for some a in L, 1 y'3 I�(a)  �( O' ) I < 2 ' I CS (a)  CS ( o' ) 1 < 2 '
whence l a  0' 1 < l .
D
9.2.10 Exercise. The PROOF above is valid if 1 is entire. 9 . 3 . The Little Picard Theorem
The preceding developments lead to the final statement in C. 9.3.1 THEOREM. (The Little Picard Theorem) a) IF 1 E E AND 1 IS 2DEFECTIVE, 1 IS A CONSTANT. b ) IF g E M(C) AND g IS 3DEFECTIVE, g
IS A CONSTANT. PROOF. Since 9.1.3 reduces b) to a) , an argument for a) suffices. Associated with 1 is F of 9.2.8. If 1 is not constant, neither is F. By virtue of B, (v. 9.2.7) , on the one hand, F ( C) contains arbitrarily large open discs, yet contains no open disc of radius one, a contradiction. D [ 9.3.2 Note. The derivation above of the Little Picard Theorem uses only the existence of some positive constant B for which A obtains. Some interest attaches to the determination of the supremum B of all such B . For .1' � { I : 1 E H (1UC ) , !' (0) = I }, and 1 in .1',
Af � sup { R and A �f sup A f. If 1 in f EF
:
1(1U) contains some
D(b, Rt }
.1',
then 1 is injective on some subregions Q of U For a subregion Q on which 1 is injective, Rf, n is the radius of the largest open disc contained in I(Q) . Then Bf �f nsup Rf, n
el[]
Section 9.3. The Little Picard Theorem
375
� clef and B clef = sup B f . Finally, if 1 E F = F n { I f then �
Af =
sup { R
1 is injective } ,
1(1U) contains some D(b, Rt } and A � = su r A f . : f EF
For the various constants above, the best estimates known to the writer are: 0.433 + 10  1 4 <
7r
< 0.472; 0.5 < A < 0.544; 0.5 :::; A :::; 4 '
B
The variational aspect of the PROOF of Riemann ' s Mapping The orem (v. 8.1. 13) suggests that a similar technique is useful in the derivation of A. In fact a variational approach yields A
�  2 (� 0.121320343),
3
2:
which is far better than A 2: "23 v2 (� 0.085786437) as found above. Ahlfors, via an improvement of Schwarz's lemma, showed B 2: (� 0.433012701). He and Grunsky conjectured, but did 
V;
not pmve ,
B
�
J
va
1 r
2
mr (�)r (H) ( � 0.4719).] 1
.
9.3.3 Exercise. a) B :::; A :::; A; b) k : 1U '3 z r+ "2 [In(l + z)  In(l  Z)] 1S 
in F; c)
7r
A :::; 4 '
9.3.4 THEOREM. IF 1 E E AND g clef = 1 0 1, EITHER g HAS A FIXEDPOINT OR I(z) = z + b AND b j. O.
PROOF. If g has no fixedpoint neither has I. Thus
g(z)  z k : C '3 z r+ '':I(z)C. z _
is entire. If k(c) = 0, then g(c) c, a contradiction. If k(c) = 1, then g(c) = I(c) , i.e., 1[J(c)] = I(c) , whence I(c) is a fixed point of I, a second =
376
Chapter 9. Defective Functions
contradiction: {O, I} C C \ k(C) . The Little Picard Theorem implies that for some c in C, k == c, whence !, (I' I  c) 1  c. Thus Z (I' I  c) u Z (I') = (/) and {O, c} C C \ !' I(C) . The Little Picard Theorem implies that for some d in C, !' I d. Hence !, is constant. D 9.3.5 Exercise. The function h : C E z r+ z + e Z is entire and has no fixedpoint. What is the set of fixedpoints of h h? 0
0
0
0
==
==
0
9 . 4 . The Great Picard Theorem
Bloch ' s Theorem makes no direct mention of defective functions. On the other hand, the burden of its message is that a function I in H (U ) and normalized by the condition I !' (0) I 2: 1 is locally not defective. The next and related result incorporates 2defectivity in the hypothesis. 9.4. 1 THEOREM. (Schottky) . IF 0 � fJ < 1, I E H (U ) , 1(0) = a, AND I IS 2DEFECTIVE, FOR SOME FUNCTION
{{Ia l
�
x
[0, 1) '3 (w, fJ) r+
1\
w} {I z l
�
fJ}} '* { 1 1 (z) 1 <
[ 9.4.2 Remark. Thus the absolute value of a 2defective function I in H (U) is, for some given bound on the value of 1 1(0) 1 (= l a l ) , bounded in D(O, fJ ) by an Ifree constant.] PROOF. The notations used below are those in the PROOF of 9.2.8. The function h may be chosen so that  21 < �[h(O)] � 21 '
If l al 2: :;1 , then I h(O) 1 lowing estimates emerge:
1+� I ln l a l l
� 2
1
� 2
+
In w clef 271" = PI (w) and the fol
�f
l u(O) 1 = VTh(6)T < 2 VTh(6)T P2 (w), I v(O) 1 = Vlh(O )  1 1 < 2 vlh(0 )  11 � P3 (w) , lu (O)  v(O) 1 < 2 Iu (0) P4 (w) , 1 = (O) + v(O) 1 < 2 Iu (0) + v(O) 1 = P5 (W) . I u ( 0 )  v(O) I lu
v(O) 1 �f
clef
The function F (in H (1U) ) may be determined so that  71" < 'S[F(O)] � 71". Then IF(O) I � l In lu (O)  v(O) 1 1 + 71" < 2(l ln lu (O)  v(O) 1 1 + 71") �f P6(W).
Section 9.4. The Great Picard Theorem
377
Fg � ��;, i z] is in H(1U) and () 1 ( (1U) contains no open disc of radius (1 _ fJ ) I F' (�) I ' Since ('(0) = 1, 9.2.4 implies (9.4.3) I F' (0 I < 1 : fJ ' As derived , (9.4.3) is valid because F'(�) j. 0; the conclusion holds a fortiori if F' (�) = O. Integration implies I F(z)  F(O) I :::; I _6 fJ · fJ < I 6 fJ ' I F(z) 1 < I F(O) I + 1 6 fJ cle=f
a) Schottky ' s Theorem implies Bloch ' s Theorem. b) ' Schottky s Theorem implies Landau's Theorem, viz.: For some map ¢ C '3 z r+ ¢(z) E (0, 00), 9.4.5 Exercise.
:
{J E H[D(O, ¢( a)] } ::::} {J is not 2defective} . c) Landau ' s Theorem implies Bloch ' s Theorem. 9.4.6 DEFINITION. A SUBSET F OF Cn ( VIEWED AS A SUBSET OF C� ) IS Coo normal IFF EVERY SEQUENCE CONTAINED IN F CONTAINS A SUBSEQUENCE THAT IS LOCALLY UNIFORMLY CONVERGENT IN C� .
{In } nE]\/
9.4.7 THEOREM. (Montel) THE SET F IS Coo NORMAL. PROOF. If a E fl and
Fa,M
�f { I
: 1 E F,
�f { I
: 1 E H(fl), D(f) = {O, I} }
I I(a) 1 :::; M < oo } ,
(9.4.8)
translations and changes of scale permit the assumptions: a = 0, 1Uc c fl, and M = 1.
(9.4.9 )
378
Chapter 9. Defective Functions
Schottky's Theorem implies that if 0 :::; fJ < 1 , then sup
{ 1 I I I D(O,l!) 11 00 : I E Fa,M } :::; <1>(I, fJ)
<
00 .
Thus, for some neighborhood W of zero, e.g., W = D(O, fJt, if w E W and I E Fa,M, then I I(w) 1 :::; <1> ( 1 , fJ) . When restrictions (9.4.9) are abandoned, the conclusion is: V �f z : z E Q, sup I I(z) 1 < oo is an open subset JEF of Q. Furthermore, a E V , whence V j. 0. If Q j. V and b E Q \ V, then { 11 (b) 1 : I E F} is not a bounded set of numbers. Thus there is a sequence {In } nE]\/ contained in F and such that Iln (b) 1 t The sequence { IIn �f gn } nE]\/ is contained in F and Schottky ' s Theorem implies that for some open neighborhood U of b and some positive P, {z E U } { l gn (z) 1 :::; P } . Vitali ' s Theorem (5.3.60) implies that for a subsequence, again denoted {gn } nE]\/ ' and a g in H (Q), the gn converge uniformly on each compact subset of U to g. Since lim gn (b) = 0, 5.4.39 implies a) g 0 or b ) 0 tt g(Q). If a) is true, for every z in Q, n+limoo In(z) = whereas, if z E V, lim 11,,(z) 1 < a contradiction. Thus a ) is false oo n+ and b ) is true. Consequently n+ limoo gn(b) j. 0, a second contradiction. In
}
{
00 .
::::}
n + CXJ
==
00
00 ,
eluctably, Q \ V = 0. If Q contains an a such that (9.4.8) obtains, then Q = V, i.e.,
{z E Q }
::::}
{ JEF I I(z) 1 } sup
< oo
.
Then 5.9.13 implies F is normal. In particular, if then { lln (a) 1 : n E N} is a bounded set of (real ) numbers. There remains the possibility that for each a in Q, (9.4.8) is false. Hence, for each a in Q, and any sequence {In } nE]\/ contained in F,
{ lln (a) 1
{
:
n
E N}
1 clef . . . I. S unb ounded. The sequence In = gn nE]\/ IS contalIled I' n F and, VIa passage to a subsequence as needed, the earlier argument shows this time
}
Section 9.4. The Great Picard Theorem
that
g O. ==
Thus
:F is Coo normal.
379
{In } nE]\/ converges to 00 on every compact subset of Q:
D
9.4.10 THEOREM. ( The Great Picard Theorem ) IF I E H [A(a; O, rt], a IS
I, AND 0 < s < r, THEN # {C \ I [A(a; O, s t]} �f N :::; 1.
AN ESSENTIAL SINGULARITY OF
PROOF. If N > 1 , it is permissible to assume
a
=
0, D(f)
:J
{O, I}, r
=
1.
Then 9.4.7 implies that the sequence
{ In : A (0 ; 0, I t '3 Z I ( � ) } nE]\/ r+
is Coonormal. Hence, via passage to a subsequence as needed, either for some positive M and every n , In c D(O, M) or, in the notation used in 9.4.7, gn c D(O, M). Passage to the limit yields
[C�(O )]
[C� (0)]
[C2� (0)] c D(O, M). The Maximum Modulus Theorem implies that for each in N, I [A ( 0; : 1 ' �] 0 ) c D(O, M) or or g
n
n
Hence I I(z) 1 or 1 is bounded near zero, which is impossible because I I(z) 1 zero is an isolated essential singularity of I · D
I,
9.4.11 COROLLARY. IF c IS AN ISOLATED ESSENTIAL SINGULARITY OF THERE IS AT MOST ONE COMPLEX NUMBER SUCH THAT IN EACH NEIGH = FAILS TO HAVE INFINITELY MANY SOLUTIONS. BORHOOD N ( c) ,
I(z) a
a
PROOF. The argument given above applies to each N ( c) . Hence, if I(z) = a has no solution in N ( c) and b j. a, then I( z) = b has a solution Z1 in N ( c) . If Z1 tt 21 N ( c) , then I ( z) = b has a solution Z2 in 212 N ( c) . Induction provides a sequence {zn } nE]\/ converging to c and such that I (zn) = b. D
Chapter 9. Defective Functions
380 9 . 5 . Miscellaneous Exercises
g g are constant functions. [Hint: In the equation ef = 1  e9 the left member is never zero. Hence { 2mri} nET C [C \ g(C)] .] 9.5.2 Exercise. If 1 and g are entire and 12 + g 2 = 1, for some entire function h, 1 = cos oh and g = sin oh. [Hint: In the equation 12 + g2 = (f + ig) . (f  ig) neither factor of the right member can be zero. Then 8.1.8h) applies.] 9.5.3 Exercise. If 1 E T, there is at most one complex number a for which I(z) = a does not have infinitely many solutions. [Hint: If g(z) �f 1 , z j. 0, zero is an isolated essential sin gularity of g.] 9.5.4 Exercise. If zero is an isolated essential singularity of I, for some c in A(O; 0, 1) and for some a in C, if ° < E < lei , then 9.5.1 Exercise. If 1 and are entire and e f + e9 = 1, then 1 and
(�)
If 1 is entire and periodic, there are infinitely many 1fixedpoints. [Hint: If g(z) �f I(z)  z, then g is entire. If p is a nonzero period of I, for all but at most one n in Z, np E g(C).] 9.5.6 Exercise. There are nondefective nonpolynomial entire functions. 9.5.7 Exercise. If x E [0, 00], there is a nondefective entire function such that p(f) = x (v. 7.3.3). 9.5.5 Exercise.
10
Riemann S urfaces
10.1. Analytic Continuation
A fundamental fact about a function 1 in H (fl) is that for each a in fl, there is a (unique!) sequence { cn,a } nE Z+ and in ( 0, 00] some radius of convergence Ta such that if I z  a l < Ta , then 00
I(z) = L cn,a ( z  a) n n=O (v. 5.3.23) . If I b  a l < Ta, the equation
(10.1.1)
L cn, a [z  b + (b  a)] L Cn,b ( Z  b) n n=O n=O serves to define uniquely the sequence { Cn, b } nE Z+ ' 10.1.3 Exercise. In (10.1.2),
(10.1.2)
=
=
00
(10.1.4) The series in the right member of ( 10.1.4) converges. For the radius of convergence Tb of the right member of (10.1.2), there obtains the inequality:
Tb 2': Ta  I b  a l · 10. 1.5 Exercise. The function T : C '3 b r+ Tb E Coo is continuous. When T a = 00, the right member of ( 10.1.1) defines a function F holo morphic in C, i.e., F E E, and F ln = I. The Identity Theorem (5.3.52) implies that if G E E and G ln = I, then F = G: F as described is unique. On the other hand, if Ta < 00, the discussion takes a more extended
form. The following Examples offer a sampling of the variety of phenom ena to be encountered. 10.1.6 Example. If fl �f C \ (00, 0] and Z E fl, then [1, z] e fl and the equat l. On I(z) cle=f  defines a function in H (fl) . Hence, if a E fl, then [ l ,z]
1 dw w
381
Chapter 10. Riemann Surfaces
382
� J ( nl (a) n cle=f � a) (Z J(Z) = � � Cn (Z  a) n and Ta = l a l . Furthermore, n . n=O n=O if z E Q, then I'(z) = � . If F(z ) �f exp[J(z)] , then [ F�Z) = O Since J(I) = 0, F(z) = z, i.e., exp[J (z)] = z, J(z) is a branch of Inrz. . If a = 1 and z E D(I, It, then dw r 1 + (w r dw J(z) = J[l,Z] w J[l,Z]  l) n (w  l) n dw =f (I) l n=O [l,z] +1 = 2: (1)" ( z n It �f 2: Cn (z  lt �f h(z). (10.1.7) + 1 n=O n=O Since T l = 1, (10. 1.7) provides no definition of h outside D(I, It. Never theless, for the Junction elements, i.e., the pairs (I, Q) and [ h , D(I, It], the I
=
(Xl
(Xl
following obtain. Q n D(I, lt j. 0; J l nn D(l,llo = h l nn D(l , l )O : J and h agree where they are both defined and J provides a natural extension of h to points outside D(I, lt· These circumstances motivate •
•
10.1.8 DEFINITION. FUNCTION ELEMENTS ( h , Qd AND (12, Q2) ARE OF ONE ANOTHER IFF:
mediate analytic continuations a) Ii E H (Qi ) , i 1, 2;
im
=
b) Q 1 n Q2 j. 0; c) h I n1nn2 12 In1nn2 =
WHEN f E N, A FUNCTION ELEMENT (II , Q J ) IS AN analytic continua tion OF A FUNCTION ELEMENT ( h , Qd IFF THERE IS A FINITE SEQUENCE OR chain {(Ii, Qi ) }l�i� J SUCH THAT (Ii, Qi) IS AN IMMEDIATE ANALYTIC CONTINUATION OF ( Ii I , Qi d , 2 � i � f,.
When ambiguity is unlikely, the function II itself is called the analytic continuation of h . The roles played by (h , Qd and (12, Q2) are symmetrical. Each of h and 12 can be an analytic continuation of the other at points where the other is not yet defined. In 10.1.910.1.12 below, J is the function in 10.1.6 and Q C \ ( 00 , 0]. =
Section 10. 1. Analytic Continuation
383
10.1.9 Exercise. If Z E fl, the closed interval [1, z] may be covered by a finite set of open discs Di �f D (Zi , ri t , 1 � i � I, such that each is con tained in fl and Z1 = 1, zi+1 E Di , 1 � i � I  1. In that event, the power series representation for I I D is the result of the next equations, i+l
n L I ( ) (Zi ) (  Zi ) n = L 1 (") (Zi ) [z  Zi+1 + (Zi+1  Zi )] n n=O n=O ( n) (zi +1 ) ( Z _ Z, ) n , _ ""' I  � +1 (Xl
, n.
z
(Xl
(Xl
n=O
v.
(10.1.2).
, n.
, n.
.
Z �f x + iy E fl and x is fixed, then :3 lim I (z) �f L + (x) and :3 lim I (z) �f L (x). yto y+O
10. 1.10 Exercise. If

Furthermore,
if x < 0 if x = 0 . if x > 0
10. 1 . 1 1 Exercise. There is no entire function F such that F l n =
I.
g �f exp (�), then g E H (fl) and g(z) 2 = Z in fl. b) There is no entire function G such that G l n = g. c) If h(z) �f Vi = )1 + (z  1) ... ' 1 = 1 + f (�  ) (� �? (�  Tn) (z _ 1) Tn �f f (z _ 1) Tn 10. 1.12 Exercise. a) If
Tn =1
n=O
Cm
then r 1 = 1. If D(a, rr c fl, then [g, D(a, rr] is an analytic continuation of [h, D(I, lr]. Despite the conclusions in 10. 1 . 10 and 10. 1. 11, if a �f a + i b E fl and a
I (n) ( ) ( Z _ ) " �f ""' cn ( Z _ ) " � I( Z ) �f ""' � (Xl
n=O
, n.
a
(Xl
n=O
a
,
(10.1.13)
then ra = 1 0' 1 ; if c < 0, then D ( a , rat n (00, 0] i 0: D ( a , rat is not contained in fl. Consequently, k denoting the function represented by the series in (10.1.13), if z E D ( a , rat and 'S(z) < 0, then k (z) i I(z), even though [k , D ( a , rat] is an analytic continuation of [h, D(I, lr].
Chapter 10. Riemann Surfaces
384
In D ( a , rat n { z of [h, D(l , 1 rl .
'S( z) < O }
there are two analytic continuations
rearrangement of series
The process of can (sometimes) yield an analytic continuation [k, D ( a , Rc )Ol of [h, D(l, 1 rl that is different from (I, Q) .
a
f
'
a
10.1.14 Example. If E H (Q), E Q, and J ( ) j. 0, the Inverse Func tion Theorem, 5.3.41, implies that for some positive in H {J [D ( , rn rl engenders == z. Furthermore, [ , D( there is a such that
r, ar g f I D(a ,r) O g a, r analytic continuations to other function elements. Local inverse functions like g above, the local inverse of exp exp, and sin constitute a particularly g
0
0
rich source of analytic continuations exhibiting the phenomena described above. In fact, 10.1.6 is based on the local inverse of the entire function z r+ exp(z).
f
a
r
a
regular f,
10.1.15 DEFINITION. WHEN E H (Q) , E 8(Q) , > 0, IS A OF f IFF SOME [ , D( rl IS AN IMMEDIATE ANALYTIC CONTINU ATION OF (I, Q) . WHEN b E 8(Q) AND b IS NOT A REGULAR POINT OF OF b IS A
point
g a, r singular point f.
The set of regular points of f may be empty, e.g., if Q C (in which case the set of singular points is also empty) or if f is such that 8(Q) is the natural boundary for f (in which case the set of singular points is 8(Q).) A singular point a of f is isolated iff f is represented by a Laurent 00 series, L cnz n and for some negative n, Cn j. O. n==(X) However, if S �f {O} U { z : mrz = I } and =
f(Z) cle=f sm. ( 1; ) , g( z) �f
{ (� ) csc o
if z tt S otherwise
then {J, g} c H (C \ S) , each point of S \ {O} is a pole of f and an isolated essential singularity of g and 0 is a nonisolated singularity of both f and g. 00
L cn zn IS NONNEGATIVE n=O AND Rc = 1, THEN 1 IS A SINGULAR POINT OF f : 1U '3 z r+ L cn zn . 10. 1.16 THEOREM. (Pringsheim) I F EACH cn IN
00
n =O
385
Section 10. 1. Analytic Continuation
Tn z �) n PROOF. Since z n = � �O (m) ( 2nTn , the rearrangement Tn
=
(10. 1.17)
00
of L cn z n diverges if I z l > 1. Owing to the hypothesis Cn 2': 0, n E N+ , n=O Fubini ' s Theorem (4.4.9) implies that if 0 < z < 1, the inversion of the order of summation is valid in (10.1. 17):
� '"z" � [t> (;,) (Z2:_� l = �O Cn [� (:) 2n�Tn 1 (z _ �) Tn �
�
(10. 1 . 18)
� :,�D (z  D If 1 is not a singularity of f , for some positive there is a function element [g , D(I, t l that is an immediate analytic continuation of (I, 1U) . Hence �
!(
�
�
r
r
the distance d between "21 and some point of the boundary of 1U U D(I, r t 1 exceeds r the right member of (10 .1.18) converges for some z in (1, 00 ) , a contradiction. D ' 10.1.19 Exercise. Pringsheim s Theorem (10. 1.16) obtains if the hy pothesis Cn 2': 0, n E N+ , is replaced by � ( cn ) 2': 0, n E N+ , and it is as00 00 sumed that the radii of convergence of both L en z n and L � (en) z n are n=O n=O one.
g ( Tn ) (�) For g : 1U 3 z r+ L � (cn ) zr> ' L m., z  I Tn Tn=O n=O diverges if z 1. Furthermore, g ( Tn ) (�) = � [f( Tn ) (�)] and 00
00
[Hint:
>
f(Tn) ( 1 ) � m! "2
(z �) Tn _
( 2)
386
Chapter 10. Riemann Surfaces
diverges if z > 1 ( whether
converges or diverges when z > 1).] 10. 1.20 THEOREM. IF THE RADIUS OF CONVERGENCE =
Re OF
P(a, z) �f L cn (z  a) n n=O IS POSITIVE ( AND FINITE ) , FOR SOME b IN ea (Re), NO FUNCTION ELE MENT [g, D(b, sr] IS AN IMMEDIATE ANALYTIC CONTINUATION OF [P(a, z), D (a, RetJ , i.e., SOME POINT b OF ea (Re) IS A SINGULAR POINT OF I : D (a, Ret r+ L en (z  a) n . n=O PROOF. Otherwise, each b of ea (Re) lies in an open annular sector 00
and there is a function element (jb, Ab ) that is an immediate analytic con tinuation of [ I, D( a, rr]. Since ea (Re) is compact, ea (Re) contains a £1M nite set { bmL <_ m<_ M such that U Ab = :J ea (Re). If s = 1�m�M max rb = and m=1 5 = 1�m�M min Rb= , then s < Re < 5 and
D (a, 5) c D (a, Re) U A(a ; s, 5).
If z E D (a, Ret n Abk n Ab" the Identity Theorem implies Ibk (z) Ibl (z). Hence the radius of convergence of P(a, z) is at least 5, a contradiction. =
D
10. 1.21 Example. The radius of convergence of
ex:> zn is one. P(O, z) d�f L n=l
n2
Furthermore P(O, z) converges for each z in T. However, 10.1.16 implies that 1 is a singular point of I 1U '3 z r+ P(O, z). :
Section 10. 1. Analytic Continuation
387
10. 1.22 Example. The opposite extreme of a function element (I, Q) that permits analytic continuation beyond Q is a function element (I, Q) such that 8(Q ) is a natural boundary for f (v. 7. 1.28). 00
The radius of convergence of L zn ! is 1. The sen=O ries defines a function f in H (1U) . Furthermore, if q E Qi, ° � r < 1, and z = re27rqi , then I f (z) I t 00 as r t 1. If F is a function holomorphic in a re gion Q meeting 1U and F l nn 1U= f lnn1U, then Q C 1U: '][' is a natural boundary for f. 00 " · lor g ( z ) cle=f ,", S I· ml· lar conc1USl· OnS 0btam � z2 . n=O The phenomena in 10.1.23 exemplify the next two results. 10. 1.23 Exercise.
C
P I < P2 < . . . ; ) 1 < M E N; d) qn ( 1 + � ) Pn , E N; e) FOR Cn zn REPRESENT ING f, = 1; f) 1 IS A REGULAR POINT OF f; AND g) Cn = 0 WHEN Pk < < qk , k E N, THE SEquENCE {Spk } k E N OF PARTIAL suMS OF L Cn zn n=O c
n
>
n
Rc
�
00
CONVERGES IN A NONEMPTY NEIGHBORHOOD OF 1 .
PROOF. Because 1 is a regular point of f, it follows that some analytic continuation of f is holomorphic in some Q containing 1U U {I}. For the entire function defined by g (z) �f � (z M + z M + 1 ) : •
•
2 if z E D(O, 1) \ {I}, then 1 1 + z l < 2, and I g(z) 1 = 21 1 z M I · 1 (1 + z) 1 < 1. g(l)
= 1;
Thus K �f g[D(O, 1)] C 1U U {I} C Q: the compact set K is a subset of the open set Q. Hence, for some positive 15, O(C) '3 U �f
U D(z, r5t C Q.
z EK
Chapter 10. Riemann Surfaces
388
E,
The uniform continuity of g on D(O, l) implies that for some positive g [D(O, 1 + t] c Q. Since h(z) �f 1 0 g(z) is holomorphic in g l (Q), it follows that for some {bn } nE l\! and a positive h(z) L brn zrn obtains in D(O, 1 + t Direct rn =O calculation shows that for k in N, (M+l)Pk rn Pk (10. 1.25) [ (z) c SPk L n g(z)t L brn z . n=O rn=O The rightmost member of (10. 1.25) converges in D(O, 1 + ) i.e., {Sp k } kE l\! converges in D(O, 1 + ) as required. D E
E,
=
00
.
=
=
E
E
E
O
O
,
10. 1.26 THEOREM. (Hadamard) IN THE NOTATIONS AND CONDITIONS
k E N, Pk +l > ( 1 + �) Pk , I(z) � Ck ZPk , AND Rc 1, THEN '][' IS A NATURAL BOUNDARY FOR I. PROOF. If E '][' and is a regular point of I, then 1 is a regular point of I(O'z). Hence the argument in 10.1.24 applies: {sPk (z)} kE l\! converges in some D(O, 1 + t In the current circumstances, {spk h E l\! {sp} P E ]\/" Hence, if is a regular point of I, in some D(O, 1 + t {Sp(z)} p E l\! converges, i.e., in some D(O, 1 + t L cn zn converges. This conclusion conn=O tradicts the assumption: Rc 1. D [ 10.1.27 Note. In 10.1.24, although the series representing 1 diverges for each z outside D ( O, 1), some sequence of partial sums of the series does converge in some nonempty neighborhood of 1: there is overconvergenCE. By contrast, for any sequence {Spk } kE l\! of partial sums of the series L zn !, if q E Qi, R > 1, and z Re27rqi , then I Sp k (Z) 1 t 00: n=O overconvergence is absent. In 10.1.24 the sizes of the gaps [the sequences of successive zero coefficients] increase rapidly, while the sizes of the nongaps [the =
k
OF 10.1 .24, IF c i 0, =
a
a
E
.
a
E
=
00
,
E
00
,
=
=
sequences of successive nonzero coefficients] may increase as well. Hadamard ' s result asserts that if the size of each nongap is one and the sizes of the gaps increase sufficiently rapidly, the boundary of the circle of convergence is a natural boundary. What follows is an illustration of what can happen when the gaps are Hadamardlike per the hypotheses of 10.1.26 and the sizes of the nongaps also increase sufficiently rapidly.
Section 10. 1. Analytic Continuation
If P(z) �f Z ( Z + 1), then
389
00
(10. 1.28)
converges and defines a function f holomorphic in each component of Q �f { z : I P(z) 1 < I }. A point a is on the boundary of a com ponent of Q iff I P(a)J = 1. One component, say C, contains, for some maximal positive r, D(O, rt. Owing to 10.1.26, the bound aries of the components of Q are natural boundaries for the restrictions of f to those components. Since S �f Co( r) n 8(C) "I 0, if a E S, then a is a singular point of f. In D(O, rt, f is repre sented by a power series P(O, z) calculable from (10. 1.28). Further more a given power of z appears in at most one term of (10.1.28) . Hence both the gaps and the nongaps in P(O, z) are Hadamard like. Overconvergence n occurs for some z in Q \ D(O, r) ("I 0). 2 If H (z) �f L �, n the radius of convergence of the right member n= l is 1. Owing to 10.1. 26, '][' is a natural boundary for h. On the other hand, whereas I f I and I g l in 10. 1.23 are unbounded in 1U, 7[2 if I z l < 1, I H( z)1 ::::: L n12 6 · ex:>
ex:>
=
n= l 1 � z n and the radius of conver If I z l < 1, then f(z) �f _ � z _1 = n=O gence of the right member is 1, while 1 + iO is the unique singular point of f. Nevertheless, if I z l > 1 and { s nk (z)} kE]\/ is a sequence of partial sums of L z n , {S nk (z)} kE]\/ diverges: overconvergence n=O is absent. 00
There is an extended discussion of the phenomena noted above in
[Di].]
For the functions f resp. g in 10.1.6 resp. 10.1.12, if (II , Q) resp. (gl , Q) is an analytic continuation of [J,z D(I, 1 t] resp. 2[g, D(I, It] ' the Identity Theorem implies that in Q, eh ( ) = z resp. g l (z) = z. The three results that follow elaborate on this theme. 10. 1.29 Exercise. For the polynomial
Tn l = l ,
...
,Tn n = l
a
'TTl l , · · · , 'TTl n
n n,
Tn 1 . . . Tn w1 w
Chapter 10. Riemann Surfaces
390
if { (II , Q) , . . . , Un , Q)} is a set of function elements such that on Q,
P (II , · · · , fn) = O, for any set {(g l , Q I ) , . . . , (gn , QI ) } of function elements that are analytic continuations of the { (II , Q) , . . . , Un , Q) } , the equation is valid on QI . [Hint: The Identity Theorem for holomorphic functions applies.] 10.1.30 Exercise. If, in the context of 10.1. 29, f is a solution on Q of the differential equation P (y, y' , y" , . . . ) = 0 and (g, QI ) is an analytic continuation of (I, Q) , on QI , P ( g, g' , g" , . . . , ) = O. 10.1.31 Exercise. The conclusion in 10.1. 29 remains valid if P is rea'm l, 'm n W � l • • • w;:' n that converges placed by a power series L CXJ , • • • ,CXl
'TTl l = l , ·· · , 'TTln =l in a nonempty polydis c X:= I D (0 , r ) 0 [ 10.1.32 Remark. The phenomena in 10. 1.2910.1.31 are ex amples of the Permanence of Functional Equations (under ana •••.
k
•
lytic continuation).
Thus, although analytic continuation permits the creation of new function elements from old, when 1 � m � n, (I'm , Q) resp. (grn, Q) are analytic continuations of one another, and
(in this instance, the second members of all the fUllction elements are taken to be the same) , nevertheless, functional relations among the f'm 1 � m � n, persist among the grn , 1 � m � n .] '
w i  w � , then (II , Q I ) � ( z r+ z, q , (12, Q2 ) �f ( z r+ z, q are function elements such that P (II , II ) = P ( 12 , h) = O. Yet neither of (II , Q I ) and (12, Q2 ) is an analytic continuation of the other. The converse of the principle of the Permanence of Functional Equations (under analytic continuation) is false.
10.1.33 Exercise. In 10.1.29, if
n = 2 and P (W I , W2 ) �f
391
Section 10.2. Manifolds and Riemann Surfaces 10.2. Manifolds and Riemann Surfaces
The developments that follow organize the study of functions such as In : D ( 1, 1 t '3 z r+ In z E C, vrD(I, It '3 z r+ Vi E C,
and other functions afflicted with ambiguity ( multivaluedness) when their original domains are extended, e.g., to C \ {a}. As the discussion in Section 10.1 reveals, a power series P( a, z ) can engender analytic continuations to power series PI (b, z ) and P2 (b, z ) such that PI (b, z ) j. P2 (b, z) . The definition of the word function as it is used in mathematics makes the term multivalued function an oxymoron, although there is a temptation to describe as multivalued a function f that is locally represented by P( a, z), PI (b, z), and P2 (b, z). Further discussion can be conducted systematically in the context es tablished by the following items. A complete analytic function is a collection CAF �f {(Iv, flv)} v EN of function elements such that: a) each is an analytic continuation of every other; b) any function element that is an analytic continuation of a function element in CAF is (also) in CAF: CAF is a maximal set of function elements each of which is an analytic continuation of any other. The function elements (II" ' fll" ) and (Iv, flv) in CAF are aequivalent, i.e. , UI" ' fll") Uv , flv), iff: a) a E fll" n flv; b) in some neighborhood N(a), fl" I ( a) = fv I ( ) " Thus to each a in the union fl �f U flv v EN there corresponds a set of "'aequivalence classes of function elements (II" ' flv) for which a E flw When (I, fl) E CAF and a E fl, the "'aequivalence class to which U, fl) belongs is the germ or a branch of (I, fl) at a, v. 10.2.1, and is denoted [I, a]. As the discussions of 10. 1 .10 and 10.1.12 reveal, it is quite possible for a function element (g , (for which a E to belong to CAF while, in the current notation, [I, a] j. [g, a]. A result proved below and due to Poincare, implies that the cardinality of all germs at a cannot exceed No . The set W �f { [I, a] : a E fl, (I, fl) E CAF } of germs is, in recogni tion of its originator, Weierstrafi, the Wstructure determined by any (hence every) function element in CAF. (A topology for W is given in 10. 2.4 below.) Associated with W are the projection p : W '3 [I, a] r+ a E fl and the map f : W '3 [f, a] r+ f(a) E C. •
•
N N "'
a
a
•
n)
•
•
n)
Chapter 10. Riemann Surfaces
392
10.2.1 Exercise. a) When a E C and P(a, z ) is a power series with a positive radius of convergence Ra, then P( a, z ) represents some function f and an associated germ [I, a]. b) For a fixed, the correspondence
{ P(a, z)
: Ra > O } +
{ [f, a] : a E fl, f E H (fl) }
between the set Sa �f {(P( a, z ) , Ra) } of all pairs consisting of a power series P( a, z) converging at and near a and the associated radius of convergence Ra and the set Hf, an of all germs at a is bijective. c) If UI" ' fll" ) and Uy , fly ) need not be "'bequivalent. Nevertheless, UIL ' fllL ) and Uy , fly ) are analytic continuations of one another. d) If a j. b two of the equations [f, a] = [g, b], [I, a] = [g, a], [I, a] = [I, b], are meaningless. For any a in fl, a function element U, fl) and one of its analytic continuations such that a E fl n the numbers f( a ) and a) may differ, v. 10.1.10 and 10. 1.12. Nevertheless, they are regarded as values of the multivalued function that arises from analytic continuation. On the other hand, f is a true (singlevalued) function on W and the range or image f(W) accurately reflects the different values f(a), g(a), . . .
(g, n)
n,
g(
,
10.2.2 Exercise. The set fl '!gf ex:>
U fly is a region. What is fl in 10.1.21?
"'EN
[Hint: The series L .;. arises by integrations and algebraic trans n=! n formations applied to L zn .] n
00
n=O
10.2.3 Exercise. The function f is welldefined on W. The description of CAF suggests a topology derivable by pasting to gether the fly used to provide analytic continuations. However, the accu
rate description of such an informal topology is beset with the complications arising from the presence in CAF of equivalent function elements. A topol ogy is more readily attached to the set W of equivalence classes, i.e., germs or branches, according to 10.2.4 DEFINITION. WHEN V IS AN OPEN SUBSET OF fl AND
[I, V] �f { [f, a] : a E V }.
f E H (V) ,
Section 10.2. Manifolds and Riemann Surfaces
393
10.2.5 Exercise. The set
T �f { [I, V]
: V an open subset of Q, for some Q,
( j, Q)
E CAr }
is a Hausdorff topology for W. 10.2.6 THEOREM. WITH RESPECT TO T, p IS CONTINUOUS AND OPEN.
PROOF. If a E V E 0(((:) and [I, a] E pl (V) , then [I, V ] is a neighborhood of [I, a] and p( [J, V] ) = V: p is continuous. For any open subset V of Q, p([I, V] ) = V, whence p is open. D 10.2.7 Exercise. If V is an open subset of Q, then p l [J, v] is injective. (Hence p is locally a homeomorphism.) [ 10.2.8 Note. For a in Q, a in C, and germs [I, a] , [g, a] , the germs [aJ, a] , [I + g, a] , and [lg, a] are welldefined. Thus, for each a in Q there is the Calgebra !i a
�f { [I, a]
: J holomorphic in some N
(a) }
of germs of functions holomorphic at and near a. Generally, for a category C, e.g. , the category of Calgebras, and a topological space X, a sheaJ S [Bre] is defined by associating to each U in O(X) an object S(U) in C. It is assumed that: When { U, V } C O(X) and U C V, there is a morphism •
Pb : S(V) r+ S( U ) •
• •
When { U, V, W } c O(X) and U e V e V w. Pw u = pu Pv ,
.
W,
then
0
p� is the identity morphism. When {U>.} ), EA C O(X), U �f U U)" k l ' k2 in S( U ) are the ), EA same iff for each A, pg>. (k I ) pg>. (k2 ) .
For U), and U above, if
=
k), E S (U), ) , A E A, { W �f U), n Ul" =j:. (/)} ::::} {p� (k),) = p� (kl" ) } , there is in S( U ) a k such that for every >., pg>. (k) = k),.
394
Chapter 10. Riemann Surfaces
When x E X, the filter V �f {V(x) } of open neighborhoods of x is partially ordered by inclusion: U(x) < V(x) iff U(x) C V(x). The stalk Sx at x is the set of all {kv } v E V such that For s E sx , if p(s) �f x, then p maps S �f
U
Sx
onto X. The
set {p  l (U) : U E O(X) } is a topology T for S; p is open and T is the weakest topology with respect to which p is locally a homeomorphism. If X is a Hausdorff space, so is S. When the objects in the category C are algebraic, e.g. , when C is the category of groups and homomorphisms or the category of Calgebras and Chomomorphisms, as x varies in X the results of the algebraic operations within the stalk Sx are assumed to depend continuously on x. For example, when X = C and C is the category of all Calgebras, for each open subset V of C, the set S(V) may be taken the set of all functions holomorphic in V, and when U C V, pi; maps each f in S(V) into f l u ' Then: S is the sheaf of germs of holomorphic functions. For a in C, Sa is the stalk at a and consists of all germs [I, a] . The map p : S r+ C is that given iri the context of W : p maps each germ [I, a] onto its second component a. A Wstructure W is a connected subset of S. The elements [oJ, a] , [I + g, a] and [ lg , a], maps from C to S are continuous.] xE X
as
•
•
•
•
•
as
10.2.9 Exercise. The topology induced on each stalk of a sheaf is discrete. (Hence, when the stalk is an obj�ct in a category of topological objects, the (discrete) topology induced by T on the stalk need not be the same the topology of the stalk viewed an object in its category.) as
as
Q
10.2.10 THEOREM. (Poincare) IF a E THE CARDINALITY OF THE SET OF GERMS [I, a] AT a DOES NOT EXCEED No . PROOF. Each germ [I, a] corresponds to some function element (such that a E If [ , a] corresponds to (again a E then is an analytic continuation of (I, Thus there is a finite chain
Qv). g
in which
Qv).
(g, QJL)
QJL) '
(f, Qv) (g, QJL)
(ik, QVk ) is an immediate analytic continuation of (ikl , Qvk _ l ) ' QVk contains a point p + iq for which {p, q} C Qi, i.e.,
2 :::; k :::; m. Each
Section 10.2. Manifolds and Riemann Surfaces
395
p + iq is a complex rational point. Thus, corresponding to each finite chain
C, there is a finite sequence of complex rational points. The cardinality of the set of all finite subsets of ((f is No · D Thus, if Z E Q and Sz �f p  1 ( z ) (the stalk over z ) , #(sz) :s; No. If r > 0 and V is an open subset of Q, each [I, z] (E sz ) is in a neighborhood [I, V] . Furthermore, p : [I, V] r+ V is a homeomorphism. Consequently, W may be viewed as consisting of sheets lying above Q and locally homeomorphic to Q. If a E Q, for some positive r, V �f D(a, rt lies under a set of homeomorphic copies of V, One copy in each of the sheets. In 10.2.11 and 10.2.12 there are precise formulations of the preceding remarks. 10.2.11 Exercise. As z ranges over Q, the cardinality #(sz) remams constant. [Hint: If #(sz) �f k E N, then #(sz) = k near z. If
analytic continuation from z to w yields a contradiction.] 10.2.12 Exercise. If 1 E E, #(sz ) == 1 for the associated Wstructure W. 10.2.13 Exercise. If a E Q, for some N(a), p  1 [N(a)] consists of (at most count ably many) pairwise disjoint homeomorphic copies of N(a). The following variant of the ideas above leads to greater flexibility of the discussion. The Wstructures introduced thus far are extended to analytic structures that include socalled irregular points [Wey] . 00 The map I : Q 3 z r+ L cn (z a ) n may be viewed as an analytic
n=O description of a complex curve
{

(z, w )
: w =
� cn (z  a)n }
in Q x C.
The same object may be described alternatively by the pair z =
00
a + t, W = Co + L1 cntn n=
(10.2. 14)
of parametric equations. Extended somewhat further, the parametric equa tions (10.2.14) are replaced by a pair 00
00
P(t) �f L an t" , Q(t) �f L (3rn t'n n =k
for which the following conditions are imposed.
Chapter 10.
396
Riemann
Surfaces
a) { k , l} C Z ; b) P and Q converge in some nonempty punctured disc .
D(O,
(if
c)
rr clef= { t
:
0
< ItI < r }
k and l are nonnegative, P and Q converge when t = 0, i.e., r
D (O , n ;
m
{ { {t l , t2 } C D (O, rr } 1\ {P (t I ) = P (t2 )} 1\ {Q (t I ) = Q (t2 )} }
::::} {t l = t2 } '
When k or l is negative, 0 is the only pole in D(O, rr of the corre sponding P or Q. Hence, if, e.g., P ( O) = 00 = P (t 2 ), automatically t 2 = O. Thus there is a uniquely defined injection
rr 3 t r+ [P(t ) , Q(t )] E C� . The parameter t is a local uniJormizer. The Junction pair (P, Q) , subjected L :
D(O,
to a)c) , is now the object of interest. Various parametric representations can represent the same curve, e.g.,
{ (cos t, sin t) : 7r < t < 7r } and { ( 11 + 77: , � ) 1+7
: 00
< 7 < 00 }
are different parametric representations of
2 y2 = 1, x
{ (x, y) : x +
>
1 } .
Function pairs [P (t ) , Q (t ) ] and [R(7), S(7)] (both subjected to a)c)) are regarded as equivalent and one writes [P (t), Q( t ) ] rv [R( 7), S ( 7)] iff the local uniformizer 7 is representable in the form 00
7
= L "Intn , "11 :=J 0, n= l
(10.2.15)
and the series (10.2.15) converges in some open disc D(O, rr. 10.2.16 Exercise. The relation rv described above is an equivalence relation. 3
The rvequivalence class containing the function pair (P, Q) is denoted (P , Q) or simply 3· The set of all 3 is 3 and the elements 3 of 3 are points.
Section 10.2. Manifolds and Riemann Surfaces
397
10.2.17 Exercise. For the maps
3(P, Q) r+ P(O) E ((:(3 : 3 3 3(P, Q) r+ Q(O) E ((:, ( (3) : 3 3 3(P, Q) r+ [P(O), Q(O)] E ((:2 , the complex numbers [3 (P, Q)] and (3 [3 (P, Q)] are independent of the representative (P, Q). Thus the notations 0' (3) and (3(3) unambiguously define complex num bers: and (3 are in ((:3 . a :
33
a,
a
a
10.2.18 DEFINITION. THE TOPOLOGY T OF 3 IS THE WEAKEST TOPOL OGY WITH RESPECT TO WHICH : 3 r+ ((:2 DEFINED ABOVE IS CONTINU OUS. 10.2.19 Exercise. If E 3 and is a nOnempty open disc L
3(P, Q)
D(O, rt
on which the map : D(O, rt 3 t r+ [P(t), Q(t)] E ((:2 is injective, a typical neighborhood N [3 (P, Q)] is (D(O, rn and consists of all 3 (15, described as follows. For some to in D(O, r t and all t such that to + t E D(O, r t, L
L
Q)
N [3 (P, Q)] consists, for all im mediate analytic continuations (15, Q) by rearrangement of the pair (P, Q), of the points 3 (15, Q) . 10.2.21 Exercise. If (P, Q) (R, S), each neighborhood N[3 (P, Q)] contains a neighborhood N[3 (R, S)]. The base of neighborhoods at a point 3 in 3 may be defined without 10.2.20 Exercise. By abuse of language,
rv
regard to the particular parametrization.
10.2.22 Exercise. As defined by neighborhoods described above, T is a
Hausdorff topology for 3. 10.2.23 Exercise. Each point 3 in 3 is contained in a neighborhood N(3) homeomorphic to 1U. 10.2.24 Exercise. The maps a and (3 are local homeomorphisms with respect to the topology T.
[� cn (z
rt1 is a function element, [I, a] the corresponding equivalence class, and (P, Q) is a function pair
10.2.25 Exercise.
If
 a) n , D ( a ,
398
Chapter 10. Riemann
created by unijormization, i.e., P(t) = a + t,
Q(t) = L cntn , the map
I : W 3 [I,
is a
(1", T) homeomorphism.
a] r+ 3 (P , Q) E 3
00
Surfaces
n=O
In many discussions of sets of equivalence classes, e.g., in LP ( X, J1 ) , the distinction between an equivalence class and one of its representatives is blurred. Thus, when ambiguity is unlikely, no distinction is made between an equivalence class 3 and one of its function pairs ( P , Q) . 10.2.26 Example. The following are some important examples of function pairs. P (t) = a + t,
P (t) = C k , k E N,
00
Q(t) = L bm tm , 00
m=O 00
m=l
Q(t) = L bm tm , l E Z. m=l
(10.2.27) (10.2.28 ) (10.2.29)
The function pair in: (10.2.27) corresponds to a function that is locally invertible near a; (10.2.28) corresponds to a function that conforms to the behavior con sidered in 5.3.47; (10.2.29) corresponds to a function with a pole.
10.2.30 Example. By appropriate reparametrizations, any function pair can be represented in one of the forms (10.2.27), (10.2.28), or (10.2.29). (10.2.27). If the original pair is antn , bm tm and a l j. 0, a reparametrization is
(� %:;0 )
(10.2.28). If k 1, ak j. 0, and the original pair is (ao + aktk + . . . , t, bmtm) , >
Section 10.2. Manifolds and Riemann Surfaces
399
for a Ck such that c� = a k , a reparametrization is
(10.2.29). If k > 0 and the original pair is a k :=J 0, and c� = a_ k , a reparametrization is ", cptp t _ � '" O'qTq , 0' 1 _ C11 , T �f p� q= 1 =1 ex:>
ex:>
Those points 3 representable in form (10.2.27) are the regular points; those representable in forms ( 10.2.28 ) and (10.2.29) are the irregular points. More particularly, points representable in form (10.2.28) are branch points of order k; those representable in form (10.2.29) are poles of order k. When k > 1 in form ( 10.2.29), the pole is branched.
analytic structure
10.2.31 DEFINITION. A SUBSET AS of 3 IS AN IFF # (AS) > AND: a) ANY TWO POINTS OF AS ARE THE ENDPOINTS OF A CURVE "/ 3 r+ E 3 ; b ) WHEN 3 E 3 AND FOR SOME CURVE E AS AND = 3 , THEN 3 E AS. Each W is a subset of some unique AS(W): W e AS(W) . If each point of AS(W) is a regular point, W = AS(W) . By abuse of language, an analytic structure AS is a 3.
,,/(0)
:
1 [0, 1] t "/(t) ,,/(1)
,,/,
curvecomponent of
10.2.32 Example. Under the convention whereby � is identified with 00,
(
�)
o
i.e., when the discussion is conducted in Coo , One neighborhood of the function pair :F �f P ( t) = t, Q(t) = corresponds to { t : t E C, I t I > O } , a neighborhood of 00, v. Section 5.6. The analytic structure engendered by analytic continuation is Coo .
Chapter 10. Riemann Surfaces
400
Owing to the generality with which function elements are defined, the analytic structure AS is larger than the analytic structure W presented earlier. The inclusion of irregular function elements permits the adjunction to W of branch points, poles, etc. Henceforth, Weyl's adaptation [Wey] of Weierstrafi ' s ideas for analytic structures is invoked where it is helpful. 10.2.33 LEMMA. EACH IRREGULAR poINT 30 LIES IN A NEIGHBORHOOD
N (30 ) IN WHICH ALL OTHER POINTS ARE REGULAR.
PROOF. If (P, Q) E 30, for some positive
r, if ° < l a l < r, then
and p' I v ¥= 0, since otherwise, the Identity Theorem implies p' 0, a con tradiction. The neighborhood N (30 ) corresponding to D(O, rt meets the requirements. D ==
10.2 .34 THEOREM. AT MOST COUNTABLY MANY POINTS IN AN ANALYTIC STRUCTURE ARE IRREGULAR.
PROOF. From 10.2.33 it follows that each 3 is contained in a neighborhood
N (30)
consisting (except possibly for 30 itself) of regular points. One of these regular points corresponds via the local homeomorphism to a p + iq in QI + iQl. Thus the irregular points are in bijective correspondence with a subset of QI + iQl (cf. PROOF of 10.2.10). D 10.2.35 Example. If clef
Cn =
(�) (� ) . . . [(2� I) ] n!
00
and I
z  1 1 < 1,
L cn (z  I) n converges and represents a function fo such that for n=O Z in D(I, It, [fo(zW = z. The two parametric representations of Vz are [R±(t),S±(t)] � (t + 1, ± cntn ) there are two square roots of z.
then
�
:
2zt
z
More generally, if :::; k E N, there is a sequence { cn } n E]\/+ ' found by formally differentiating and evaluating the results when = 1, and for 00 in D(I, It, L cn( z  It converges and represents a function fo(z) such
z
z
n=O
that for in D(I, It,
[Jo(zW
=
z: Parametrizations of (fo, D(I, 1) ° ) are
Section 10.2. Manifolds and Riemann Surfaces
[Rq(t), Sq (t)] �f (t + 1, (different) kth roots of
401
� ( � ) cn t n )
z.
exp 2 7ri
, 0 :::; q :::; k
 1: there are k
Via 10.2. 19, there are k pairwise disjoint neighborhoods
each homeomorphic to 1U. The corresponding AS provides a kfold cover of C, v. Section 10.3. If 0 :::; q :::; k 1, "I : [0, 1] 3 s r+ "I(s) E 3 is a curve, and for s in [0, 0.5), "I(s) (Ps , Qs ) is regular, v. (10.2.27) , while "1 (0 . 5 ) = (Rq , Sq ), there are k possible continuations of "1 ( 0 . 5) along the rest "I ([0 .5, 1]) of the curve. If "I : [0 , 1] 3 t r+ "I(t) E 3 is a curve and, when
�f

:::; t :::; to < 1, each "I(t) is a regular point but "I (to) is a branch point of order k, according to which of the k different choices of representation defines "I (to), the curve "I is One of k curves "11 , . . . ,"Ik , say "Ii such that if 0 :::; t < to if to :::; t :::; t 1 :::; 1 . The phenomenon just described is the genesis of the term branch point and the particular "Ij is a branch. That "Ij may lead to another branch point for some t l in (to, 1). One of the branches corresponding to a t l in (to, 1) can be one of the "Ii discarded at to · 10.2.36 Exercise. In the context above, if 5 is the set of t such that "I(t) is an irregular point, #(5) E N+ . [Hint: The argument in the PROOF of 10.2.33 applies.] o
The further study of analytic structures is facilitated by the next dis cussion of related topological questions. 10.2.37 DEFINITION. A TOPOLOGICAL SPACE X IS: a) IFF ANY TWO POINTS OF X ARE THE ENDPOINTS "1 0 AND "1 1 OF A CURVE r+ X; b) IFF FOR EACH x IN X, EACH NEIGHBORHOOD N(x) CONTAINS A NEIGHBORHOOD V(x) SUCH THAT ANY TWO POINTS IN V(x) ARE END POINTS OF A CURVE SUCH THAT "1 * C N(x) ; c) IFF X IS CURVECONNECTED AND EACH LOOP
curveconnected () () locally curveconnected
simply connected
"I : [0, 1]
"I
"I : [0, 1] r+ X
Chapter 10. Riemann Surfaces
402
d)
IS LOOP HOMOTOPIC IN X TO A CONSTANT MAP;
locally simply connected IFF X IS LOCALLY CURVECONNECTED AND FOR EACH x IN X, EACH NEIGHBORHOOD N(x) CONTAINS A NEIGH BORHOOD V(X) SUCH THAT EACH LOOP "( FOR WHICH "((0) x ( THE LOOP STARTS AT x), "((I) x ( THE LOOP ENDS AT x), AND "( * C V(X) ( THE LOOP IS CONTAINED IN V(X)) IS LOOP HOMOTOPIC ( VIA SOME CONTINUOUS MAP F OF [0, 1] 2 ) IN N(x) TO THE CONSTANT MAP 15 : [0, 1] 3 t r+ r5(t) x. ( IN THE NOTATION OF 1.4.1, =
=
=
"( "' F,N(x)
15.)
curvecomponent.
MAXIMAL CURVECONNECTED SUBSET OF X IS A 10.2.38 Example. The punctured open disc 10 is curveconnected, locally curveconnected, locally simply connected, but simply connected. The union 1Ul:JD(2, is locally curveconnected, simply connected, locally simply connected, and connected. The A
•
•
•
not 1r not topologist's sine curve { (x, y) : y sin (�) , ° < x � 1 } l:J { (0, y) :  1 � y � 1 } ' in the topology inherited from ffi.2 is connected, not curveconnected, and not locally connected ( no point on { (O, y) :  1 � y � I } lies in =
a connected neighborhood ) .
n chart at
10.2.39 DEFINITION. A CONNECTED HAUSDORFF SPACE X IS AN IFF: FOR EACH POINT IN X THERE IS A PAIR (A CONSISTING OF A NEIGHBORHOOD OF AND, FOR SOME NONEMPTY OPEN SUBSET U OF THERE IS A HOMEOMORPHISM
dimensional cumplex manifold x x) •
•
{N(x), ¢ } N(x) x en , ¢ : N(x) 3 x r+ U. FOR CHARTS {N(x), ¢} AND {N(y), 1jJ } SUCH THAT N(x) n N(y) j. (/) THERE IS THE transition map t<jJ¢ �f 1jJ ¢l : ¢[N(x)] n 1jJ (N(y) r+ ¢[N(x)] n 1jJ (N(y). WHEN EACH t<jJ¢ IS IN H (W) , i.e., WHEN a �f (a l , . . . , a n ) E W, AND THERE ARE n POWER SERIES c( k ) Tn l .... ,Tn n (Zl  a l ) Tn l . . ( n  an ) Tn n 1 _< k _< n 0
, z
Section 10.2. Manifolds and Riemann Surfaces
403
t7jJ¢ ON EVERY COMPACT SUBSET OF (a , r t CONTAINED IN W, X IS AN n dimensional complex analytic manii fiold. THE SET A � {{N(x), ¢} L E x IS AN atlas. Two charts {N1 (x), ¢d and {N2 (x) ' ¢2 } are holomorphically compat ible iff the map ¢2 ¢� 1 : ¢ 1 [N1 (x) n N2 (X)] r+ ¢2 [N1 (x) n N2 (X)] is bi holomorphic. CONVERGING UNIFORMLY TO A NONEMPTY POLYDISC X�= l D
0
Two atlases � and � for a complex analytic manifold X are ana lytically equivalent 1(�1 rv �22 ) iff for each x in X, every chart {N1 (x), ¢ 1 } in � 1 is holomorphically compatible with every chart {N2 (x), ¢2 } in �2 . An atlas � is complete iff any chart holomorphically compatible with some chart in � is in �: � is saturated. [ 10.2.40 Note. The assumption that X is a Hausdorff space is not redundant [B e]. For k in N+ U {oo}, there are Ck compatible charts, and C k _ equivalent atlases. c ) The map t7jJ ¢ may be regarded as a ffi.2 n_ valued function on ffi.2 n . clef . If Zj = X2j  1 + �X j and t<jJ¢ clef= (U 1 , U2 , · · · , U2n 1 , U2n ) the 2 sign of the determinant of the Jacobian matrix J [ 8(X 8 (U 1 , . . . , U2n) ] det ( J) �f det 1 , . . . , X2n ) may vary from point to point of X. When det ( J) 0 everywhere , X is oriented by the atlas and X is orientable. When there is no atlas that orients X, X is nonorientable. The Riemann sphere L 2 is an orientable ( Idimensional ) complex manifold. If 0 < a < b < 1, f z a } , V �f L2 n { (x, y, z) z < b } , u � L 2 n { (x, y, z) then U U V = L 2 . Under the convention � = 0, 1 8 P P U 3 r+ 8 ( P ) and (3 : V 3 r+ ( P ) provide charts {U, a} and {V, (3} that constitute an atlas for L 2 . The Mobius strip M S is the set S �f [0, 1] (0, 1) reduced modulo the equivalence relation rv defined by if a j. O (a, b) rv (c, d) {:} { {a{a == O}c} /\/\ {b{c == dI}} /\ {b = 1  d} otherwise · >
:
>
00
a :
x
404
Chapter 10. Riemann Surfaces
More intuitively, the Mobius strip is S with O x (0, 1) and 1 x (0, 1) identified according to the rule (0, == (1, 1 For the maps
x)  x). ¢ : [0 , 0.6] x (0, 1 ) 3 (x, y) r+ (x, y), 'ljJ : [0 . 4, 1] x (0, 1 ) 3 (x, y) r+ (1  x, 1  y), an atlas A for M S is the pair of charts {[O , 0.6] x ( 0, 1 ) , ¢} and {[0. 4, 1] x (0, 1 ) , 'ljJ}. The map t<jJ viewed a map of as
W �f (0.4, 0.6) x ( 0, 1 )
+ + i(l into itself takes the form = The deter minant of the corresponding Jacobian matrix is  1: M S is not oriented by
t<jJ(x iy) x
 y).
A.]
A2
10.2.41 Exercise. a) If Al and are equivalent atlases for a manifold, the determinants of the corresponding Jacobian matrices are of the same sign. ( Hence the Mobius strip is not orient able. b) If A l and are equiv alent atlases for an oriented manifold, are associated and charts such that the determinant of the Jacobian matrix for n j. 0, . cle . l t he transItIOn map <1>, <1>2 =f ' 0f constant sIgn . IS
{ Nl , ¢1 }
)
A2
{N2 , ¢2 } N N l 2 . t '/' 1 0 ¢2 [Hint: b) For charts { N1 , ¢ 1 } resp. { N2 , ¢2 } in Al resp. A2 such that Nl n Nl n N2 n N2 j. 0 , a calculation of the determinant of  the Jacobian matrix for ¢ 1 0 ¢:; 1 0 ¢ 1 0 ¢2  1 applies.] The set of charts endows X with a topology and with the capacity to support a notion of holomorphy for certain maps defined on X: X is an (ndimensional) complex analytic manifold. When X and Y are ndimensional complex analytic manifolds, a func tion F in Y is holomorphic near x iff for some chart { N (x), ¢} and some chart { N[F(x)], 'ljJ}, 'ljJ o F o ¢  1 is holomorphic on ¢[N(x)]. If F is holo morphic near each x, F is analytic. For the special case Y = Coo , F is X holomorph'ic iff F is analytic and F(X) c C; F is X meromorphic iff F is analytic and F(X) c Coo . When E CX , is harmonic, subharmonic, superharmonic resp. har monic, subharmonic, superharmonic near x iff for each chart resp. some chart { N ( x ), ¢ } , 11 0 ¢ 1 E Ha{¢[N(x)]}, 11 0 ¢ 1 E SH{¢[N(x)]},  11 0 ¢ 1 E SH{¢[N(x)]}. /"

x
1l
11
The use of charts permits many discussions of complex analytic man ifolds and functions on them to be carried out locally. In particular, the
Section 10.2. Manifolds and Riemann Surfaces
405
immediate analytic continuation
analytic continuation
notions and are for mulable for appropriate function elements defined for ndimensional com plex analytic manifolds. Remark. A analytic structure AS (determined by some function element (I, Q)) is a special kind of When E AS, is a neighbor hood of r+ AS is the associated map, and D(O, is a chart.
[ 10.2.42 lytic manifold. 3(P, Q), {N[3 (P, Q)],  I } L
:
3(P, Q) rr
L
N[3 (P, Q)]
complex ana
Although the study of complex analytic manifolds is a large and important field of mathematics, only the definition of a complex analytic manifold is offered here. Useful references are [BiC, Nar, SiT,
St].]
10.2.43
Riemann surface
DEFINITION. A IS A IDIMENSIONAL COMPLEX ANALYTIC MANIFOLD. In what follows, most of the discussion will deal with Riemann surfaces. The Wstructures W and analytic structures AS have served the purpose of motivating the treatment. Exercise. If R is a Riemann surface determined by an atlas A, there is a holomorphically compatible atlas Al such that each chart o. (The atlas of Al is i.e., 1U and Al is a canonical atlas.) applies.] Riemann's Mapping Theorem
10.2.44 {N(x), ¢} canonical, ¢[N(x)] ¢(x) 8.1.1 [Hint: 10.2.45 Exercise. a) If W is a Wstructure and [J, V] E T, then {[J, V],p} =
=
is a chart. With respect to the set of all such charts, the Wstructure determined by the function element (I, Q) is a Riemann surface. Exercise. A Riemann surface is orientable. A transition function is a (vector) function
10.2.46 [Hint:
t'j;.p
(lie t2 The Jacobian matrix .J for t<jJ.p is the
( Ux ) uy
vx
Vy
.
(�)
2 2 matrix x
m
8.1.68.1.7.] 10.2.47 Exercise. For each convergent series L cn (z  a) n and its circle 'n=O of convergence D( a, r r, the formul32 f(z) �f L cn (z  a) n , P(t) �f + t, and Q(t) �f L cn tn nO 71=0 The CauchyRiemann equations apply, v.
=
=
a
00
=
406
Chapter 10. Riemann
Surfaces
[I, a
establish a correspondence between the germ ] and the equivalence class 3(P, Q). Then I : W 3 ] r+ 3(P, Q) E 3 is an injective and holomorphic map of the Riemann surface W into the Riemann surface AS, the Riemann surface determined by 3(P, Q) (and contained in 3). 10.2.48 Example. The open unit disc 1U is a Riemann surface. If E H (1U) and 8(1U) is a then 1U is the analytic structure de termined by the function element (I, 1U) . Hence , if Q) is an immediate analytic continuation of (I, 1U) , then Q c 1U. Thus, if E Q, any neighbor hood of the germ ] is a subset of the neighborhood 1U] . 10.2.49 Example. The complex plane C is a Riemann surface. If is entire, C is the analytic structure determined by the function element q . One atlas consists of the single chart {C, id }. 10.2.50 Example. The set Coo viewed as the onepoint compactification of C, is a Riemann surface. An atlas for Coo is given in 10.2.40. The function element (id , Coo ) determines Coo . The function id is holomorphic in the sense that it is analytic on Coo and id (Coo ) = Coo . Liouville's Theorem (5.3.29) implies that H (Coo ) consists entirely of constant functions. In 10.3.21 and 10.3.23 there are precise formulations of the U niformization Theorem, a significant generalization for simply connected Riemann surfaces of 8.1.1. Two of the closely related consequences, 10.3.24 and 10.3.26 are discussed as well. In Sections 10.5 and 10.6 there are sketches of the proofs of the U niformization Theorem.
[I, a
natural boundary for f, [g, a
f
(g , a
[I,
f (f,
10.2.51 Exercise. The Riemann surface R corresponding to 10.1.6 is homeomorphic to C. If Sn �f { z : z = < < 00, mr < e � mr and S �f { awl + then R is homeomorphic : (a, E to: a) U Sn resp. b) S.]
[Hint:
(3W2
reiO , 0 r (3) [0, 1) 2 },
}
nEZ
10.2.52 Exercise. A Riemann surface is locally compact and curve connected, locally curveconnected, and locally simply connected. For each chart {N ( x ) , ¢ }, ¢ is a homeomorphism.]
[Hint:
10.2.53 DEFINITION. FOR AN ANALYTIC STRUCTURE AS DETERMINED BY A FUNCTION ELEMENT (I, Q) , AN OPEN SUBSET OF U OF AS , SUCH THAT P cle=f P u IS INJECTIVE IS A
patch.
I
10.2.54 Exercise. For a region 11 of AS, a function F in CQ is
phic on 11, i.e., F E H ( 11 ) , iff for each patch U contained in 11, F (p l ) E H [P(U)]. 0
holomor
Section 10.2. Manifolds and Riemann Surfaces
407
10.2.55 THEOREM. IF AS IS DETERMINED BY THE FUNCTION ELEMENT (I, Q) THE CORRESPONDING FUNCTION f IS HOLOMORPHIC ON AS.
, a]
=
PROOF. If U is a patch, some [ I is in U and p is injective on U. Fur thermore p(U) �f V is an open subset of and f p  l I l v . D There is a parallel between the study of functions 1 holomorphic in a region Q of and functions F AS r+ holomorphic in a region 11 on the analytic structure AS associated to I . A local uniformizer t maps a neighbor hood N ( ) into 10.2.56 Exercise. A function F : AS r+ is holomorphic in a region 11 iff for each 30 in 11 there is a local uniformizer t such that in some neighborhood of 30 , F is representable by a convergent power series:
C
C
:
0
C
C
C.
a
00
C,
n=O
z Iz C  z g z 2 2 z C g(z).
[ 10. 2.57 Remark. If Q c a function I : Q 3 r+ ( ) E of one complex variable gives rise to an equation w I( ) = 0 in volving two complex variables. The function 1 in 10.1.6 satisfies e f( z ) 0 and the function in 10. 1 . 12 satisfies ( ) o. Each equation is an instance, for some subset £ of and a function F £ 3 (w, ) r+ F( w , ) E of an equation of the form F w ) = 0 in which w is replaced by resp. Just as
z
g
=
:
z
=
z C,
I(z) { (x, y) : ( x, y) E ffi? , x (1 y2 ) ! } � { (x, y) : (x , y) E ffi? , x2 + y 2  1 0 } , the inclusion { ( w, z ) : w  I (z) o } � { ( w, z ) : F ( w, z ) = o } ( ,z
_
=
=
=
when w is replaced by 1 can obtain. Both sides of the latter inclusion describe parts of a CAY, hence of a W, hence of an R. In the cited illustrations, if ( b) E £, F is representable at and
a, near ( a, b) by a power series L 00
'TTl,n=O
CTnn (w
a,
the series converges absolutely near ( b) , 00
rTL , n=O 00
clef ", Cr' (w ) ( z  b) n . =
� n=O
 a) Tn m(z  b) n . Since
408
00
Chapter 10. Riemann Surfaces
( a) j. 0, there is a power series L dk (w  a) k such that near k=O a, F (w, � dk (w  a) k ) 0 (v. 5.3.43). If F(w, z) �f w  z2 and a b 1, then C (1) 2 and when w is near 1 there are two power series If C
1
=
=
=
1
=
(
Z = ±2 l + �2 (W  l) + m 21( � ) (W  l) ' +
) �f ±P(I, w)
F[w , ±P(I, w)] 21Tiot . The continuing (P, D(I, lr) along [0, 1] e results in ( P, D(I, lr). The same (P, D(I, In and (P, D(I, lr). F(w, z) �f w 2  z2 (cf. 10.1 .33), then C (1) 2 and z = ±[(w  1) + 1] [�f ±P(I, w) ±w]. In this instance, both P and P are entire and analytic con tinuation along "I : [0, 1] 3 t e 2 1T it of either of (P, D(I, lr) or ( P, D(I, lt) does not result in the other bnt ouly in the func tion element with which the continuation was begun. The CAY associated with (P, D(I, lr) differs from that associated with (P, D(I, ln· When F(w, z) L cmn (w  a) m (z  bt , a series converging rn.n=O at and near (a, b), as a subset of ([2 , { (w, z) : F(w, z) O } may be regarded as the graph Qp of the equation F(w, z) = o. The topology of ([2 induces a topology on Qp . When (I, Q) is a func tion element such that for w in Q, F[w, f(w)] 0, for each w in Q, [w , f(w)] E Qp . The CAY associated with (I, Q) corresponds such that the curve "I : 3 t r+ CAr is engendered by O n the other hand, if 1
=
=
=
r+
00
=
=
=
to a ( possibly proper ) subset of Q F . Consequently a CAY, a W, and more generally an R may be viewed as a ( possibly proper ) subset of some Qp . Owing to the method of topologizing a CAY, a W, or an R, these subsets, like Qp itself, are for the most part locally conforrnally equivalent to 1U. However, as the consideration of the case reveals, Q F and one of the corre sponding CAY, W) or R are not necessarily homeomorphic. The global topological character of CAY and W is determined by the process of analytic continuation whereby any two neighborhoods may be connected. When analytic continuation is attempted at
F( w, z) �f w2  z2
Section 10.3. Covering Spaces and Lifts
409
some neighborhood of OF , not every other neighborhood of OF is necessarily accessible. For example, the graph OF of is a connected = set, namely the union of two complex intersecting at Nevertheless their equations, = engender two (dif ferent!) complete analytic functions CAF± that lie in disjoint components of the sheaf S, v. 10.2.8.]
F(w, z) w2  z2 stra'ight lines w ±z,
(0, 0).
1 0 . 3 . Covering Spaces and Lifts
p)
The triple (W, Q, consisting of a Wstructure W and its associated region Q resp. local homeomorphism W r+ Q is an example of a conforming to the general pattern (X, Y, p) described below. The contents of 10.2.18  10.2.25, treating the topological properties of a Riemann surface, are central to the discussion. To simplify the presentation, unless the contrary is stated, each topological space introduced below is assumed to be a curveconnected, locally connected ( whence locally curveconnected ) , and locally simply connected Hausdorff space.
p
triple
covering space
:
10.3.1 DEFINITION . A covering space tr'iple (X, Y, p) CONSISTS OF TOPO LOGICAL SPACES X AND Y AND A MAP p : X r+ Y SUCH THAT: a) p IS A CONTINUOUS SURJECTION ; b) FOR EACH IN Y AND SOME OPEN NEIGHBORHOOD N(y), p l [N ( y )] IS THE UNION OF PAIRWISE DISJOINT XOPEN SUBSETS, EACH HOME OMORPHIC TO N(y) (N(y) IS evenly covered). FOR A CURVE [0 , 1] r+ Y AND A POINT A IN p l b(O)] , A lift of through A IS A CURVE ;:y : [0 , 1 ] r+ X SUCH THAT ;:y(0) A AND p o ;:Y [ 10.3.2 Remark. Conventionally, X is a covering space of Y. The DEFINITION above is most convenient for the purposes below. Y
=
"I :
= "I .
"I
Alternative definitions, some more and some less restrictive, can be found in the extensive literature of topology.]
10.3.3 Example. a) As remarked above, a Wstructure W is a covering space of the underlying region Q. b) For any topological space X , (X, X , id ) is a covering space triple. c) For X
(_ �f (0, 1) 2 , Y d_ �f ( 0, 1), and p : X 3
(
a, b) r+ a E Y,
Chapter 10. Riemann Surfaces
410 P
1
which is not homeomorphic to a union of pairwise dis = x joint Xopen subsets homeomorphic to ( X, is a covering space triple. 10.3.4 Exercise. If X is a covering space of and is a covering space of Z, then X is a covering space of Z. 10.3.5 Exercise. If X is a covering space of and for some is homeomorphic to each component of
( a) a (0, 1),
Y, p) not Y Y Y y E Y, N(y), p  l [N(y)] N(y). 10.3.6 THEOREM. FOR A COVERING SPACE TRIPLE (X, Y, p) AND A CURVE "I : [0, 1] Y SUCH THAT "1 (0) �f a, IF A E pl (a), THERE IS A UNIQUE LIFT ;:y : [0, 1] X "I THROUGH A. 10.3.7 Note. The result 10.3.6 is central to much of what [ ft
ft
a:
of
follows, in particular to 10.3.12.]
of ;:Y. N(A) p I N(A) p[N(A)] �f N(a) t l (0, 1] s tl , "Is(t) �f "I(st), N(a) N(A) N(a) "I(s) E N(a). p s t "Is A. ;:ys �f "Is s ;:Ys "Is A, f � 0, [0, t Il [0, 1, N b,,(l)] N p, N [;:y,, (1)]. ( 1], {a < s < (} hs(l) E N b,,(l)]} , and ¢" "Ie, is a lift of "Ie, through A. Thus a i sup S, a contradiction: a = l. Uniqueness of ;:Y. If i] is a second lift of "I through A and
PROOF. Existence Some neighborhood is open and curve is a homeomorphism. Hence connected, and is open and curveconnected. Consequently, for some in if O :::; < then Thus, if ¢o is the ft local inverse of and < then ¢o is a lift of through If S is the (nonempty) set of for which there is a lift of through S then a > and S = a] . C S, is connected, sup S If a < some curveconnected neighborhood is homeomor phic, say via ¢,,' an b,,(l)]local inverse of to a curveconnected neigh borhood For some in (a, 1,
0
:
'*
0
E T i 0, and continuity considerations imply T is closed. If t E T, N [i](t)] (= N [;:Y (t)]) is phomeomorphic to a neighborhood N [1] (t)] (= N {p[i](t)]}) . Some neighborhood N(t) is mapped by "I into N[1J (t)]. If r E N(t), then p [i](r)] = p [;:Y (r)]. Since p I NF(t)] is a homeomorphism, it follows that r E T: T is open. Because [0, 1] is connected, T = [0, 1]. D
then ° some
Section 10.3. Covering Spaces and Lifts
411
In the following paragraphs there is described the construction of the
universal covering space X for a topological space Y. For a fixed point ao in Y and a (variable ) point a in Y, the set Sa of all curves "I : [0, 1] r+ Y for which "1(0) ao and "1(1) a is decomposable =
=
into equivalence classes {Sa ,OI } OlEA for the following equivalence relation. Curves "I and 15 are equivalent ("I ev a 15) iff: "1(0) = 15(0) = (both curves start at "1(1) = 15(1) = (both curves end at ) for some continuous F [0, 1] 2 3 { x , t } r+ Y,
ao a
•
•
•
ao); a;
:
F(x, 1) = F(x, O) = F(O, t) = "I(t), F(l, t) = r5(t), t E [0, 1] ,
ao,
a,
("I and 15 are homotopic ) .
clef
There emerges the set X = {Sa p } OlEA ' An element A of X is determined by: a point of Y; a curve "I such that "1(0) = and "1(1) = equivalence class of "I is A.)
a
•
•
ao
a
a. ( The homotopy
If A E X, there is in Y a ( unique) such that some homotopy equiv alence class of curves starting at and ending at engenders A. Hence there is defined a map p X 3 A r+ E Y. The set X is given the topology T for which a neighborhood base is the totality of all neighborhoods as described next. For a point A Sa ,OI in X, a "l in A, and a neighborhood N( ) , a neighborhood N( A ) consists of the union, taken over all b in N( ) of the set of all homotopy equivalence classes of curves that are products "115 of "I and curves 15 such that 15(0) = 15(1) = b, and 15* c ) In brief, each point of Y gives rise to a set of homotopy equivalence classes. For each point b in some neighborhood N ( ) of there is a set Sb of homotopy equivalence classes, each consisting of all pairwise homotopic curves for which the curveimages start at pass through thereafter remain in N( ) and end at b. The neighborhood of A is
ao a
:
a
�f
a
a,
N(a .
a,
a
a a,
ao ,
a,
N(A) �f
U Sb . b EN( a )
Thus a neighborhood N(A ) of an A in X is determined by: •
a curve "I such that "1(0) =
=
ao and "1(1) a �f p(A);
a,
Chapter 10. Riemann
412 •
Surfaces
a neighborhood N(a) . The set X as described above is the for the space Y. 10.3.8 Example. If Y = 'lI' , then Y is not simply connected. The infi nite helix X � { (x, y, z ) : x = cos 27rt, Y = sin 27rt, z = t,  00 < t < 00 } is the universal covering space of 'lI'. For (X, 'lI',
universal cov ering space
p
:
X 3 (cos 27rt, sin 27rt, t) r+ cos 27rt + sin 27rt E 'lI',
i
p) is the covering space triple.
10.3.9 THEOREM. a) THE UNIVERSAL COVERING SPACE X IS A HAUS DORFF SPACE; b) (X, Y, IS A COVERING SPACE TRIPLE; c) X IS CURVE CONNECTED , LOCALLY CONNECTED (HENCE ALSO LOCALLY CURVECON NECTED ) , AND LOCALLY SIMPLY CONNECTED . PROOF. z, there a) If W and Z are two points in X and w exist disjoint neighborhoods N ( w) and N ( z ) . Hence the corresponding neighborhoods N(W) and N(Z) are disjoint. = there are curves 'Yw and 'YZ On the other hand, if w = connecting and w and such that:
p)
�f p(W) I p(Z) �f
def p(W) p(Z),
ao
'YW : [0, 1] r+ Y, 'Yz : [0, 1] r+ Y, 'Yw (O) = 'Yz (O) = 'Yw (l) = 'Yz (l) = 11) , 'YW is not homotopic to 'YZ .
ao,
Moreover, w lies in an open, curveconnected, and simply connected neigh borhood N(w) . The pairs bw , N(w)] and b z , N(w)] determine neighbor hoods V(W) and V(Z) of W and Z. If E E V(W) n V(Z), there are curves I' : [0, 1] r+ Y resp. 15 : [0, 1] r+ N(w) for which =
ao, 1'( 1) w resp. 15(0) = w, 15(1) = p (E) and such that 'YE �f 1' 15 determines E. Since E E V(W) n V(Z), the ho 1'(0)
=
motopy equivalence classes bE } , bwr5 } , and bzr5 } are the same, i.e., br5 } = bwr5}
=
bzr5 } , bw }
=
bz } ,
a contradiction, since 'YW and 'Y Z are not homotopic. The remaining axioms for a topological space are directly verifiable, particularly in light of 1. 7.2. b) If A E X and (A) = any neighborhood N(A) is mapped by onto some neighborhood of is open.
p
a, a: p
p
Section 10.3. Covering Spaces and Lifts
413
If A E X, p(A) = a, and N(a) is a neighborhood in X, N(a) may be assumed to be simply connected. Then N (a) and some curve "( such that "((0) = ao and "((I) = a determines a neighborhood N(A) and, by definition, p[N(A)] = N(a) : p is continuous. If A E X and p(A) = a, because Y is locally simply connected, for some curveconnected open neighborhood N(a) , any loop containing a and contained in N(a) is null homotopic in Y. The neighborhood N(a) and some curve "( such that ,,((0) = ao and "((I) = a determines a neighborhood N(A). If W and Z in N(A) are determined by curves "(8 and "(( and if p(W) = p(Z) b [E N(a)] , then "(8 and "(( are homotopic, and thus W Z: p is injective. As an injective, continuous, and open map, p I N (A) is a homeomor phism. If A I W E p l (a) , N(W) is the associated N(a)induced neighbor
�f
=
�f
hood of W, and K E N(W) n N(A) , then p(K) C E N(a). If the curves that determine K, W, A are resp. "((, "(1] , ,,(e, then "(( and "(1] are homotopic, "(( and "( e are homotopic, whence "(1] and "( e are homotopic. Thus A = W, a contradiction. Thus different N (a )induced neighborhoods are disjoint: p [N (a)] is the union of pairwise disjoint homeomorphic copies of N ( a). c) If 1]s : [0, 1] 3 t ft Y is a set of curves, each starting at ao and de pending continuously on the pair (s, t), for each s, 1]s determines a point As in X: p ( As ) = 1]s ( l ) . If N (As ) is a neighborhood of As, there is some curveconnected open neighborhood N [1]8 (1)] to which is associated a neighborhood U ( As ) contained in N ( As ) . The map p I N (As) as IS a homeomorphism. If I hl is small, 1]s + h ( l) E N [1J s ( l ) ] , whence
1
�f
depends continuously on s. In particular, if "( determines some A in X, i.e., if ,,((0) ao, "((I) = a, then "(s : [0, 1] 3 t ft ,,((st) E Y depends continuously on the pair (s, t) and thus As depends continuously on s. The curve ;:Y : [0, 1] 3 s ft As E X con nects A to the point Ao corresponding to the loop : [0, 1] ft ao that starts and remains at ao : X is curveconnected. Since p is a local homeomorphism, the local properties of Y are local properties of X: X is locally connected and locally simply connected. D 10.3.10 Exercise. The universal covering space X, modulo homeomor phisms, does not depend on the choice of ao . [ 10.3.11 Note. For a given curveconnected space Y and its uni versal covering space X, there is an action, described next, of the group 7r l (Y) (v. 5.5.4) on the universal covering space X. Al though X does not depend on the choice of ao and 7r l (Y, Yo ) does As
=
f£
Chapter 10. Riemann
414
Surfaces
not depend on the choice of Yo , the discussion below is phrased in terms of some and a particular Yo.
ao
For ao fixed in Y and A in X, there is a curve "I starting at ao , ending at "1(1) and engendering the homotopy equivalence class that is the element A of X. If p E 7r dY, "1(1)], then p is represented by a loop 1] starting and ending at "1(1). The product 1]"1 is a curve starting at ao and ending at "1(1). The homotopy equivalence class A of 1]"1 is the result of the action of p on A: A = p . A. Since the constructs employed are independent of the choices ao and "1(1), P . A is welldefined in terms of p in 7rl (Y) and A in X.
�f a,
fixed in X , the set O(A) �f { p . A : p E 7r l (Y) } is the orbit of A. The projection p A r+ a carries each point of the orbit of A into a. The set of orbits and Y are in bijective
For A 7rl (Y)
0
:
correspondence via p. The customary notation for this situation is Y X/7rl (Y).] rv
Paraphrased, the next result asserts that lifts of homotopic curves are homotopic and that their starting and ending points are the same. It is not only intuitively appealing but is the basis for a number of important conclusions, e.g., 10.3. 14 10.3.17. 10.3.12 LEMMA. IF a) (X, Y, p) IS A COVERING SPACE TRIPLE; b) Ao E X, p (Ao )
t
(s, t)
=
ao
;
c) F : [0, 1] 2 3 r+ "Is ( ) E Y IS CONTINUOUS ON [0, 1] 2 , i.e., GENER ATES A HOMOTOPY BETWEEN "10 AND "1 1 ; AND d) "Is(O) == "18 (1) THE LIFTS ;;;0 AND ;;;1 THROUGH Ao ARE HOMOTOPIC, AND END AT THE SAME POINT: ;;;0 (1) = ;;;1 (1) B.
ao ,
�f
==
b;
PROOF. Each "Is has a unique lift ;;;s through Ao. Naturally associated to F is F [0, 1] 2 3 r+ ;;;s ( ) E X. Because X and Y are locally homeomor phic, the technique used in the PROOF of 10.3.6 applies: F is continuous and the curves ;;;0 and ;;;1 are homotopic. By definition, the connected sets ;;;o ([0, 1] ) resp. ;;;1 ([0, 1] ) are contained in the sets p l ( o) resp. p l (b) . Thus ;;;o ([0, 1] ) = Ao resp. ;;;1 ([0, 1]) is a point: ;;;0 (0) = ;;;1 (0) = Ao and ;;;0 (1) = ;;;1 (1) = B. D :
(s, t)
t
totally disconnected
a
10.3.13 THEOREM. EACH UNIVERSAL COVERING SPACE IS SIMPLY CON NECTED (cf. 10.2.37c) ) .
Section 10.3. Covering Spaces and Lifts
415
PROOF. For a point p in Y there is the homotopy equivalence class P (in X) of loops beginning and ending at p and loop homotopic to the constant curve l1 : E X is a t r+ l1(t) == p. If ;::; : r+ loop beginning and ending at P, for each ( ) E Y and = p(P) = p. For each in the curve ( ) = po
[0 , 1] 3
[0, 1] 3 7 ;::; ( 7) 7, p o ;::; (7) �f 1' 7 [0, 1], ;::;s : [0, 1] 3 7 r+ ;::; ( S7) E X
;::; (0)
1' 0
s
is a lift through Pby virtue of 10.3.6, the unique lift through Pof the curve = p. If ;::; ( ) then r+ E Y, and i.e., is the homotopy equivalence class of q = p o ;::; ( )p the curve in particular, P, which is the homotopy equivalence class of ;::; = l1. r5 p l1 is also the homotopy equivalence class of ( = D The next results show in what sense the universal covering space X of a space Y is universal.
I's : [0, 1] 3 7 I'(S7) (Qs), Qs I's(l) �f s I's, �f
I's(O)
s
�f Qs,
s
1'1 1'):
0
10.3.14 THEOREM. IF Y IS SIMPLY CONNECTED AND (Z, Y, ) IS A COV ERING SPACE TRIPLE, Z AND Y ARE HOMEOMORPHIC. PROOF. By definition, is continuous, open, and surjective. The next lines show that is also injective (whence a homeomorphism). If {p, q} c Z, there is a curve r : r+ Z such that = p and r(l) = q . If (p ) = ( q) y, then r l' is a loop, and, since Y is simply connected, l' is null homotopic. The (unique) lift of l' through p is r, and the unique lift through p of the constant map r5 r+ (p ) is l1 r+ p. Thus 10.3.12 implies p = q. D a
a
a
a
a
�f
[0, 1] �f
r(O)
a 0
: [0 , 1]
: [0 , 1]
a
10.3.15 THEOREM. IF X IS THE UNIVERSAL COVERING SPACE OF Y AND (Z, Y, ) IS A COVERING SPACE TRIPLE SUCH THAT Z IS SIMPLY CON NECTED, Z AND X ARE HOMEOMORPHIC. PROOF. Each homotopy equivalence class in Z is mapped by into a ho motopy equivalence class in Y and thus to an element of X. Thus, W denoting the universal covering space of Z, there is a map F W r+ X. If: a) is the basis of the construction of W; b) and is the basis of the construction of the universal covering space X, c) 10.3.12 implies, by abuse of language, that each homotopy class in Y lifts to a unique homotopy class (through of Z: there is a map G X r+ W. Direct examination of the maps reveals that F o G and G F are the identity maps. By the same token, both F and G are continuous: W and X are homeomorphic. 10.3.14 implies that W and Z are homeomorphic. D 10.3.16 Exercise. The universal covering space X of a space Y may be characterized as the simply connected covering space of Y. a
Yo
Zo
o·
zo)
0
:
a
(zo) �f Yo; :
Chapter 10. Riemann Surfaces
416
10.3.17 Exercise. For a given space Y, the set C of all covering spaces is a poset with respect to the order: Xl > X2 iff for some P12, (Xl , X2, P12) is a covering space triple. Relative to > , C has a maximal element. This maximal element is the universal covering space of Y. The results 10.3.6, 10.3.12, and 10.3.15 apply when Y is an analytic structure AS (with its attendant set of function elements and region Q) and X is its universal covering space S. More generally, the same results apply when Y is a Riemann surface and X is its universal covering space. In the SAS context, a denoting the projection of S onto AS, there are two covering space triples, (AS , Q , p) and (S, AS, a) .
If Y is a Riemann surface with its attendant atlas A and X is the universal covering space of Y, for X there is an atlas B such that with respect to B : a) X is a Riemann surface; b) the map p X r+ Y is locally biholomorphic. = y , and p is a chart at y, a) If N(x) is a homeomorphism, ( ) , can serve as a chart ( ), 0 at b) If l ( ) , ;3 is a second such chart at then 10.3.18 Exercise.
:
[Hint: x.
p(x) {N(y), ¢} {N x ¢ p} �f {N x a} {N x } x, a ;3 1 E H {a (N(x) n ;3 [Nl (x) J) } .J
I
0
Any two simply connected covering spaces of AS are holomorphically equivalent. An important consequence of the simple connectedness of S is a con siderable generalization of Riemann's Mapping Theorem. The latter may be stated as follows. If Q � e and Q is simply connected, Q is holomorphically equivalent to 1U. From this point forward, the context is a Riemann surface R with its attendant complete atlas � defined via canonical charts n i.e., charts such that ( n ) = 1U. When 0 < < 1 and is a chart, ;; l (D(O, a denoted . By extension, the complement of the confor < R\ mal disc < is 2': < and a punc tured conformal disc is Vn . In the spirit of the same : 0< <1 convention, for u in en and 3 in R, u ( ) denotes u(3) . As clarity requires and circumstances permit, both kinds of notations are used in what follows. 10.3.19 Exercise.
r {z : disc, { z : Iz l
{{Va, ¢ } a EA } ' ¢n V ¢ rt) conformal {Va, ¢a} Izl r } r } { z : I z l r } �f { z : I z l r } {z I z l } �f z
417
Section 10.3. Covering Spaces and Lifts
TYPE
TRUE FALSE FALSE
FALSE TRUE FALSE
hyperbolic elliptic parabolic
Table 10.3.1
10.3.20 D EFINITION. A RIEMANN SURFACE R IS; • •
•
hyperbolic WHEN 5) ; FOR EACH a IN R, THE GREEN ' S FUNCTION AT a, G ; R \ {a} 3 z  In I z  a l EXISTS; elliptic WHEN R IS COMPACT ; parabolic WHEN ,5) 1\ R IS NEITHER HYPERBOLIC NOR ELLIPTIC. IE;
ft
, IE ;
Table 10.3. 1 shows how the adjectives are paired with the paradigms 1U, Coo , and C. More details of the pairings are given in 10.6.18. In the course of the remainder of the Chapter, the distinguishing features noted in 10.3.20 and valid for the simply connected Riemann surfaces 1U, Coo , and C are shown to characterize, modulo biholomorphic equivalence, simply connected Riemann surfaces.
all
10.3.21 THEOREM. IF R IS A SIMPLY CONNECTED RIEMANN SURFACE, DENOTING ONE OF 1U, C, OR COO , THERE IS A BIHOLOMORPHIC MAP H ; R ft M . [ 10.3.22 Note. The result above is variously called the Uni formization Theorem or the (General) Riemann Mapping Theorem. Owing to the fact that no two of 1U, C, and Coo are holomorphically equivalent, the M in 10.3.21 is uniquely determined by the adjectivehyperbolic, parabolic, or elliptic describing R. For any Riemann surface R and its universal covering space R there is the group 9 of (Germ�n; ) biholomorphic automorphisms g ; R + R for which P (g(3t= P (3 ) · A function holomorphic on R can, by a kind of 9averaging such as that used in 8.4.14, be converted to a function that is 9invariant; is defined on R .
M
transformationen f f
cover transformations
�eck

f
418
Chapter 10. Riemann Surfaces
In view of 10.3.21, a function holomorphic on one of the three paradigmsV, C, CcJO is converted to a function on the under lying R. Owing to Liouville's Theorem, there is no nonconstant holomorphic function on Coo . There are however nonconstant meromorphic functions on Coo . Averaging these via 9 provides nonconstant meromorphic functions on R.] Among the important consequences of the Uniformization Theorem is the following, given here without proof. 10.3.23 THEOREM. A NONCOMPACT RIEMANN SURFACE IS THE ANA LYTIC STRUCTURE FOR SOME NONCONSTANT FUNCTION ELEMENT (I, Q). A COMPACT RIEMANN SURFACE IS THE ANALYTIC STRUCTURE FOR A NONCONSTANT MEROMORPHIC FUNCTION. Some regard 10.3.23 as the Uniformization Theorem, since behind it is the following important assertion also given here without proof. 10.3.24 THEOREM. IF R IS A RIEMANN SURFACE, R IS ITS UNIVERSAL COVERlNG� SPACE, AND 9 IS THE GROUP OF COVER TRANSFORMATIONS, THE SET RI 9 OF 90RBITS, ENDOWED WITH THE CANONICAL QUOTI� NT TOPOLOGY, ADMITS A COMPLEX ANALYTIC STRUCTURE WHEREBY RI 9 AND R ARE HOLOMORPHICALLY EQUIVALENT. By definition, every complex analytic manifold R is i.e., at each point x of R there is a countable set {Nn (x) } nEN of neighborhoods of x so that if N(x) is any neighborhood of x, for some no , Nno (x) C N(x) . 10.3.25 Exercise. A Riemann surface is second countable: there is a countable set {Un (x) } nEN such that if U E O(R) , then U = U Un . ( The
first countable, Un C U
second countability of a complex manifold is not implicit in its definition [Be] . ) 10.3.21 applies. ] 10.3.26 Note. The second countability of a Riemann surface R can be derived without using the Uniformization Theorem. In [Nev2] differentials, i.e., Iforms, are used to endow R with a metric ( cf. 10.5.3) and, via Schwarz's and Harnack's Theorem (6.2.24) , to show that R contains a countable dense subset. Yet another approach to the second countability of R is found in 10.6.9.
[Hint: [
alternating process
In [Nev2] ' one of the ingredients in the proof of the Uniformiza tion Theorem is the second countability of R.] Section 10.5 is devoted to a sketch of the principal results leading or related to the Uniformization Theorem 10.3.21.
Section
lOA.
419
Riemann Surfaces and Analysis
1 0 . 4 . Riemann Surfaces and Analysis
Analytic continuation of a function element can lead to the phenomenon exhibited in 10. 1. 10. On the other hand, for and as in 10.1.6, if 1 ::;; k :s: K, are chains, and = = 1 :s: j � � ( then = = and = What lies at the basis of the conclusion above is the simple connect edness of The general phenomenon, is detailed in the next paragraphs. When "/ [0, 1] ft C is a curve, is a chain of function
f
Q
II g l f, ( fJ , Qj) , gk ' n k ) Q 1 Q� l , QJ Qk , fJ gK · Q. monodromy, {( lj, Qj)} 1 < . < J J elements such that U Qj ::) * , (0 ) E Q l , and ( 1 ) E QJ , the function j= 1 element ( h, QI) is an analytic continuation along the curve 10.4.1 Example. For the function f of 10.1.23, if * C 1U, (0 ) = 0, and 0 E Q c 1U, there is an analytic continuation of (I, Q) along On the other hand, if J (t) 2t, t E [0, 1] ' since J* rt. lU, there is no analytic continuation of (I, Q) :
_
,,/
,,/
J_
,,/
"/ .
,,/
"/
T
=
along J.
Q
(I, Q I ) ( a Ql ; Q.
10.4.2 LEMMA. IF: a) IS A FUNCTION ELEMENT; b) W AND HAVE THE MEANINGS ASSOCIATED WITH SUCH A FUNCTION ELEMENT; c) cle E "/ IS A CURVE SUCH THAT ,,/ 0 ) =f AN ANALYTIC CONTINUATION OF ALONG "/ EXISTS IFF "/ * C
(I, Q l ) PROOF. If there is an analytic continuation, e.g., corresponding to the chain {(fJ , Qj)} l «'j«'J ' of (I, Q I ) along then * C Q. Conversely, if * C Q and �f [J, a] , through there is a unique lift ;Y of For each ;Y (t) in ;yo there is a neighborhood N [;Y(t)] on which is a homeomorphism. Because ;Y* is compact, there are ti such that J o �f to < t l < . . . < tJ �f 1 and ;Y* C U N [;Y (tj )] . If j= 1 {N [;Y (tj )] } �f Qj , 1 :s: j :s: J, ( tj) �f Tj, 1 :s: j :s: J, [Jj , Tj ] = ;Y (tj) , then {( lj, Qj )} 1 < . < J is a chain yielding an analytic continuation along D a
"/
p
"/.
p
_
J_
,,/,
"/
a,
p
"/ .
420
Chapter 10. Riemann Surfaces
Qj)} I «J J and { (gk , Qk ) } l < k < K are chains along a curve i.e., "1* C Qj n Qk ) , and II I : n�, = g l l n,nn" =J 1 k= 1 then fJl n ] n n K = g Kl n J n n K• 10.4.3 Exercise.
If { (Ii,
_
_
(U ) (U
"I,
[ 10.4.4 Note. The result in 10.4.3 may be paraphrased as fol lows. There is at most one analytic continuation along a curve. ]
The conclusion in 10.3.12 lies at the heart of the next result. In broad terms it says: All analytic continuations along curves that start at some point a and end at some point b are the same if the curves are pairwise homotopic in a region that admits analytic continuation along all such curves contained in the region. 10.4.5 THEOREM. (Monodromy Theorem) FOR A WSTRUCTURE W AND ITS ASSOCIATED REGION (cf. 10.2.2 ) , IF: a) "Ii : [0, 1] ft i = 1, 2, ARE CURVES, EACH STARTING AT "1 (0) = "12 (0 = a (E AND ENDING b; AT "1 (1) "12 (1) b) FOR SOME CONTINUOUS MAP F : [0, 1] 2 ft "12 AND "12 ARE HOMOTOPIC (VIA F) IN c)
=
1 "11 "' F, n hI
Q
�f
1
) clef
Q)
Q,
Q);
ARE CHAINS ENGENDERING ANALYTIC CONTINUATIONS
J K ALONG "11 resp. "12 AND U Qj U U r2 k C Q ; d) j = 1 k= 1 E (G 1 n r2 1 ) , b E (G J n r2 K ) , g1 1 G1 nnl h 1 I G, nn,; a
=
ANALYTIC CONTINUATIONS OF A FUNCTION ELEMENT ALONG ALL CURVES IN A HOMOTOPY EQUIVALENCE CLASS YIELD THE SAME RESULT.
Q,
Section 10.4. Riemann Surfaces and Analysis
421
1'1
1'2
PROOF. From 10.3. 12 it follows that the unique lifts of and through a] a]) in W are homotopic and end at the same point. One of the lifts ends at b], whence b] b] , from which the desired result follows. D 10.4.6 Note. If R is a Riemann surface and is holomorphic in a neighborhood N (30) Q, the analytic continuation of the func tion element Q) is definable in a manner analogous to that used for defining the analytic continuation of a function element Q). The corresponding analytic structure AS induces a covering space triple (AS, R, p).]
[II , (= [h I ,
[gJ ,
[gJ, = [hK ,
[
�f
(F,
F
(I,
10.4.7 Exercise. a) The analog of the Monodromy Theorem holds in the context described in 10.4.6. b) If the Riemann surface R is simply connected, analytic continuation of a function element is independent of the choice of the underlying curve.
u
F �= u .
10.4.8 THEOREM. IF IS HARMONIC ON A SIMPLY CONNECTED RIEMANN SURFACE THERE IS A HARMONIC CONJUGATE v SUCH THAT + lV IS HOLOMORPHIC ON R. PROOF. If 3 E R and {N(3) , q;} is a canonical chart, � I�,
U cle=f U 0 q;  1 E HaIR (1U) , and there is a v such that 11 + iii E H (1U) . Thus, if v
�f v 0 q;, then
If Q is the set of points \1) t� which f can be continued analytically along a curve starting at 3 , then Q is open and nonempty. Owing to the Monodromy Theorem in the context of R, there is a unique function holo morphic in Q and such that f. The analytic continuation may be carried out by means of a sequence of neighborhoods drawn from canonical charts. For each such neighborhood, the argument of the first paragraph shows that �he real part of the corresponding analytic continuation is throughou�Q, If � E QC , there is some canonical chart {N (�), 'ljJ}, and again
F I N(3 ) =
F
u:
�(F ) = u.
 cle=f u 0 'ljJ  1 E HaIR (1U) .  + iV E H (1U) and if V cl�f V 0 'ljJ, then For some V, H cl�f U U
Chapter 10. Riemann Surfaces
422
HF
In W � Q n N(�) , assumes only imaginary values: 5.3.62 applied to 0 'ljJ l shows that for some constant whence L E Q, i.e., Q is closed (v. also PROOF of 6.2.24b)). Since R is connected, Q R: is holomorphic on R and is a harmonic conjugate of u. D
(H  F) = F
c, H l w= F l w + c,
'S(F)
1 0 . 5 . The Uniformization Theorem
The current Section covers the concepts and arguments leading to the Uniformization Theorem and its associated statements. The Uniformization Theorem deals with simply connected Riemann surfaces and their three paradigms, 1U, Coo , and C. Some of the results established in the course of the derivation of the Uniformization Theo rem apply more generally to arbitrarynot necessarily simply connected Riemann surfaces. In Section 10.6 there are Exercises that lead to the proofs of some of the assertions stated without proof in the outline itself. Since entire books are devoted to the derivation of the Uniformization Theorem, at best, only suggestions of the arguments that constitute its proof are given below. The interested reader is urged to consult [AhS, Be, Jo, Nev2, Re, Spr] for further details. The theme of the development is that for each kindhyperbolic, ellip tic, parabolicof Riemann surface R there is a nonempty set of functions of some special type  subharmonic, harmonic, holomorphic, meromorphic, or admissible. The last two types are described in
FK, Fo,
10.5.1 DEFINITION. A FUNCTION : R r+ C ON A RIEMANN SURFACE R IS IFF FOR EACH CHART {Va , q)a } , 0 q)� l IS MEROMORPHlC, = i.e. , FOR THE CHART AND SOME k IN Z,
F
meromorphic
z
F F(z) L Cn zn . n>k
A FUNCTION R r+ C IS AT A POINT \1) OF R IFF IS BOUNDED OFF EACH NEIGHBORHOOD OF \1) AND HAS A SIMPLE POLE AT \1) .
H
H H 10.5.2 Example. When u in Cn is such that eU ( z ) z, for some (local) determination of In, if z E 10, u(z) = In( z ) . The function u is meromorphic :
admissible
=
and admissible at 0, while l u i is harmonic in 10. The mechanisms and conceptsbarriers, Perron families, Dirichlet re gions, and Green's functionsof Sections 6.3 and 8.5 are transferred (and modified as needed) so that they are applicable in the context of R. The significant results that lead to the Uniformization Theorem are organized in the following manner.
Section 10.5. The Uniformization Theorem A)
423
a) For a canonical chart z and r in (0, 1 ) , there is a barrier for the Riemann surface Rr R \ { z : I z l ::;; r } at every point of the set Cr � { Z : I z l = r }. b) If I E C (Cr, lR) , in HaJR (Rr) there is a func tion UJ such that for each a in Cn lim UJ (z) = I(a) : UJ is a solution
�f
z+a z E 'R r
of Dirichlet's problem for the region Rr and the boundary values I.
�) The Riemann surface R is hyperbolic iff R supports a nonconstant
negative harmonic function u iff Green's function G( · , a) exists for some a iff Green's function G( · , a) exists for each a . C) If R is not hyperbolic (hence is compact or parabolic) , 11.1 E Q c R, and 1 E H(Q \ {11.1 }), there is a unique function u: a) harmonic On R \ {11.1 }; b) bounded off each neighborhood of 11.1 ; c) and such that E) u

'iR (f) E HaJR (Q \ {11.1 }) and u( l1.1 )
= O.
[ 10.5.3 Note. The impact of B) and C) is the following. If R is a Riemann surface and K is a nonempty compact subset of R, there is a function 1 harmonic on R \ K. If K is the (compact) closure of a conformal disc, "( is rectifiable, connecting two points a and b,
�f R \ K, and 1 E Ha (Ro) , Jdj2 + (* df) 2 �f dl' (a, b) 0 and d(a, b) �f inf dl' (a, b) is "( *
1
c Ro
> then a metric on Ro. This metric is an essential tool in the argument mentioned in 10.3.26 about the proof that every Riemann surface is second countable. I'
I'
The second countability of every Riemann surface is a consequence of the Uniformization Theorem, proved by the methods outlined below and which do not establish first that a Riemann surface is second countable, v. 10.6.9.] D) If a and b are two points on R, there is a function 1 meromorphic on R and such that a E P(f) and b E Z (f) . The results B) D) are roughly summarized by: Each Riemann surface R supports a function 1 harmonic at every point save one and that at the exceptional point, say a, 1 has a 1 IS singularity. Furthermore, if R is hyperbolic, 1(3)  In 13  a l 
424
Chapter 10. Riemann Surfaces
(_1_)
is harmonic near a , while if R is not hyperbolic, J(3)  � 3a harmonic near a. E) If R is simply connected and hyperbolic, for each chart z and Green's function G, G(z) + In I z l is harmonic and for some F in H (R) ,
G(z) + In I z l = In I F(z) l .
z Furthermore, for each chart z, the function h : R 3 z r+ IS a F(z) biholomorphic map and h(R) C 1U. (Riemann's Mapping Theorem (8. 1.1) implies that for some conformal map
g o h(R)
=
g,
U)
F) If R is simply connected and elliptic or parabolic, and a E R, the following obtain. Fl) There exist admissible functions at a. F2) If fa resp. fb is admissible at a resp. b, for some Mobius transfor mation T, fa = T fb . F3) An admissible function is injective . If f is admissible, f(R) C Coo and R is biholomorphically equiv alent to Coo resp. C according as R is elliptic resp. parabolic. The conjunction of E) and F) is the essence of .the Uniformization Theorem. In summary, for any simply connected Riemann surface R: R is hyperbolic iff there is a biholomorphic map R r+ 1U; R is elliptic iff there is a biholomorphic map R r+ Coo ; R is parabolic iff there is a biholomorphic map R r+ C.
o
•
•
•
•
1 0 . 6 . Miscellaneous Exercises
10.6.1 Exercise. If: a) {( Ii , ni ) } l
a

I * "/ C U n i . For such a ,,/ there is a chain i=1
a
a (1) { [gj, D (aj,r i n L ::S;j :::::J such
(Analytic continuation can always be achieved by analytic continuation along a curve and via function elements.)
circular
Section 10.6. Miscellaneous Exercises
425
10.6.2 Exercise. The Schwarz Reflection Principle (v. 6.2.17 and 8.2. 10) provides a means for analytic continuation. 10.6.3 Exercise. What is a useful and natural definition of a pole at of a map f : M \ 3 3 r+ N between the complex analytic manifolds M and N? 10.6.4 Exercise. The complement C 3 \ I (W) (cf. 10.2.47) in 3 of I (W) is discrete, i.e., each point 3 of C is in some neighborhood N(3) containing no other points of C: N(3) n C = {3} . 10.6.5 Exercise. If X is curveconnected and h : X r+ Y is a continuous surjection, h(X) is curveconnected. 10.6.6 Exercise. If X is simply connected and h : X r+ Y is a homeo morphism, h(X) is simply connected. 10.6.7 Exercise. If a
a
�f
�f { z �f X �f { z �f X
�f
x
x
} 7r } ,
+ iy
 00 < x <
0, 7r < y < 7r ,
iy
00 < x <
0, 7r < y ::;
+
Y
�f D(O, It \ (1, 0) , Y �f D(O, It \ {O} ,
h
:
C 3 z r+
eZ E C :
a) H � h l x and ii h l x are continuous bijections; b) X and X are simply connected; c) H(X) = Y and ii = d) Y is simply connected; e) is n�t simply connected: the image under a continuous injection of a simply connected set need not be simply connected. 10.6.8 Exercise. Since 10.6.6 implies H in 10.6.7 is not a homeomor not continuous? phism, where is The following items are offered as a guide to the proofs and cOnse quences of the assertions in A)F ) .
Y
(X) Y;
ii  I
For A ) : a) The result 6.3.29, proved in 8.6. 16, applies. b) The methods of Section 6.3 apply. 10.6.9 COROLLARY. A RIEMANN SURFACE R IS SECOND COUNTABLE. PROOF. In the notations of A ) , f may be chosen to be nonconstant. If (T, RT, p ) is the universal covering space triple for RT the function
Chapter 10. Riemann Surfaces
426
is harmonic on T. Hence in H (T) there is a nonconstant F such that �(F) = Uj, v. 10.4.8. Furthermore, if w is a chart for T, then [F(w ) , 1U] is a function element for an analytic structure AS of which T is a covering space: T is a covering space of a region Q in C. Since Q is second countable, 10.2.10 implies T is second countable. Hence Rr and R = Rrl.,J { z : I z l :::; r } are second ���. D For
B):
10.6.10 Exercise. If R is a Riemann surface, G( · , 11.1) is a Green's function at 11.1, and k > 0, then 0 > U  min{k, G( . , I1.1) } , U is subharmonic, and U is not a constant: R is hyperbolic. Conversely, if R is hyperbolic, the following items provide the argu ment for the existence of a Green's function (which the Maximum Principle implies is unique) .
�f
B1) For
a
in R and a chart z at
a,
the set
{ w : 0 :::; w E SH (R \ ) , supp w E K(R ) , w (z) + In Izl E SH(z) } a
B2)
�f
�f
is a nonempty Perron family. If K { z : Izl :::; r }, Q R \ K, u E C (QC ) n HaIR (Q) , u l a(n) = 0 r :::; M
In< 1
clef
0
1,
�f I [Hint: The maximum principle implies In r ] B4) If g �f sup v, Perron's method implies g HaIR(R \ {O}) . Further In r } . more, { 0 < Iz :::; r } { G(z) cle f g(z) In zl 1 In r : G
B3)
Izl=1
:F
B5)
r1
Vr +
:::; VI :::; vrMr.
E
1
::::}
=
+
1
:::;
Mr _
+
IS
harmonic and bounded in A(O; 0, r t . Then 6.2.28 implies there is in HaIR [D(O, rt] a harmonic extensionagain denoted Gof G. G > 0 and if O (z) + In Izl E HaIR [D(O, rt] ' then 0 2': G. Hence the Green's function at is G. a
For C): Sections 5.8 and 5.9 and 6.5.8 provide the basis for the argument.
Section 10.6. Miscellaneous Exercises
427
�f
10.6.11 Exercise. If R is compact and K { z 1 z 1 :::; 1 } is a ( closed ) conformal disc on R, r < 1, f E Ha ( R \ { z : I z l 2': r } ) and I f(z) 1 :::; M, Stokes's Theorem and Green's formula are applicable:
r
fa ( K )
* df
=
r
fa (R\ K )
* df =
r
fR\ K
d * df = o.
(10.6.12)
The items C1)C8) below are devoted to establishing (10.6.12) when R, which is not hyperbolic, is also not compact. C1) For the approximants Wn to Uf in the PROOF of 6.3.15, if {p, q} e N, 8p+ q Wn 8PH Uf then converge uniformly on compact subsets of Q to �x� y �x 8q y [Hint: 5.3.35 and 6.2.25 apply. ] C2) If R is not compact it contains relatively compact Dirichlet subregions Q such that K is a relatively compact in Q. For each such Q there is a function fn in Ha IR ( Q n { z : I z l 2': r }) and such that In ( z ) =
{ t(z)
if I z l = r if z E R \ Q '
The set of all such fn constitutes a Perron family F ( parametrized by the set of relatively compact regions Q containing K). If F = sup fn :F and I z l 2': r, then f(z) = F( z). [Hint: The Maximum Principle and the fact that R is neither compact and nor hyperbolic apply. ] C3) If gn E HaIR ( Q n { z : I z l 2': r }) and
{I
clef
if l z l = r gn (z) = 0 if z E 8 ( Q ) '
clef
the set 9 of all gn is, like F, a Perron family. If G = sup gn and 9 I z l 2': r, then G(z) = 1. C4) In F resp. 9 there are sequences {fn} nEN resp. {gn} nE N associated with relatively compact regions Qn and converging together with their derivatives uniformly on compact sets to F resp. G and their deriva tives, v. C2). C5) gn t:J. fn  fnt:J.gn = 0, n E N , ( v. 5.8. 18).
1
nn \ K
�
C6) If 0 < p < r < and Up E Ha IR [R \ D(a, p t ]' while up I Ca p = 1R(f) ( ) up is the solution of Dirichlet's problem for the boundary values 1R(f) then for k (r ) as in 6.5.7, max D r O up :::; k ( r ) .
R\ (a , )
Chapter 10. Riemann Surfaces
428
1 C7) If 5 > rn ..l 0, for each n there I. S a sequence {Unrr,} m E N such that {Unm } ,n EN ::) {un+ l.m } m EN and for fixed n, Unm converges uniformly on A (0; rn , 1). For every positive r, the diagonal sequence {u mm } m E N converges uniformly on A(O; r, 1) to a function U harmonic in A(O; r, 1); the Maximum Principle implies convergence is uniform on A(O; r, (0 ) : {U mm } m E N converges uniformly on A( 0; 0, (0 ) to a function U harmonic in A(O; O, oo) . , C8) The argument in 6.5.7 leads to I [u � ( f )] (t �il! ) I :::; k(r) 1�t which converges to 0 as t + O.

( )
For D): w
10.6.13 Exercise. For fixed charts z resp. at a resp. b, if R is hyper bolic, Green's functions Ua resp. Ub exist. If R is elliptic or parabolic, there are functions Ua resp. Ub harmonic in R \ {a} resp. R \ {b} and such that are harmonic near a resp. b. resp. Ub(W)  � ua ( z)  � � (ua)x  i ( ua)y meets the require In either event the function f (Ub)x  i (Ub)y ments of D). [Hint: If Ua is a Green's function, for S me h holomorphic near 0, is harmonic u ( z) + In I z l = �[h(z)]. Similarly, if ua ( z)  �
(�)
(�)
O
(� )
(�) �(g) . h(z)  � resp.
near 0, for some g holomorphic near 0, ua (z)  �
=
In those respective cases, (ua)x (z)  i (ua)y (z) = 1 ( ua)x (z)  i (ua)y (z) = g(z)  2 : the numerator resp. denomiz nator of f has a pole at a resp. b . ] For E) : 10.6.14 Exercise. a ) For a in R, there is in H (R) an f that in each chart z, u ( z) = G(z) + In I z l , v. 10.4.8. If
�f U + i v such
F(z) � e xp [G(z) + In I z l + iv(z)] ,
I I
clef
z =  In I ha(z) l · then In I F(z) 1 = G(z) + In I z l and G(z) =  In F(z) b ) I ha l < 1.
Section 10.6. Miscellaneous Exercises
429
ha(b)  ha(z) IS . holomorphlc . on R and I ¢ < 1 c) For b fixed, ¢ : R 3 Z r+ I 1  ha ( b » ha(z (v. Section 7.2 and Chapter 8) . d) If Ord f (a) �f n, off Z(¢) , u �f  '! ln l¢1 is harmonic and positive off n the zeros of ¢. e) If ( is a chart at b, then :F consisting of functions w such that: el) w is subharmonic and nonnegative off b; e2) supp (w) is compact; e3) w (() + In 1(1 is subharmonic at b, is a Perron family. f) u ( ( ) � G(() �f  In I hb l and hb (a) � 1¢(a) l * � 1¢(a) l . g) 1 == l. hb h) Z(¢) = b and ¢ is injective. Riemann's Mapping Theorem implies the Uniformization Theorem for hyperbolic Riemann surfaces. For F): 10.6.15 Exercise. 1 Fl) For some meromorphic f, g Z r+ f(z)   is holomorphic near a, f :
Z
is holomorphic off each neighborhood of a, 'iR(f) �f u is bounded, and u(a)  'iR o. F:r some meromorphic 1, g Z r+ 1(z)  is holo
(�)
�
:
=
morphic near a, and f is holomorphic and bounded off each neighbor hood of a. Near a, both f and 1 are injective. If b is near but different 1 1 cl f from a, then g clef = are h 1omorp h'lC ff b and g =e f  f(b) f  f(b) while b is a simple pole of g and ?j. Some linear combination ag + fig is holomorphic and bounded. Since R is not hyperbolic, ag + fig is con stant on R: for some Mobius transformation T, f = T Finally, 0
0
(1),
(1).
1 = if, 2s(f ) = 'iR whence f is admissible. F2) The end of the argument in Fl) implies that the set S of points b such that for some Mobius transformation T, f = T (fb) is both open and closed: S = R. F3) If f is admissible at a and f(a) = f(b), for a g admissible at a and con structed as in Fl), for some Mobius T, g = T(f), whence g(a) = g(b) . Since a is the unique pole of g, a = b. F4) If R is elliptic, i.e., compact, for an f admissible at a, f(R) is both compact and open, whence f(R) = Coo . b
Chapter 10. Riemann Surfaces
430
If R is parabolic, and f is admissible, f (R) c Coo and f is not sur jective. For some Mobius T, T o f(R) c C. If T o f(R) �C, then T o f(R) is conformally equivalent to 1U: R is hyperbolic, a contradic tion. 10.6.16 Exercise. The fundamental group 7r l (R) of a Riemann surface is finite or countable. [Hint: The result 10.2.10 and the construction of the universal covering space apply.] 10.6.17 Exercise. a) For some f in H (1U) , if Z E 1U, then 2 exp { [ J ( z W }
J ;
=1+
z,
1 Z and is holomorphic in 1U. b) Ana i.e., f is some branch of ln lytic continuations of f along the curves 'Y± [0, 1] 3 t r+ ±1 + e2rrit lead to different function elements. c) The functionals 'Y± are homologous in Q � C \ ({l} u {I}) . d) The Q� homotopy equivalence classes h+ } h } and h } h+ } are different, i.e., h+ } h } h+} l h_ r l j. 1. Hence 7r l (Q) is not abelian. 10.6.18 Exercise. Without reference to the Uniformization Theorem, for any Riemann surface, the conjunction 5j 1\ IE is impossible. (Thus the listing of possibilities in the second and third columns of Table 10.3.1 is exhausti ve.) 10.6.19 Exercise. If Q c C, f E H (Q), and for each a in C and some nonempty open neighborhood N(a) , f may be continued analytically to a function element (fa , N( a ) ) , there is an entire function F such that F l n = f. 10.6.20 Exercise. When Q � [0, 1] 2 is regarded as a subset of C, if f E H(Q) and f[8(Q)] C lR, there is an entire function F such that F I Q = f. [Hint: The Schwarz Reflection Principle applies.] :
11
Convexity and Complex Analysis
1 1 . 1 . Thorin's Theorem
In a number of disparate contexts, e.g. , topological vector spaces, probabil ity theory (measure theory confined to measure spaces (X, 5, p,) for which X E S and p,(X) = 1 [Kol]) , von Neumann's theory of almost periodic func tions on groups as well as his theory of games [NeuM] , linear and convex programming, etc. , the role played by convexity is central. The following discussion attempts to illustrate that role in complex analysis. 11.1.1 DEFINITION. WHEN {Vd 1 < k < n IS A SET OF VECTOR SPACES, Fk E [0, (0) Vk , 1 :::; k :::; n, AND G E ilR � ¢ IN lRV, x " , x Vn IS (G; F1 , " ' , Fn )n CONVEX IFF WHENEVER 0 :::; tk , tk = 1 , AND G o ¢ IS DEFINED, k=l
L
G O ¢ (V l , . . . , Vn ) :::;
n
L tk Fdvk) '
k=l
[ 11.1.2 Note. In the context of Jensen's inequality (3.2.35), a function ¢ convex in the ordinary sense is (id ; id , id )convex. When V1 Fl
=
�f U (X
clef In II II
p,
, p"
) V2 �f U' (X , p, ) , F2 cle=f In II II G cle=f In,
Vi E V;, i = 1, 2, f (v l , v2 ) �f
p' ,
I x Vl (X)V2 (X) dp,(x) I '
Holder's inequality states that ¢ �f ln f is (G; F1 , F2 )convex:
431
Chapter 11. Convexity and complex analysis
432 2 7T n i n
� enSince 2 7T kt
1e = ,c;2=w' = 0, for h in MVP [D( a, r t] n in N, � 1  e� k =O and s in [O , r ) , � denoting approximately equal to, h ( a)
=
=
h
[nl� :;;: ( 1
'
a + se
2 7r k i n
rIr h (a + st) dT (t)
I1l'
= 0
1�
)
1
� :;;: h
k=O
2Wk,
(a + sen ) . (11. 1.3)
Thus, when = and � are read :::; , (11.1.3) may be viewed as a convexity property of h. By abuse of language, subharmonic func tions, may be viewed as a class of convex functions.] When X is a set, B, D c X, and I E jRx , the Maximum Principle in D relative to B obtains for I iff sup I (x) :::; sup I(x) . xED
xEB
11. 1.4 Example. The Maximum Modulus Theorem asserts, i.a., that if
D(a, r) C Q and I E H(Q) , the Maximum Principle in D � D( a, r t relative to IJ �f 8[D( a, r )] obtains for I II . 11. 1.5 Exercise. If D e B, the Maximum Principle in D relative to B obtains for all I in jRx . 11. 1.6 Exercise. If a function I in jRlR is convex or monotone, for every finite interval [a, b] , the Maximum Principle in [a, b] relative to the set {a, b} (consisting of the two points a and b) obtains for I . 11. 1.7 Example. If if ° :::; x :::; 27r I (x) �f { �Os x otherwise for every finite interval [a, b] , the Maximum Principle in [a, b] relative to { a, b} obtains for I in jRIR . However, I is neither convex nor monotone: the converse of 11.1.6 is false. 11.1.8 LEMMA. FOR A VECTOR SPACE V, A ¢ IN jRV IS CONVEX IFF FOR EACH A IN jR, EACH PAIR {x , y } IN V, AND EACH t IN [0, 1] , THE MAXIMUM PRINCIPLE IN THE INTERVAL [x , y] �f { z : z = tx + (1  t)y, t E [0, 1] } RELATIVE TO {x , y } OBTAINS FOR ¢ [tx + (1  t ) y]  At .
433
Section 11.1. Thorin's Theorem
The map ¢ in lRv is convex iff for each A in lR
sup {¢[tx + (1  t )y]  A t } ::;; max{¢(y), ¢(x)  A }.
O<:: t <:: l
PROOF. If ¢ is convex , ¢[tx + (1  t )y]  A t ::;; t ¢(x ) + (1  t)¢(y)  A t = t [¢( x)  A] + (1  t)¢(y) ::;; max {¢(x)  A , ¢(y) } . Conversely, if the Maximum Principle
as
described obtains, for all A in
lR, sup ¢[tx + (1  t )y]  At ::;; max {¢(y), ¢(x)  A }. Hence t E [O ,I]
sup ¢( tx + (1  t )y) ::;; max {¢(y) + A t , ¢(x)  A + At}
O<:: t <:: l
clef
= max {¢(y) + A t , ¢(x)  A(l  t )} = max {
If
A
max { , B} =
{AB,,
A, B} .
then ¢( tx + (1  t )y) ::;; ¢(y) + At then ¢( tx + (1  t)y) ::;; ¢(x)  A(l  t) .
In each case when A = ¢(x)  ¢(y), ¢[tx + (1  t)y] ::;; t¢(x ) + (1  t)¢(y).
D
11. 1.9 Exercise. If each element of {¢,x } ,X EA is convex, ¢ � sup ¢,X is ,X EA convex. 11.1. 10 Exercise. If g in lRlR is a monotonely increasing function contin uous on the right and the Maximum Principle in D relative to B obtains for j, the Maximum Principle in D relative to B obtains as well for g o j. The development above leads to the next result, which implies a num ber of important conclusions. 11.1.11 THEOREM. (Thorin) HYPOTHESIS: a) X IS A VECTOR SPACE; b) C IS A CONVEX SUBSET OF X; c) Y IS A SET; d) j E lRx x Y ; e) FOR EVERY LINE SEGMENT [xo, x d �f { xo + t ( X l  xo )
0 ::;; t ::;; 1 }
Chapter 1 1 . Convexity and complex analysis
434
CONTAINED IN C AND EVERY A IN JR, THE MAXIMUM PRINCIPLE IN o
[x , xd
X
Y RELATIVE TO
OBTAINS FOR FXQ ,Xl ,>' : [0, 1] CONCLUSION: M(x)
X
( XO
Y 3 (t, y)
X
r+
Y) U (X l
X
Y) o
I [tX l + (1  t )x , y]  At .
�f yEY sup I (x , y) IS A CONVEX FUNCTION OF
X.
PROOF. As formulated, the result is equivalent to the statement that for all A, the Maximum Principle in [0, 1] relative to {O, I} obtains for
I [tX l + (1  t )xo , y]  At . From 11. 1.8 it follows that for each y, I (x , y) is a convex function of x. Then 11. 1.9 applies. D 11.1.12 THEOREM. IF X AND Z ARE VECTOR SPACES , K C [Z, X] , C IS A CONVEX SUBSET OF X, AND Z E n L  l (C) �f r THEN, IN THE CONTEXT LEK
ABOVE, M(z; K) �f sup { F(L(z) , y) : L E K, y E Y } IS CONVEX ON r . PROOF. Since L is linear, L  l (C) is convex. Then 11.1.11 applies. D [ 11.1.13 Note. If r = (/), the conclusion above is automatic. In any event, since each L is linear and C is convex, r is convex. The result asserts in particular that if M(z; K) is finite at both endpoints of a line segment J lying in r, then M (z; K) is bounded above on J.]
1 1 . 2 . Applications of Thorin 's Theorem
The applications of Thorin's Theorem are not limited to the field of complex analysis. They appear as well in classical functional analysis and in the theory of harmonic and subharmonic functions. 11.2.1 Exercise. Thorin's Theorem implies Holder's inequality. In Section 7.3 the focus is on some entire function I and the manner in which I I(z) 1 grows as I z l t oo . For convenience of description, the growth of various logarithmic functions of I I I rather than the growth of I I I itself is estimated. The THEOREMs that follow are concerned with some function I holomorphic in a region Q that is a strip or an annulus and the behavior in Q of some logarithmic function related to I I I .
Section 1 1 .2. Applications of Thorin's Theorem
435
11.2.2 THEOREM. (Hadamard's ThreeLines Theorem) IF

Q �f ( a, b)
x
lR, I E H(Q) n C ( Qc , q , I(Q) c D(O, K)
)
AND M(x; I) d�f sup { 11(x + iy) 1 00 < y < oo } , THEN In M IS CON( VEX. PROOF. For each ), in lR, eAz I(z) � F(z) conforms to the hypotheses for I. The Maximum Modulus Theorem implies, in the present context,
:
M(x; F) :::; max {eAa M(a; I), eA b M(b; I)
} and
}
In M(x; F) = ),x + In M(x; I) :::; max { ),a + In M(a; I), ),b + In M(b; I) . As in the PROOF of Thorin's Theorem, if ), =

In M(b; 1)  ln M(a; I) . . , I.e., If ),a + In M( a; I) = )'b + In M(b; I), ab
then direct calculation shows that for t in [0, 1] , In M [ta + (1  t) b; Il :::; t In M ( a; I) + (1  t) In M (b; I) .
D
11.2.3 THEOREM. (Hadamard's ThreeCircles Theorem) IF
:
I E H [A(a, r Rtl n C[A(a, r : R) , ej , FOR t I N [r, R] , In M(t; I ) IS A CONVEX FUNCTION OF In t. PROOF. If ° < r < R, a = In r, b = In R, and Q = (a, b) x lR, for a suitable real the map ¢ QC 3 z r+ eC>z carries QC onto A( a, r R) . Hence I 0 ¢ conforms to the hypotheses of 11. 1. 11. D [ 11.2.4 Remark. It is the method of proof of Thorin's Theorem rather than the theorem itself that leads to the last two results. In [Thl the author attributes his attack on his general theorem to Hadamard's original approach! 0',
:
:
For In M(x; I) resp. In M(r; I), one may substitute g o M(x; I) resp. g o M(r; f) when g is a monotonely increasing function con tinuous on the right. A notion of the strength of Hadamard's Theorem and of the last remark can be derived from the following considerations. The Maximum Modulus Theorem implies merely M(x; I) :::; max{M(a; I), M(b; I)}
Chapter 1 1 . Convexity and complex analysis
436
resp.
M(t; I) � max{M(r; I), M(R; I)} .
These inequalities do not imply the convexity of either M(x; I) or M I); a fortiori, they do not imply any of the convexity properties of In M(x; I), In M(t; I), g o M(x; I), or g o M( ; I).
(r;
r
The replacement of, e.g., M by In M, leads to a function like ¢ in 11.1.1.] 11.2.5 THEOREM. IF I E H(1U) ,
r E (0, 1 ) ,
p
E (0, 00 ) , AND
r;
a) ON (0, 1 ) , Ip(r) IS A MONOTONELY INCREASING FUNCTION OF b) ln lp(r) IS A CONVEX FUNCTION OF ln r. PROOF. a) If ° � r 1 < < the Maximum Modulus Theorem implies
r2 r,
that for some
r2 ei02 , � t I I (r 1 e 2""w i ) I P � � t I I (r2 e 2""W ' ei02 ) I P . k= l
k=l
b) Thorin's method of proof as exhibited in the PROOF of 11.2.2 and as used in the PROOF of 11.2.3 applies. D 11.2.6 THEOREM. (M. Riesz) IF
FOR THE ( BILINEAR ) MAP B : em
x
0',
f3
>
0,
Tn
n f en 3 (x, y) r+ L L ajkXjYk � B(x, y),
j=l k=l
THE CONDITIONS 0, (J'k > 0, 1 :::; j � m , 1 :::; k � n, def def P = ( P1 , · · · , Pm ) , 0" = ( (J'l , . . · , (J'n ) ,
Pj
>
AND S THE SET OF ALL (x, y, P, 0" ) SUCH THAT m
L j IXj � :::; 1, j=l P l
and
n
L (J'k IYklfr :::; 1, k=l
Section 1 1 .2. Applications of Thorin's Theorem
437
THE LOGARiTHM OF Mu(3 �f
sup I B(x, y) 1 IS A CONVEX FUNCTION (x.y . p.CT)ES OF THE ( a , /3) IN THE ( OPEN ) QUADRANT Q �f { ( a , /3) a , /3 > o } . [ 11.2.7 Remark. Thus, if :
{ (I' , J), (1], () ) C Q , 0 � t � 1 ,
and ( a , /3) = t (1', J) + ( 1  t)
(1], ( ) ,
(MO:(3 is a multiplicatively convex function in Q).] PROOF. The notations below are useful in the discussion that follows:
For >'1 , ), 2 real and fixed and (0'0, /30) in Q, owing to 11.1.11 and 11.1.12, it suffices to prove that the logarithm of
is a convex function of t on (0, 00 ) . If the real variable t is replaced in the right member above by the complex variable t + i u , each ¢j resp. 'l/J k is translated by U), l In rj resp. U), 2 In S k , and the value of the right member is unchanged:
If all the variabl�save t and u are fixed, Hadamard's ThreeLines Theorem is applicable to M (0'0 + )'It, /30 + ), 2 t) . Thus In M (0'0 + )'It, (30 + ), 2 t) is a convex function of t on (0, 00). D [ 11.2.8 Note. a) When 0' = 0 or /3 = 0, 1
1
a
0
clef 00 or
1

/3
=
1

0
cle=f 00.
Chapter 1 1 . Convexity and complex analysis
438
n Tn Concordantly the conditions L I Xj I i :::; 1 or L (J"k I Yk I � :::; 1 are k= l
j =l
interpreted I Xj I :::; 1, 1 :::; j :::; m , or I Yk I :::; 1, 1 :::; k :::; n. The ar gument given for 11.2.6 remains valid when a = 0 or f3 = 0, v. 3.2.6. b ) M. Riesz showed that for the bilinear map BIR : ]R2 x ]R2 3 (x, y) ft (Xl + X 2) Y l + ( Xl  X 2 ) Y2 , 1 when a = 2 ' the logarithm of the minimum of
is concave. Thorin extended M. Riesz's result by showing that if the logarithm of the maximum is concave [Th] . Thus, o < a :::; if B is restricted to ]RTn ]Rn the conclusion in 11.2.6 is invalid un less 0' + f3 z 1 and {a, f3} C [0, 1] ' i.e., unless (a, f3) E Q, v. Fig ure 11.2.1.]
�,
X
11.2.9 Exercise. In terms of j= l
an equivalent formulation of 11.2.6 is the following.
(0,
I,
1)
o
1)
L� a
Figure 1 1 .2.1.
Section 11.2. Applications of Thorin's Theorem
The map M<>(3 : Q r+
(x,y,p,CT) E S V (
sup
439
[U(p, X)] (3 is a multiplica[V(lT, x)] a
tively convex function of the pair {a, ;3} in the quadrant Q. [Hint: 3.3.6 applies.] An important application of 11.2.6 is found in functional analysis, particularly in the treatment of continuous linear maps
T : £P 3 f r+ Tf E U , 1 :s; p < 00, 1 :s; q :s; 00.
. For such a map, a cle=f , and ;3 cle=f , the functlOn 1 q
1 p


should behave like the map M<>(3 considered above. The justification for such an expectation follows. 11.2.10 THEOREM. THE FUNCTION Na(3 IS MULTIPLICATIVELY CONVEX IN THE QUADRANT Q . PROOF. Since the linear span S of the set of characteristic functions of sets of finite measure is dense in LP �f LP(Z, p,), the arguments below about Na(3 when f is confined to S are transferable without change to LP. If f E S, for some characteristic functions X F ' 1 :s; j :s; m , of pairwise disjoint sets Pj of finite measure and some constants X l , . . . , X rn , J
171
171
j =l
j=l
f = L Xj XFj , Tf = L XjT (XFJ
rrt
�f L Xj gj . j =l
There is a set { Ei h :'S: i:'S:n of pairwise disjoint sets of fini te measure such that n
each gj is I Ilqapproximable by a gj � L bj;X Ei Hence, if ; == 1
'Tn
X; (x)
�f L bij xj , 1 :s; i :s; n, j == l
440
Chapter 11. Convexity and complex analysis
o
v
11.2.11 Exercise. The argument is valid if, for counting measure and some (Y, S, ) Lq (� Lq ( y, ) 1 1 1 11 .2.12 Exercise. If 1 :::; q and +  1 clef = > 0: a) 1 < q < p'; b) p q r for some t in (0, 1 ) , r = tp + (1  t) q (v. 4.9.7) . c) if p = 00 or q = 00 , the result remains valid. The next paragraphs show that (4.9.7) is a direct consequence of 11.2.6 and its generalization for No:(3 . v ,
v .

11.2.13 LEMMA. (4.9.7 repeated) IF 1
(f, g)
E LP(G, p,)

:::; x

q,
1 1 1 +  1 clef = r q P



>
0, AND
U (G, p,) ,
*
THEN Il f g il T :::; Il f ll p . Il g ll q · PROOF. If {j, g} C Coo (G, q the inequalities *
Il f g ll oo :::; Il f ll p' . Il g ll p , Il f g ll p :::; Il f l l l . Il g ll p , *
imply that T,q is a map both from LP to Coo (G, q (c L 00) and from LP to LP. The multiplicative convexity of No:(3 and 1 1 .2.12 apply. 0 The M. Riesz Convexity Theorem has many consequences. In particu lar, it leads to the theorems of Hausdorff/Young, and F. Riesz in functional analysis. Since their proofs use the M. Riesz Convexity Theorem for which the proof above involves complex function theory, they appear at this junc ture. The setting is a locally compact abelian group G equipped with Haar measure p,. The central facts relevant to the current discussion are Pon trjagin's Duality Theorem (v. 2) in Section 4.9) and its ramifications in functional analysis. These are treated in detail in [Loo, N ai, We2 ] .
441
Section 1 1.2. Applications of Thorin's Theorem
a) According as G is discrete, compact, or neither, i.e., locally compact and neither discrete nor compact, !he dual group of G is compact, . discrete, or neither. Furthermore, = G. b) To each Haar measure p, for G there corresponds a dual Haar measure for O. bI) If G is discrete, L 1 (G, p,) C L 2 (G, p,) . b2) If G is compact, L 2 (G, p,) C L 1 (G, p,) . b3) If G is discrete, say G = {g,X L E A ' and e ,X cle=f X{ } ' E A , the Fourier transforms tG., ). E A, are a CON for L2 b4) If G is compact, {¢,X L E A is a CON in L 2 (G, p,) , and f E L 2 (G, p,) , then 2 ) (1, ¢,X ) 1 2 = l f(x W dp,(x) (Parseval's equation) . ConG 'x E A 2 versely, if L i c,X 1 < 00 , for some f in L 2 (G, p,) , (I, ¢,X ) = c,X . 'x E A The last is a generalization of the classical FisherRiesz Theorem (v. 3.7.14) . c) If G is neither discrete nor compact, neither of the inclusions
0
11
0
g"
\ /\
(0,11) .
l
need obtain. However, L 1 (G, p,) n L 2 ( G, p,) �f S is a dense subspace of L 2 (G, p,) . If f E S, then 1 E L 2 and
(0, 11)
(Plancherel's Theorem) . Thus there is definable an extension, again denoted , to L 2 ( G , p,) of the Fourier transform and
is an isometric isomorphism (v. 4.9.6). d) The statement in a) is logically equivalent to the conjunction of the statements in b) and c): {a)} {} {{b)} 1\ {c) } } . e) If 1 :::; p < 00, by abuse of notation, P � Coo(G, q Coo (G, q (the subspace generated in Coo( G, q by functions arising from convolution of functions in Coo(G, C) ) is II lipdense in LP(G, p,) . Furthermore, if and the Fourier inversion formula f E P, then 1 E L 1 *
(0,11)
442
Chapter 11. Conve�ity and complex analysis
is valid. 11.2.14 Example. The discussion below is based on items a) e) above and the interpretations and extensions in 5) of Section 4.9. For the map
the previous observations imply
,
1 8 (I, g) 1 � Il f ll l . Il gll l = I l f 112 , 1 8 ( I , g) 1 � II f l12 . II gl1 2 .
11 � 1 2
The multiplicative convexity of No:(3 implies whence if 1 � p � 2, then
11 � l p' � Il f ll p·
11 .2.15 THEOREM. (Hausdorff/Young and F. Riesz) a) IF 1 < p � 2 AND THE CON{¢n} nE ]\/ (DEFINED ON [0, 1]) CONSISTS OF UNIFORMLY BOUNDED FUNCTIONS (FOR SOME M, II ¢n lloo � M, n E N, ) FOR f IN
THERE OBTAIN
{ cn �f 1 1 f(x)¢n(X) dX } nE]\/ E £P' , AND (� len IP' )
b) IF {cn } nE ]\/ E £P , FOR SOME f I N U' ([0, 1] , q ,
1 1 f(x)¢n(X) dx {e2mrit}
=
Cn, n E N, AND Il f ll p'
�
1
[7
�
(� ) len I P
Il f ll p . 1
p
PROOF. The discussion of 1 1 .2.13 suffices for the CON consisting of the functions n EZ appropriately reenumerated as a sequence {¢n } nE]\/ ' For the general CON the observations 00
2 L I cn l � Il f ll� n =l
len I
�
(Bessel's inequality),
M ll f ll l ' (M nfree, )
443
Section 1 1.2. Applications of Thorin's Theorem
LP LLP L
n 2 3 f r+ { cn } n E N ' regarded as a map from imply that the map T the function space n 2 to the function space eN , is one to which the multiplicative convexity of No: f3 applies. 0 [ 1 1 .2.16 Note. As the counterexamples below demonstrate, the condition 1
1 1 .2.17 Example. a) If f3 > 1, c E lR \ {O}, and f( x ) �f
OCJ
L n� (In n)f3 e 2mrix ,
n= 2
icn In n
'i
then f is continuous but if p > 2, then b) The trigonometric series
f
I
�icn ln n
n= 2 n 'i (In n)f3
I
P' =
00 .
(11.2.18) is not a Fourier series, i.e., there is no f in L 1 such that if m j. 2 n otherwise [Zy] . Nevertheless, if q
>
2,
00
L1 ( n1� )q
n=
<
00 .
(After the substitution e 2 7ri x + Z , the lacunary series in (11.2.18) is related to the lacunary series in 10.1.23.) In a circle of ideas studied by Phragmen and Lindelof, there are gener alizations of the Maximum Principle in its various forms for holomorphic, harmonic, and subharmonic functions. Thorin's method implies many of their results. 1 1 .2.19 Exercise. A function f in lRfl is subharmonic in the wide sense iff for each subregion Q 1 such that Q 1 u a (Q 1 ) c Q and for each h in the Maximum Principle in Q 1 relative to a (Q 1 ) obtains for f [Hint: The argument for 6.3.37 applies.]
 h.
444
Chapter 11. Convexity and complex analysis
11.2.20 THEOREM. (PhragmenLindelof) IF: a)
Q � { (x, y) : a < x < b, y E lR } ; b) f E usc (rl) ; c) f E SHW(Q) ; d) sup f(a, y) � M, sup f(b, y) � M ;
YE�
YE�
_7r_.
f) FOR SOME CONSTANT K , I N rl , f(x, y) � K e a l y l ; THEN e) a < ba' sup f(x, y) � M. (x,Y) E fl
IF f IS BOUNDED ON 8(Q) AND GROWS AT A CONTROLLED RATE IN Q, THEN f IS BOUNDED IN Q.
7r
PROOF. If a < {3 < _ ' E > 0, and b a
(
)
a+b ff (X, y) clef = f(x, y)  E COS {3 x  cosh {3y, 2
(11.2.21)
the subtrahend in the right member of (11.2.21) is harmonic: ff E SHW(Q) . Then 1 1 .2.19 implies that the Maximum Principle in Q relative to 8(Q) obtains for ff' Since ff � 00 as I y l t oo, the Maximum Principle in Q relative to 8(Q) obtains as well for f · 0 [ 11.2.22 Note. If 
f(x, y)
=
clef
7r
(
)
then
7r
cosh  y < e ba f(a, y) = f(b, y) = 0,
f(x, y) {a
<
7r
a+b cos  x  cosh  y, ba 2 ba
�
x < b} ::::}
{ lim f(x, y) l y l+oo
wlYI
ba ,
=
}
oo .
Hence, if the condition e) in the statement of 11.2.20 is relaxed to a �  , the conclusion is false.] ba
7r
12
Several Complex Variables
1 2 . 1 . Survey
The discussion of complex analytic manifolds, v.Section 10.2, provides an introduction to the possibilities of studying analytic functions of several complex variables. The particular case, when n > 1 and Q is a subregion of en , of functions in e� and analytic in Q is of great interest. At first blush the theory seems to be a simple generalization of what is known about holomorphy as introduced in Chapter 5. A closer look reveals that parts of the more extended theory are intrinsically different from their cC counterparts; furthermore, some n of the most useful theorems about elC have no natural extensions to elC • The typical element z of en is a vector ( Z I , . . . , zn ) and when en is viewed as ]R2 n , then Zj �f X 2j  l + i X 2j , 1 :::; j :::; n. When Q c en , a Bf , 1 :::; k :::; 2n, exists function f is in C 1 (Q) iff each partial derivative BX k and is continuous. The concept of analyticity can be introduced via the introduction of the following operators ( cf. 5.3.5):
12.1.1 DEFINITION. A FUNCTION f IN C 1 (Q) IS HOLOMORPHIC IN Q, i.e., f E H (Q) , IFF 8f = O. [ 12.1.2 Remark. The condition 8f = 0 is the nvariable version of the CauchyRiemann equations.] When n > 1, the study of functions holomorphic in a region of en IS simplified by the introduction of specialized vocabulary and notation. 445
446
Chapter 12. Several Complex Variables
When Q l , . . . , Qn are regions in c., their Cartesian product is a polyregion . When each Qk is an open disc D (a k , rk t , Q is a polydisc . ( + ) n �f clef �f When a = ( a 1 , • • • , an ) E Z , z  ( Z l , . . . , Zn ) , a  (a l , . . . , an), and r clef = ( r l , . . . , rn ) , Z
= Zl
a clef
"'1
a! = a l ! ' "
clef
'"
. . . Zn n , an ! ,
aa
=
clef
a"" , . . . , "' n ' clef I a I = 0' 1 + . . . + an ,
ra clef II ri'" n
==
i= l
t ,
a"' n clef al a i a clef a"" . . "' n = a J = "' 1 . "' 1 aZ l ' aZn aZ l ' " aZna n '
12.1.3 Exercise. For a polydisc b.(a, rt, how do aD [b.(a, rtl
and a [b.(a, rtl
differ? Many of the theorems about functions in CC have, when n > 1, their natural counterparts for functions in c.c n • There follows a systematic listing of these counterparts. ' 12.1.4 THEOREM. IF J E C[b.(a, r) , C] AND, AS A FUNCTION OF zk , WHILE THE OTHER COMPONENTS OF ( Z l , . . . , Zn ) ARE HELD FIXED, THEN IN b.(a, rt,
1
1 J(z) = ( 27r i ) n ao[� ( a,r ) Ol ( Z l
J(w) dW · · · dwn . ) · · · ( Zn  w n ) l wI 
12.1.6 COROLLARY. IF J E H (Q) , THEN J E C = (Q, q .
(12.1.5)
447
Section 12. 1. Survey
12.1. 7 THEOREM. IF I E H (Q) , K (Cn ) THERE ARE CONSTANTS Co. SUCH THAT
3
K c Q, AND 0 ( Cn )
3
U ::) K,
12.1.8 COROLLARY. IF {In} nE]\/ C H (Q) AND FOR EACH COMPACT SUBSET K OF Q, In I K � 1 1 K ' TH EN I E H (Q) . 12.1.9 COROLLARY. THE nVARIABLE VERSION OF VITALI'S THEOREM (5.3.60) IS VALID. 12.1 .10 THEOREM. IF I E H [b.(a, rt] ' THEN
THE SERIES CONVERGES UNIFORMLY TO I IN EVERY COMPACT SUBSET OF b.(a, r t . 12.1.11 THEOREM. ( Cauchy's estimates ) IF I E H (b.(0 , rt) AND I I(z) 1 � M IN b.(a, rt, 12.1.12 THEOREM. ( Schwarz's lemma ) IF I IS HOLOMORPHIC IN A NEIGH � ORHOOD OF b.(O, r) AND II(z) 1 � M IN b.(O, r) , FOR SOME k IN Z+ AND k ALL z IN b.(O , r) , I I (z) 1 � M .
I�l
12.1.13 THEOREM. ( Jensen's inequality) IF b.(0, r) C Q, I E H (Q), AND In II(z) 1 dA 2n � In 1 1(0) 1 · 1(0) 1 0, THEN A 2n b. 0, r ) ) �(O,r) [ 12.1.14 Remark. When n > 1, Jensen's inequality implies that if I E H(Q) and I =t= 0, then
(\
1
A 2n ({ z : z E Q, I(z)
= O } ) = O.
Results like ( ) above are valid for holomorphic maps between complex analytic manifolds, v. [Gel].] *
When n > 1, there are theorems that have no nontrivial counterparts when n 1. =
448
Chapter 12. Several Complex Variables
12.1.15 THEOREM. (Hartogs) IF n > 1, Q c en , I E en , AND AS A FUNC TION OF EACH VARIABLE Zk , AS THE OTHER COMPONENTS OF ( Z 1 , Z ) ARE HELD FIXED, I IS HOLOMORPHlC, THEN I E H (Q) . [ 12.1.16 Remark. The contrast between 12.1.15 and 12.1.4 deserves attention.] .
. ·
,
n
n
PROOF. (Sketch) a) The formula in (12.1.5) implies that if II ri > ° and I I I
i= 1
n
is bounded in a polydisc Q �f II D (0, ri t , then I is continuous, whence 12.1.4 applies.
n
b) An argument relying on Baire categories shows that if II ri > ° and
i= 1 Q, then D(a, r ) contains a polydisc l1 such that l10 j. (/) and I I I is bounded in l1 . c) Mathematical induction, Cauchy's estimates (12.1 .11) (for the case of n 1 variables) , and 6.3.39 conclude the proof [Ho] . 0 D(a, r)
c

n
12.1.17 THEOREM. (Polynomial Runge) IF Q �f II Qk IS A SIMPLY CON k= l
NECTED POLYREGION IN en AND K IS A COMPACT SUBSET OF Q, EACH I IN H (K) IS UNIFORMLY APPROXIMABLE ON K BY POLYNOMIALS. [ 12.1.18 Note. If b is sufficiently small and positive and Q is the polyregion [D(O, 1 + bt] 2 D(O, bt 3 ( Z 1 ' Z2 , Z3 ) , the map X
is injective and holomorphic. Nevertheless, on F(Q) Polynomial Runge fails to hold [Wer] .] On the other hand, there are no generally valid counterparts to remov able singularities (cf. 5.4.3) nor to the phenomenon of natural boundaries as exemplified by 7.1.28 and 7.1.29. 12.1.19 THEOREM. IF n > 1, Q IS A REGION CONTAINING THE BOUNDARY OF D(a, rt, AND I E H(Q) , THERE IS IN H (D(a, rt) A UNIQUE 1 SUCH THAT ll nnD(a.r)O = I l nnD(a.r)o ' PROOF. (Sketch) For z in D ( a, r t the formula 1
I ( z ) = 27ri

�
1
I w  a n l =r
I ( Z 1 , . . . , Zn  1 , W ) W  Zn
dw
449
Section 12. 1. Survey
o
defines the f as described.
12.1.20 COROLLARY. IF n > 1 , r > 0, AND a IS AN ISOLATED SINGULAR ITY OF AN f HOLOMORPHIC IN D(a, rt \ {a} , THEN a IS A REMOVABLE SINGULARITY OF f. ALL ISOLATED SINGULARITIES ARE REMOVABLE, CF. 5.4.3 . 12.1 .21 THEOREM. IF 1 � k <
S �f { Z
n, E >
0,
'. Z l = . . . = Zk = 1, Zk +1 < 1 + E , . . " Zn < 1 + E } ,
II
I I I I S, ::)
I I
D(O, 1t u r 1 = . . · = rk = 1, rk+ 1 = . . · = rn = 1 + E, r cle=f ( r 1 , . . · , rn ,
Q
)
AND f E H (Q), THERE IS IN H [Q U D(O , rn A UNIQUE f SUCH THAT ll = f · n PROOF. ( Sketch ) The formula
defines a function as described. o There arises the question of characterizing an open set in en as a domain of holomorphy, i.e., roughly described, an open set U for which some f in H (U) has no holomorphic extension to a proper superregion U1 , cf. 7.1 .29. 12.1.22 DEFINITION. AN OPEN SET U IN en IS A domain of holomorphy IFF FOR no OPEN SETS U1 , U2 : a ) (/) j. U1 C (U2 n U) ; b) U2 I S CONNECTED AND U2 ct U; c ) WHENEVER f E H (U) THERE IS IN H (U2 ) A ( NECESSARILY UNIQUE ) h SUCH TH AT f l u! = h l u! ' 12.1 .23 DEFINITION. FOR A COMPACT SUBSET K OF AN OPEN SET U, THE H (U) hull OF K IS
Ku �f
{z : z
E Q, U E H (U)}
'*
{ I f(z)1
� s�p
I f (z ) l } } .
[ 12.1.24 Note. The set Ku is closed but need not be compact.]
450
Chapter 12. Several Complex Variables
12.1.25 THEOREM. IF U IS OPEN IN en , THE FOLLOWING ARE EQUIVA LENT: a ) U IS A DOMAIN OF HOLOMORPHY; b ) IF K IS A COMPACT SUBSET OF U, THEN Ku IS RELATIVELY COMPACT IN U; c ) SOME f IN H (U) HAS NO ANALYTIC CONTINUATION BEYOND U, i.e. , THERE ARE NO U1 , U2 CONFORMING TO a) AND b ) IN 12.1.22. 12.1.26 COROLLARY. IF U IS CONVEX, U IS A DOMAIN OF HOLOMORPHY. 12.1.27 DEFINITION. THE SET OF ALL POLYNOMIALS IS Pol. A Runge domain U IS A DOMAIN OF HOLOMORPHY SUCH THAT IF f E H (U) , THEN f IS UNIFORMLY APPROXIMABLE ON EACH COMPACT SUBSET OF U BY ELEMENTS OF Pol. WHEN K IS COMPACT,
WHEN K = K, K IS polynomially convex. 12.1.28 THEOREM. A DOMAIN OF HOLOMORPHY U IS A RUNGE DOMAIN IFF FOR EACH COMPACT SUBSET K OF U, K = Ku . Further details can be found in [Ho] and [GuR] . They provide exten sive treatments of the results cited above and relate them to the theory of partial differential equations, the study of Banach algebras, complex ana lytic manifolds, etc.
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Symbol List The notation a.b. d indicates Chapter a, Section b, and page d. a.e.: 2.2. 60
A \ B: for two subsets of a set X, { x A(a; r, R) : 5.4. 234 A(a : r, R; 8, 4;) : 5.4. 234 [B, F]c: 3.3. 105 Bn ( f ) : 3.2. 100 BP (x, r): 1.2. 10 BW: 3.4. ' 112 (Bft : 3.4. 112
x E A, x tJ. B }, 1.1. 3
c:
1.1. 5 10.2. 395 Cabed: 8.4. 359 c : 2.2. 69 C: 1.1. 4 cn : 1.1. 5 C= : 5.6. 252 C* : 4.9.192 C: 8.4. 359 CF: 3.5. 129 C k (C, lR): the set of functions f having k continuous derivatives, 5.1. 203 C = (C, C): the set of functions f having k continuous derivatives, k E N, 5.1. 203 Co: 1.2. 13 Co(X, C) : 3. 1. 90 Cu (r): 8.2. 344 Coo(X, C)) : 2.3. 61 Coo (X, lR) : 2.3. 76 Cubed: 8.4. 359 Ck , Tn : 4.6. 165 CON: 3.2. 97 Cont (f) : 3.7. 134 Conv (A): of a set A, the intersection of the set of all convex sets containing A, 3.7. 136 cos: 2.4. 80 div : 6.3. 287 DLS: 2.1. 50 D: 2.2. 59 D( a, r)o: 5.1. 203 Sz :
455
456
D: D (L 1 ) , 2.2. 56 D(L): 2.2. 56 D( f ) : 9.1. 369 Rt : 5.5. 234; 10: 5.4. 251
D(O,
Symbol List
det : determinant, 4.7. 183 dP,' dP: 5.8. 261 diam (S): 1.2. 15 dim : dimension, 3.4. 112 Discont (
Symbol List
Ha(Q) : 5.3. 227 HaJR(Q) : Ha(Q) n IRQ , 5.3. 227 H (A): 2.2. 68 H b (Q) : 8.1. 337 H (S) : 5.3. 209 'S(z): 1.1. 4 IT:
1.3. 18
I : 10.2. 398
im (T): 3.6. 129, 4.8. 185 ind y(a): 2.5. 83 Ind y(c) : 5.2. 206 J: the DLS integral, 2.1. 50 J(I) : the Jacobian matrix of f, 4.7. 182 K < f : 1.2. 15 K(X) : 1.2. 12 KG,\: 2.3. 77 Ku : 12.1. 449 ker (T): 3.6. 129 £k : 2.1. 55 £(y ) : length of the curve ,,( , 4.8. 185 lim , lim 1.2 9, 2.5. 86 Iz : 2.4. 81 L: the function lattice at the heart of DLS theory, 2.1. 43 LCTVS: 3. 1. 89 Ld: £ 1 , 2.1. 55 Lf: 4.6. 167 LRN: 4.2. 144 lsc ( S ) : 1.2. 9 Lu: 2.1. 48 Lui: 2.1. 52 £ 1 : 2.2. 62 £: 2.4. 81 mid (a, b, c): 2.2. 55 M: 8.2. 343 M 1 : 8.2. 343 M(a, r): 5.3. 217 M(R; I); 7.3. 323 Mod: 8.4. 358 Mo: 8.2. 343 M(Q) : 5.4. 231 MON: 4.6. 170 NBV: 4.10. 197 N : 2.2. 62 n � n': 1.2. 11
457
458
Symbol List
0: orthogonal, 3.2. 96 0: 1.1. 5, 3.1. 89 ON: 3.2. 96 Ord f(a): 5.4. 241
' p :
for p in [ 1 , 00 ), p' �f
{
�� 1 if 1 < p , 3 . 2 . 9 2
if p = 1 Pol: 12.1. 450 P(k): 6.2. 275 P(f): 5.4. 231 Pr((}): 6.2. 274 s.p(X ) : the power set of the set X, i.e., the set of all subsets of X: 1.2. 6 p . A : 10.3. 414 Q: 1 . 1 . 3 Qpqrn : 7.1. 302 Q k, rn : 4.6. 165 Rc , Rs : for the power series S �f L (z  a) n and �f { } the 00
Cn n=O radius of convergence r", q.v.: 5.2. 205
Rr: 8.4. 359 JR.: 1.2. 3 R([O , 1] ): 2.2. 75 �(z): 1.1. 4 JR." : 1.1. 5 ra : 10.1. 381 S ( k ) : 5.2. 206
�f (Xl
(X, S, /l) S X K : 4.4. S0K : 4.4.
X
153 154 153154 S: 10.2. 393 s n+ l : 5.6. 253 sgn : 2.4. 81 S H : 6.3. 285 SHW: 6.3. 293 sp (x ) : 3.5. 120 Sp ( A ) : 3.5. 124 Sa,o: : 10.3. 411 S8 : 8.3. 351 Sch : 8.2. 347 S ( U): 10.2. 393 supp : 1.2. 15 Tabed: 8.2. 343 T: 1.2. 6 T A : 1.2. 6 T': 3.4. 116
X2 , S I
X
S2 , /l1 x /l2 ) :
4.4. 153154
c
cn nE]\/+ '
Symbol List '][': 1.1. 5 Tf (x) : 4.10.
197 TVS: 3.1. 89 U a , R : 6.3. 286 usc( S): 1 .2. 9 1U: 5.3. 224 V': 3.3. 104 V* : 3.3. 104 (Va , 4>a ) : 10.3. 416 [V] C : 3.3. 104 X I : 3.5. 117 (Xo , . . . , Xn ) : 1.4. 21 [Xo , . . . , Xn ] : 1.4. 21 (X, d) : 1.2. 8 (X, Y, p) : 10.3. 409 yX : the set of all maps f : X H Y, 1.2. 7 { z : Izl < r }: 10.3. 416 3: 1.2. 9, 10.2. 396 ,,: 10.2. 396 ,6(X): 3.7. 136 * "( : 2.4. 79 "( "'A J: 1.4. 20 "( '" J: 1.4. 20 "( "' F, A J: 1.4. 20 ;y, r: 5.5. 237 l1u: 5.3. 226 Ja : 2.1. 55 J(a, b) : 5.6. 252 J(F) : 1.2. 14 (: 5.9. 269 8 : 5.6. 251 Il([a, b] ): 4.10. 196 XK: Il 4.4. 154 1l0K : 4.4. 154 * Il , Il* : 2.2. 68 3.2. 97 n + : 5.7. 254 7r(x) : 5.9. 269 7rl (Y, Yo) , 7rl (Y): 5.5. 245 p(T) : 4.7. 178 P(x): 3.5. 122 L: 2 : the Riemann sphere, 5.6.251 a (B, B') , a (B', B) : 3.4. 112 a�k) : 1.4. 27 v:
459
Symbol List
460
T: 3.5. 127 Q: the equivalence class containing the wellordered set of all ordinal num bers corresponding to countable sets, 1.1. 5; a region contained in C, 5.1. 204 w ( k ) , *w ( k ) : 5.8. 258259 N o : 1.1. 5 (0, Q) : 2.2. 67 #: 1.1. 5 aj; (X) : 4.7. 182 aXj l � i .j <:: n 0, QSJ: 4.4. 154 X E : 2.1. 54 �: isomorphic, 5.5. 245 =: equal almost everywhere, 2.2. 62 8 , 0 : 4.4. 151 EB : direct sum, 3.6. 129 8: 5.3. 186 a: 1.2. 11, 5.3. 210 aoo(Q): 5.6. 252 < : 1.2. 10 A x B: for sets A and B, the set { (a, b) a E A, b E B }, 1.1. 3 X: 4.4. 153 �: converges uniformly, 1.2. 17 m=rs: converges in measure, 3.7. 134 II�p: converges in the I I lipinduced topology, 3.1. 91 varh(P) 4.6. 169 j l\ g, j V g: 1.7. 39 {A 1\ B}: logical and, i.e., for the assertions A and B, the assertion A and B: 2.1. 48 ALJB, U ),E A ), : unions of pairwise disjoint sets {A, B}, {A)'h E A : 2.2. 61 A (P, Q): 3.2. 100 ( , ) 3.2. 96 3.2. 9294 II lip, II II II Q : 3.5. 106 a(u, v ) : 5.8. 260 a( x , y) �n : homologous, 5.5. 249 �: approximately equal to: 11.1. 432 ,G: the denial (negation) of G: 10.3. 417 p l q: for p and q in ;2;, their greatest common divisor: 1.7. 39 0: the empty set, i.e., for any set X, 0 = X \ X, 1.2. 6 lxJ resp. rxl : for x in JR., the greatest integer not greater than x resp. the least integer not less than x: 7.3. 325
(
)
:
1 00 :
Glossary jlndex The notation a.b. d indicates Chapter a, Section b, page d. A
ABEL, N. H . : 4.10. 193 Abel summation: 4.10. 193 abelian: of a group G that the binary operation is commutative: ab = ba, 4.5. 160 abelianization: For a group G , the subgroup generated by the set of all elements of the form xyx  1 y  l is a normal subgroup, the commutator subgroup C. It is the smallest normal subgroup modulo which G is abelian. The quotient group G/C is the abelianization of G. 5.5 251 absolute topological property: a topological property P such that if (X, T) is a topological space, S e A e X, then S has property P with respect to T iff S has property P with respect to T, 1.2. 17 absolute value: 1.1. 4 absolutely continuous component: 4.2. 144 function: 3.2. 96, 4.6. 148, 4.6. 171 measure: 4.2. 143 summable: 3.3. 111 absorbent: 3.4. 113 actioninvariant: for a set X and a group G acting on X, of an attribute A of subsets of X that if A obtains for E and g E G , then A obtains for g(E) , 6.2. 274 additiondistributive: for a vector space V, of a map · V 2 '3 (x, y) H X . Y that X · (ay + bz) = ax · y + bx · z, 3.5. 116 adjoint: 3.5. 116 admissible (function on a Riemann surface): 10.5. 422 advertible: 3.5. 118 AHLFORS, 1 . V . : 9.1. 370 ALAOGLU , L: 3.4. 100 Alaoglu's Theorem: 3.4. 115 ALEXANDER, J. W.: 7.1. 305 algebra: a vector space A (over a field) on which there is defined an addition distributive map . , v.additiondistributive, 6.5. 296 algebrahomomorphism: between algebras, a map that commutes with the algebraic operations: 3.5. 124 almost everywhere: 2.2. 62 A


 

:
461
Glossary /Index
462
almost periodic function (on a group G): a map f : G H e such that for each g in G, the closure of U f(ag) is compact, 11.1. 431 aEG
alternating process: a technique for deriving from the solutions of Dirichlet's problem for each of two intersecting regions a solution of Dirichlet ' s problem for the union of the regions [Nev2] : 10.4. 418 analytic: 10.2. 404  continuation: 5.4. 232, 10. 1. 382 along a curve "/: 10.4. 419  structure: 10.2. 395 analytically equivalent: 10.2. 403 anharmonic ratio: 8.2. 342 annular sector, annulus: 5.4. 234 antipodal point theory: for Sn �f E ffi.n+1 , I } , the study of maps Sn H ffi. such that for some = 1.4. 20 approximate identity: 6.5. 296 arc: 2.4. 82 Archimedean: 1.1. 4 arclength: for a metric space (X,'d) and a curve "/ [0, 1] '3 t H ,,/(t) E X ,
T
{x : x
:
I xl12 x, T(x) T( x), =
:
6.2. 274 Argument Principle: 5.4. 243 ARZELA, C.: 1.6. 32, 9.1. 370 ArzelaAscoli Theorem: 1.6. 37, 9.1. 370 ASCOLl, G . : 1.6. 37, 9.1. 370 associate (of a 8.4. 359 atlas: 10.2. 403 auteomorphism: for a topological space X, a bicontinuous bijection X H X, 3.5. 119 autojective: of a map X H X that it is bijective, 1.2. 7 Axiom of Choice: If { Sa E A } is a set of sets, some set S has exactly one element in common with each Sa , v.Zorn's Lemma, 1.5. 32
Tabed:
:
a
B
BAIRE, R.: 1.3. 19 Baire category: 1.3. 19 set: 2.3. 76 space: 2.3. 75 ' s Category Theorem: 1.3. 19 BANACH, S.: 3.1. 89, 3.3. 110, 3.5. 116, 3.5. 124 
Glossary /Index
463
Banach algebra: 3.5. 116 field: a Banach algebra F that is a field in which the map

is continuous, 3.5. 124 space: 3.1. 89 Steinhaus: 3.3. 110 barrier at a: 6.3. 291 barycenter, barycentric subdivision: 1.4. 23 base: 1.2. 6 of neighborhoods at a point: 1.2. 8 for a space: 1.2. 8 BERGMAN, S . : 8.3. 353 Bergman's kernel: 8.3. 353 BERNSTEIN, S . : 3.2. 100 Bernstein polynomials: 3.2. 100 BESSEL, F. W . : 3.2. 97 Bessel's inequality: 3.2. 97 bicontinuous: of a map T : X H Y between topological spaces, that T is injective and both T and T  1 are continuous, 1.7. 39 biholomorphic: of a map H : X H Y between complex analytic manifolds, that it is injective and that both H and H  1 are holomorphic, 8. 1. 341, 10.2. 403 bijection, bijective: 1.2. 7 binary markers: 1.2. 16 biorthogonal pair: 3.3. 108 Birkhoff, G. D.: 4.6. 173 BLASCHKE, W. : 7.1. 301, 7.2. 320 Blaschke product: 7.1. 301, 7.2. 320 BLOCH, A . : 9.1. 370, 9.2. 371 Bloch's Theorem: 9.2. 371 BONK, M . : 9.1. 370 BOREL, E . : 2.2. 59 Borel measurable: 2.2. 59 subset: 2.2. 59 boundary: 1.2. 11 value problem: 6.3. 288 bounded (set in a topological vector space) : 3.7. 136 variation: 4.6. 170 bounding cycle: 5.5. 249 branch: 10.2. 391 point: 10.2. 401  of order k: 10.2. 399 BROUWER, 1. E. J.: 1.4. 25, 1.5. 31, 2.5. 83 ��


��



Glossary /Index
464 Brouwer degree of a map: 1.4. 25, 2.5. 83 Brouwer's Fixed Point Theorem: 1.5. 31 c
CALGEBRA: an algebra over C, 3.5. 1 16 Cvector space: a vector space over C, 3.1. 89 calculus of variations: the study of extrema (maxima, mlmma, saddle points) of functionals, e.g., for a vector space V, maps V H JR., 8.1. 341 canonical atlas, canonical chart: 10.2. 405  product: 7.2. 318, 7.3. 328   representation: 7.2. 318 CANTOR, G . : 1.2. 13, 2.2. 6667 Cantor function: 2.2. 67  set: 2.2. 66 CARATHEODORY, C.: 2.2. 69 Caratheodory measurable: 2.2. 69 cardinality: for a set X, the set of all sets Y such that for some bijection bxy , bxy (Y) = X, 1.1. 5 Cartesian product: 1.1. 5 CASORATl, F . : 5.4. 230 category: a complex consisting of: a) a class C of o bjects , A, B, . . . ; b) the class of pairwise disjoint sets [A, B] (one for each pair {A, B} in C x C) of morphisms; c) an associative law 0 :
[A, B]
x
0
[B, G] ", (f, g) H g o f E [A, G] �f [A, B] [B, G] ;
of composition of morphisms; d ) for each A in C, in [A, A] a morphism l A such that if f E [A, B] and g E [G, A] , then f l A = f and l A g = g, 3.5. 125, 4.9. 189 CAUCHY, A . 1. DE: 1.2. 8, 1.6. 35, 5.3. 211212, 5.3. 215, 12.1. 447 Cauchy estimate: 5.3. 218, 12.1. 447  Hadamard Theorem: 5.2. 205  net: 1.6. 35  Riemann equations: 5.3. 210  's formulre: 5.3.211  's integral formula, basic version: 5.3. 215    , global version: 5.4. 235  sequence: 1.2. 8   Theorem: 5.3. 212   Theorem, basic version: 5.3. 212 tECH, E . : 3.7. 136 compactification: 3.7. 136 central index: 7.3. 324 0
0
Glossary /Index
465
chain: 10.1. 382 character group: 4.9. 189 characteristic function: 2.2. 59 chart: 10.2. 402 circled: 3.4. 112 circular: 10.6. 424 closed ball: 1.2. 8 curve: 4.8. 185, 5.2. 182  degenerate nsimplex: 1.4. 21  interval: 1.1. 4  set: 1.2. 11  simplex: 1.4. 21 closure: 1.2. 11 operation: 1.7. 42 coarser: 1.5. 32 cofinal diset: 1.2. 11  net: 1.2. 11 commutator: in a group, the product of elements each of the form 5.5. 250  subgroup: 5.6. 251, v.abelianization compact: 1.2. 12 complete analytic function: 10.2. 391  atlas: 10.2. 403  measure space: 2.2. 62 metric space: 1.2. 8  ordered field: 1.1. 3  orthonormal set: 3.2. 97  aring: 2.2. 62 completion of a measure space: 2.2. 62 of a metric space (X, d) : for the set CS of Cauchy sequences in X and the equivalence relation ", defined by { }
xyx  1 y  l :
xn nE]\/
the set of ",equivalence classes; if � resp. 1] is an equivalence class containing { } resp. { } the distance between them is
xn nE ]\/
Yn nE]\/
nE]\/ xn, Yn ) , 2.5. 87
D(� , 1]) �f inf d (
complex analytic manifold: 10.2. 405 conjugate: 5.1. 204  curve: 10.2. 395  integration: 5.1. 204  measure: 4.1. 137
Glossary /Index
466
  space: 4.1. 138  numbers: 1.1. 4  projective line: 5.7. 254  rational point: 10.2. 395 component: in a topological space X, a maximal connected subset of X , 2.5. 83, 5.2. 208 concave: of a map f : V H JR. of a vector space V into JR., that  f is convex, q.v., 11.2. 438 Conformal Mapping Theorem: 8.1. 336 conformal disc: 10.3. 416  selfmap: 8.1. 341 conjugate bilinear: for a vector space V over C, of a map
f : V 2 '3 {x, y } H f(x, y )C, that f(x, y ) = f(y , x) and f(ax + (3y , z ) = af(x, z ) + (3f(x, z ) , 3.6. 128  linear: of a map T bet.ween vector spaces, that
T(ax + by ) = aT(x) + bT(y ), 3.6. 128, 5.8. 261  pair: 3.2. 92 connected: 1.1. 3, 1.2. 11 continuous: 1.2. 7  on the right: of a function f in JR.IR , that for all x, lim f(y) f( x ) , ytx 11.1. 433 contour: 5.7. 254 converge: 1.2. 8, 1.2. 10, 1.2. 17, 1.5. 32 in measure: 3.7. 134  uniformly: 1.2. 17 convergence inducing factor: 7.1. 312 convex combination: for a set X l , . . . , Xn of vectors and a set aI , . . . , an of =
nonnegative numbers such that 103
 function: 3.2. 100
n
ak L k =l
=
1, the vector
n
L a kxk , 3.3. k= l
set: in a vector space V, a subset 5 that contains every convex combination vectors in 5, 1.4. 22, 3.1. 89 convexity theorem of M. Riesz: 4.9. 191, 11.2. 436 convolution: 4.4. 157, 4.10. 195, 6.2. 274 countably additive: 2.2. 61, 4.1. 137 count ably subadditive: 2.2. 68 counting measure: 3.2. 97 �
Glossary /Index
467
covariant functor: for categories C 1 and C2 , a map F : C 1 H C2 of each ob ject to an object, each morphism to a morphism, each l A to I F (A ) , and such that: when g l II is defined, F (g l ) F ( II ) is defined and F (gd F ( II ) = F (gl II ), 3.5. 125 covering space, covering space triple : 10.3. 409 cover transformation (German: Decktransformation: 10.4. 418 cross ratio: 8.2. 342 curve: 2.4. 79, 4.8. 185, 5.1. 204 image: for a curve ,,(, the set "(* : 2.4. 79 component: 10.2. 399 connected: 10.2. 401 starting at yo: 5.5. 245 customary topology (for JR.) : 1.2. 6 cycle: 5.5. 249 cylinder: 1.2. 9 0
0
0
0



D
#(D(f))DEFECTIVE: 9.1. 326 DANIELL, P . J . : 2.1. 44, 2.1. 45 Daniell (DLS) functional: 2.1. 45 measurable function: 2.2. 56   subset: 2.2.59 decomposable: 4.2. 146 defections: 9.1. 369 defective function: 7.3. 330, 9.1. 369 degenerate ksimplex: 1.4. 21 degree of a map: 1.4. 25 DENJOY, A . : 4.7. 177 dense: 1.2. 11 DE RHAM, G.: 5.5. 237 De Rham's Theorem: [SiT] , 5.5. 249 derivative: 4.6. 166, 4.7. 178 diagonal: 4.5. 163 diagonalization method: 5.3. 228 diameter: of a subset S of a metric space (X, d) , 
sup { d(x, y) : {x, y} C S } , 1.4. 22 differentiable: 4.7. 177 differential forms: 5.8. 261 dimension: of a subspace W of a vector space V, the cardinality of (any) maximal linearly independent subset, i.e., basis, of W, 4.7. 179 DINI, D . : 1.2. 16 Dini's Theorem: 1.2. 16
468
Glossary /Index
DIRAC, P . A . M . : 2.2. 55 Dirac functional: 2.2. 55 directed: 1.2. 10 DIRICHLET, P. G. 1.: 3.7. 133, 6.3. 285, 8.2. 342 Dirichlet region: 6.3. 285 's problem: 6.3. 285 's kernel: 3.7. 133 series: 5.9. 269 's Principle : 8.2. 342 discrete topology: 1.2. 6 diset: 1.2.10 diverge: 5.2. 205 divergence: 6.3. 287 domain of holomorphy: 12.1. 449 Dominated Convergence Theorem: 2.1. 53 dual: 3.3. 104 Haar measure: 11.2. 441 pair: 3.3. 104 space: 3.3. 104 dyadic rational number: for some n in N and some M in ;2;, a number of 


n
the form M + L Ek T k , E% k= l
=
Ek : 1.2. 16 E
EDGE: 1.4. 21 EGOROV, D . F.: 2.5. 87 Egorov's Theorem: 2.5. 87 electric charge: 6.3. 287 elliptic: 10.3. 417  function: 8.5. 363 endomorphism: 3.5. 117 entire: 5.3. 255, 7.1. 301 epimorphism: a surjective homomorphism, 2.4. 82, 3.3. 108 equicontinuous: 1.6. 36 equivalence classes: 1.1. 3 equivalence relation: for a set 5, in 5 2 a subset R such that: a) {a E 5} ::::} {{a, a} E R} (R is reflexive); b) {{a, b} E R} ::::} {{b, a} E R} (R is symmetric) ; c) {{{a, b} E R} 1\ {{b, e} E R}} ::::} {{a, e} E R} (R is transitive); 1.2. 11 equivalent function elements: 10.2. 396
Glossary /Index
469
 norms: 3.5. 117 ERDOS, P.: 5.8. 269 essentially equal: 1.2. 11 essential singularity: 5.4. 230  at 00: 5.6. 252 ESTERMANN, T . : 9.1. 370 Euclidean metric: 1.2. 8, 5.1. 204 EU L ER, 1 . : 2.4. 80, 5.7. 255, 8.1. 341 Euler's equations: If U ( . . . ' xn ) minimizes Xl '
, then
[CoH] .
8.6. 337
formula: 2.4. 80
evaluation map: 3.7. 136, 5.8. 257, 8.3. 353 evenly covered: 10.3. 409 eventually in: 1.2. 11 exponent of convergence: 7.3. 327   divergence: 7.3. 327 exponential function: 2.4. 80 extended real number system: 1.2. 8  JR.valued functions: 1.2. 8  Schwarz Reflection Principle: 8.2.346 exterior calculus: 5.8. 257  (wedge) product: 5.8. 261 F
FACTOR SPACE: 4.4. 153 FATOU, P . : 2.1. 53 Fatou ' s Lemma: 2.1. 53 FEJ E R, 1 . : 3.7. 133 Fejer's kernel: 3.7. 133  Theorem: 3.7. 133 Fhomotopic in A: 1.4. 20 field: a commutative ring (q.v.) containing a multiplicative identity with respect to which each element is invertible, 1.1. 3 filter: 1.5. 32  base: 1.5. 32  corresponding to a net: 1.5. 32
Glossary/Index
470

generated by a filter base, 1.5. 28 finer (filter): 1.5. 32 finite cylinder: 1.2. 9 intersection property: 1.2. 12 measure space: 4.2. 146 subset (of I,p(X)): a subset S such that each finite set of elements of S has a nonempty intersection, 1.2. 13 finitely additive set function: for a ring 5 of sets, a map

¢
: 5 '3 E H ¢( E) E
ffi.
such that if En , 1 � n � N are pairwise disjoint elements of 5, then ¢
(u := 1 An) = t, ¢ (En ) , 2.2. 61
first (Baire) category: 1.3. 18 FISCHER, E . : 3.7. 135 FischerRiesz Theorem: 3.7. 135 fixed point property: 1.4. 26 form(s) (Oform, Iform, 2form): 5.8. 258 FOURIER, J . : 3.7. 115, 4.9. 169, 4.9. 172 Fourier coefficients: 3.7. 132 series: for a function f having the sequence { cn } nEZ of Fourier cn e 2n7rit , 4.10. 192 coefficients, the series nE Z transform: 4.9. 189 FRAENKEL, A . : 1.1. 3, 4.5. 160 frequently: 1.2. 11 Fubinate: 4.4. 153 FUBINI, G . : 4.4. 153 Fubini's Theorem: 4.4. 155 functional calculus: a calculus for defining functions of a finite set of ele ments of [SJ] c ' 3.7. 131 function element: a pair ( I, Q) consisting of a region Q and an f in H (Q): 10.1. 382 lattice: 2.1. 44 pair: 10.2. 396 functor: v. category fundamental group: 5.5. 245, 10.6. 430 set: 8.4. 362 Fundamental Theorem of Algebra (FTA): 9.1. 369 Fundamental Theorem of Calculus (FTC) : 4.6. 164

L

G
Glossary /Index
471
GAP, NONGAP: 10.1. 388 GAUSS, C. F . : 5.9. 269 rautomorphic: 8.4. 361 GELFAND, I. M . : 3.5. 124125, 4.9. 189 GelfandFourier transform: 4.9. 189 map: 3.5. 125 Mazur: 3.5. 124 generated (aring) : 2.2. 58 genus: 7.2. 318 germ: 10.2. 391 global Cauchy integral formula: 5.4. 235 globally injective: of a function in H (fl) , that it is injective on fl, 5.3. 223 gradient: 6.3. 287 GRAM, J. P . : 3.2. 98 GramSchmidt process: 3.2. 98 graph: 3.3. 109 Great Picard Theorem: 9. 1. 369, 9.4. 376 G REEN, G . : 5.8. 262, 10.4. 372 Green's formula: 5.8. 2.6 2 function: 8.5. 363, 10.3. 417  Theorem: 5.8. 262 group: a set G and a map · : G2 '3 {g, h} H g . h E G such that · is associa tive; G is assumed to contain an identity e such that e . g g; for each g there is an h such that h . g = e, 2.5. 84 algebra: 4.9. 189 GRUNSKY, H.: 9.4. 375 GUTZMER, A . : 5.3. 218 Gutzmer's coefficient estimate: 5.3. 218 

==

H
HAAR. A . : 4.9. 187 Haar measure: 4.9. 187 HADAMARD, J.: 7.3. 328, 10.1. 388, 11.2. 435 Hadamard's Gap Theorem: 10.1. 388 Threecircles Theorem: 11.2. 435 Threelines Theorem: 11.2. 435 HAHN, H.: 4.1. 139 HahnBanach Theorem: 3.3. 105 Hahn decomposition: 4.1. 140 halfline: for two points x and y in a vector space, the set

{ x + ty : 0 � t < oo } , 1.4. 26 halfopen: 1.1. 5
472
Glossary /Index
cube vertexed at (k, m ) : 4.6. 165  interval:l.l. 5, 4.5. 163 ndimensional cube: 1.1. 5 ndimensional interval: 1.1. 5 HAMEL, G : 3.3. 104 Hamel basis: 3.3. 104 HARDY, G . H.: 7.3. 334 Hardy spaces: 7.3. 334 harmonic: 5.3. 227  analysis on locally compact abelian groups: 4.9. 192  conjugate: 5.3. 227 measure: 6.5. 300 HARNACK, A . : 6.2. 281 Harnack's Theorem: 6.2. 281  inequality: 6.5. 280 HARTOGS, F . : 6.3. 257, 12.1 394 Hartogs's Theorems: 6.3. 294 HAUSDORFF, F . : 1.2. 8, 1.5. 32, 1.6. 34, 4.9. 187, 11.2. 442 Hausdorff Maximality Principle: Every nonempty poset contains a maximal ordered subset, v.Zorn's Lemma. 1.5. 32  space: 1.2. 8 topology: 1.2. 8  uniformity: 1.6. 34 /Young and F. Riesz Theorem: 11.2. 442 hereditary: 2.2. 68 aring: 2.2. 68 HILBERT, D . : 3.2. 82, 3.5. 1 10, 8.6.337 Hilbert space: 3.2. 97, 3.5. 117 H O LDER, 0 . : 3.2. 77 Holder's inequality: 3.2. 93   extended: 3.7. 135 holomorphic: 5.3. 185  autojection: 8.1. 341  function: 5.3. 209  near x : 10.2. 404  on AS: 10.2. 406 holomorphically compatible: 10.2. 403 homeomorphic: of two topological spaces X and Y, that there is a homeomorphism X H Y, 1.4. 22 homeomorphism: 1.2. 7 homologous: 5.5.249 homology: 5.5 245 homotopic, homotopy (in A): 5.5. 245 H(U)hull: 12.1. 449 HURWITZ. A . : 5.4. 243 





473
Glossary /Index
Hurwitz's Theorem: 5.4. 243 hyperbolic: 10.3. 417 hyperplane: in a vector space, the translate of the span of a finite number of vectors, 1.4. 22 of dimension n: the translate of the span of n linearly independent vectors, 4.7. 178
��
I IDEAL:
3.5. 124
Identity Theorem: 5.3. 255 identity modulo a left (right) ideal: 3.5. 124 image: 2.4. 79, 3.6. 129, 4.8. 145, 5.6. 253 immediate analytic continuation: 10. 1. 382 index of a curve with respect to a point: 2.4. 83 subgroup H in a group G: the cardinality of the set { gH : g E G }, of cosets of H, 4.5. 159 indivisible: 2.5. 84 induced by T: 1.2. 6 inner measure: 2.2. 68 outer measure: 2.2. 68 topology: 1.2. 12 infinitedimensional: of a vector space V, that for each n in N there are n linearly independent vectors in V, 3.3. 104 infinitely differentiable: of a function f in ffi.1R that for each n in N, f ( n) exists, 2.4. 80 infinite product: 7.1. 311, 7.2. 313 infinity: 6.3. 287 injection, injective: 1.2. 7 inner measure: 2.2. 68 product: 3.2. 96 regular: 2.3. 78 integers: 1.1. 3 integrable: 2.2. 63 integration by parts: 4. 10. 193 interior: 1.1. 5, 1.2. 7 intermediate value property: of a function f in ffi.IR , that if a j. b and � is between f(a) and f(b) , for some c between a and b, f(c) = � , 1.7. 43 theorem for derivatives: If f E ffi.1R and f is a derivative, then f enjoys the intermediate value property, q.v., 4.6. 176 interval: 1.1. 5 inversion: 8.2. 344 irregular point: 10.2. 395 isolated point: 1.2. 10 ��
��
�
��
���
474
Glossary /Index
 essential singularity: 5.4. 230  singularity: 5.4. 230
isometric: of a map f : X H Y between metric spaces that J[J(a), f(b)] == d(a , b), 8.4. 362  circle: 8.4. 359 isometrically isomorphic: 3.2. 97, 3.6. 130 isometry: an isometric map, 3.4. 112
(X, d) and (Y, J),
J
JACOBI , C . G . J . : 4.7. 182 Jacobian matrix: for a differentiable map
I I
x n
8y . 8 ( Y , . . . , Yrn ) matnx in which the ij entry is ] , 4.7. 182 8x, 8 (X , . . . , Xn ) JENSEN, J . 1 . W . V . : 3.2. 102, 7.3. 332, 11.1. 431, 12.1. 447 Jensen's formula: 7.3. 332  inequality: 3.2. 102, 11.1. 431, 12.1. 447  Lemma: 7.2. 321 JORDAN, C . : 2.5. 83, 4.1. 142, 5.7. 255, 7.1. 305 Jordan curve: a homeomorphism "/ : 1I' '3 t H "/(t) E C, 5.8. 234 Curve Theorem: If f : 1I' H ffi.2 is a homeomorphism, ffi.2 \ f (1I') consists of two (open) components, one bounded, and one unbounded, and of which f(1I') is the common boundary. 2.5. 83, 7.1. 305  decomposition: 4.2. 142  inequality: 5.7. 255 jump function: 4.10. 196 the m
KERNEL: 3.6. 128 Kfold Fubinate: 4.4. 154 Kfold preFubinate: 4.4. 154 KOEBE, P . : 8.1. 342 K O NIG, R.: 8.1. 342 KRONECKER, 1 . : 3.3. 108 Kronecker's function: 3.3. 108
K
L 00
E,
L Cn , that for some positive and a sequence n=O Pk+ l , then Cn 0 (the {Pd kEN ' PPHk I 2: 1 + k E N, and if Pk
LACUNARY: of a series
E,
series has gaps) ,
11.2. 443
< n <
=
475
Glossary/Index LANDAU , E . : 9.1. 370, 9.2. 371, 9.4. 377 Landau's Theorem: 9.4. 377 LAPLACE, P . S . DE: 5.3. 227 Laplacian: 5.3. 227 last maximal term: 7.3. 324 lattice: 2.1. 44 LAURENT, P. A . : 5.4. 237 Laurent series: 5.4. 237 LEBESGUE, H . : 1.2. 15, 2.1. 52, 2.2. 67, 2.4. Lebesgue integrable: 2.4. 78 integral: 2.4. 78 measurable: 2.2. 67  function: 2.4. 78   set: 2.2. 67 measure: 2.4. 78 numbers: 1.2. 15 's Covering Lemma: 1.2. 15 's decomposition: 4.2. 144  set: 4.6. 168 's decomposition: 4.2. 144 's Monotone Convergence Theorem: left adverse: 3.5. 118 atranslate: 2.5. 84  inverse: 3.5. 117 closed interval: 1.1. 5  continuous: 4.10. 196
78, 4.6. 168







2.1. 52



hand derivative: of an
3.2. 102
f in ffi.IR , lim f(b)b xf(x) , when it exists, btx


length: 4.8. 185 lift: 10.3. 409 limclosed: 2.2. 57 lim closed: 2.2. 57
2.2. 57 limit point: 1.2. 10 LINDELOF, E . : 11.2. 444 limclosed:
linear and convex programming: [ZAJ , 11.1. 431 functional: for a vector space V, a map ¢(ax + by ) a¢(x) + b¢(y ) , 3.3. 104  combination: for a set S of vectors, =
¢:VHC
such that
476
Glossary /Index
v.span linearly dependent: not linearly independent, q.v. independent: of a set {x>,} ), E A of vectors, that for each finite subset a
of 1\, if
L a),x),
), E "
=
0, then a), I" 0, 1.4. 21 =
LIOUVILLE, J . : 3.5. 122, 5.3. 218 Liouville's Theorem: 3.5. 122, 5.3. 218 LIPSCHITZ, R.: 4.10. 192 Lipschitz function: 3.2. 96, 4.10. 172 Lipschitian: of an f in JR.( a , b ) , that for some Lipschitz function (q.v. ) L( x, y ) and some positive D , if a < x < y < b, then
I f ( x )  f ( y ) 1 � L (x , Y) l x  Y i n , 3.2. 101
Little Picard Theorem: 9.3. 374 lives: 4.2. 143 local inverse: 10.1. 384 locally compact: 1.2. 12  connected: 1.2. 12  convex topqlogical vector space: 3.1. 89  curve connected: 1.2. 12, 10.2. 406  injective: of a map f, that it is injective in some neighborhood of each point, 5.3. 223  representable: 5.1. 204  simply connected: 10.2. 402 local uniformizer: 10.2. 396 logarithmic function: 2.4. 80 logoid derivative: 8.2. 347 loop: 5.5. 246  homotopic: 5.5. 246  homotopy: 5.5. 246 lower semicontinuous: 1.2. 9 M
MACLAURIN, C . : 2.4. 79 Maclaurin polynomial: for an f in
N
e N (JR., JR.)
when
( o ) n , 2.4. 79 L f (nn.) _x ,
n=O  series: for an
__
f in eOO (JR., JR.) ,
00
f ( n) (0)
'" __ xn , 2.4. L, n!
n=O
79
0 � k � N,
477
Glossary/Index majorization: 5.3. 223 majorize: 5.3. 223 maximal set of function elements: 10.2. 391 maximal simply connected subregion: 5.9. 265 Maximum Modulus Theorem: 5.3. 220 in Coo : 5.6. 253 Principle: 6.1. 271 in D relative to B: 11.1. 432 MAZUR, S . : 3.5. 124 Mean Value Property: 6.1. 271 measurable cover: 2.2. 68 kernel: 2.2. 68 partition: of a set E in a aring 5, in 5, a countable subset 



{ En} nE ]\/
such that E U nE ]\/ En , 4. 1. 138 measure: 2.2. 61 space: 2.2. 61 meromorphic: 5.4. 231 , 10.5. 422 in an open set: 5.4. 231 Metric Density Lemma: 4. 10. 192 space: 1.2. 8 midclosed: 2.2. 56 midpoint convex: 6.3. 284 MINDA, C. D . : 9.1. 370 Minimum Modulus Theorem: 5.6. 253 MINKOWSKI, H . : 3.4. 114 Minkowski functional: 3.4. 114 's inequality: 3.2. 93 MITTAGLEFFLER, G . : 7.1. 301 MittagLeffler's Theorem: 7.1. 301 MOBIUS, A . F . : 8.2. 343 Mobiusinvariant: 8.2. 348 strip: 10.2. 365  transformation: 8.2. 342 modular: 3.5. 124 group: 8.4. 357 monodromy: 10.4. 420 Monodromy Theorem: 10.4. 420 monotone: of a sequence {an } nE ]\/ ' that for each n, an :::; an +l or a n 2': an +l : 1.2. 9; of a class C of functions, that together with every monotonely increasing or monotonely decreasing sequence {In } nE ]\/ contained in C, the function limoo In is in C, 2.2. 57 =


n+
 ring of sets:
2.5. 85
Glossary /Index
478 monotonely increasing function: in
ffi.1R
an
f such that
{a < b } '* {J(a) ::::: f(b) } , 1.2. 13 MONTEL, P . : 5.9. 267, 9.4. 335 Montel's Theorem: 9.4. 377 MORERA, G . : 5.3. 224 Morera's Theorem: 5.3. 224 MROWKA, S.: 1.7. 42 f.l* afinite: 2.2. 69 multiplicative linear functional: for an algebra A, in A', an element x' such that x'(x · y ) = x'(x) . x'(y) , 3.5. 124 multiplicatively convex function: 11.2. 437 multivalued function: for a set X, in X 2 , a subset S such that for each x in X there is in S a subset Sx contained in { x } x X, 10.2. 392 multivaluedness: 10.2. 391 mutually singular: 4.2. 143 N
NATURAL BOUNDARY: the boundary of a region in which a function f is holomorphic and incapable of extension to a function holomorphic in a proper superregion, 6.2. 279, 7. 1. 312, 10.2.406  numbers: 1.1. 3 ndimensional complex analytic manifold: 10.2. 403  Lebesgue measure: 4.5. 162 n + Icell: 1.4. 21 negative set: 4.1. 138 neighborhood: 1.2. 7, 5.6. 252, 10.2. 397 net: 1.2. 10  corresponding to the filter: 1.5. 32 N EUMANN, C . : 8.1. 342 N EUMANN, J . VON: 4.2. 143, 11.1. 431 NEVANLlNNA, R.: 7.3. 323 nicely shrinking: of a sequence { Ern }"nE N of sets in ffi.n , that for some fixed x, each Ern is contained in some B (x, rrn t and for an mfree constant
a(x), >. (Ern ) � a(x)>. [B (x, rrn t J , 4.6. 169
NIKODYM, 0.: 4.2. 144 nonisolated singularity: 5.4. 232 nonnegative count ably additive set function: 2.2. 61  linear functional: 2.1. 45 nonorientable: 10.2. 403 nonsingular: of a linear map T between finitedimensional vector spaces, that T  1 (O) { O } , 4.7. 178 norm: 1.1. 5, 3. 1. 89 =
479
Glossary/Index
normable: 3.7. 136 normal element of [SJ l c : 3.6. 130 family: 1.6. 38 subgroup: in a group G , a subgroup H such that for all 9 in G , g 1 Hg H, 8.4. 358 normalized function of bounded variation: 4.10. 197 normbicontinuous: of a bijection between normed vector spaces, that it is bicontinuous with respect to the topologies induced by the norms, 3.2. 
=
98
normdecreasing: of a map
T between normed vector spaces,
that
I I T(x) 1 1 � Il x ll , 3.5. 123 normed ring: v.Banach algebra, 3.5. 1 16 space: 3.1. 89 norminduced: of a topology in a I I IInormed vector space, that
{ x : Il x ll
< E, E >
0}
is a base of neighborhoods at 0, 3.4. 111 separable: of a norminduced topology, that it is separable, q.v.,

3.2. 98
nowhere dense: 1.3. 18 nsimplex: 1.4. 21 null function: 2.2. 62 homotopic: 5.5. set: 2.2. 62
245
�
o
IFORM: 5.8. 258 oneparameter family of graphs: 5.5. 245 point compactification: 1. 7. 42, 5.6. open: 1.1. 5, 1.2. 6, 1.2. 6 annular sector: 5.4. 234 ball: 1.2. 8  disc: 5.1. 205 interval: 1.2. 6 Open Mapping Theorem: 3.3. 108, 5.3. 221 operator: 3.6. 128 orbit: 10.3. 414 order of a function: 7.3. 324 pole: 5.4. 231   growth: 7.3. 324 

��
��
252
480
Glossary /Index
ordered: of a subset of a poset, that any two elements of S are <related:
1.1 3
ordinal number: an equivalence class of wellordered sets, orientable: 10.2. 403 oriented boundary: 7.1. 302 complex interval: 1.1. 5 cycle: 7. 1. 304 real interval: 1.1. 4 orthogonal, orthonormal: 3.2. 96 OS TROW S K I, A.: 10. 1. 387 Ostrowski's Theorem: 10. 1. 387 outer measure (induced) : 2.2. 68 regular: 2.4. 78 overconvergence: 10. 1. 388
1.1. 5




p
PARABOLIC: 10.3. 417 parameter of regularity: for a subset S of ffi.n , and Rs
�_ef { R
.. R a cube and S C R } ,
sup � ; [Sak] , 4.6. 169 RE Rs An ( R ) parametric equations: 10.2. 395 PARSEVAL( D ESCHENES ) , M . A . : 3.2. 97 Parseval's equation: 3.2. 97 partially ordered set: 1.2. 10 partition: 2.2. 73 of unity: 1.2. 16 patch: 10.2. 406 period: for a group G , a set S, and an J in S G , in G an element p not the identity of G and such that J(px) == J(x), 2.4. 80  parallelogram: 8.6. 366 periodic: for a group G, a set S, and an element p not the identity of G, of an J in S G , that J(px) == J(x), 3.7. 133 Permanence of Functional Equations: 10.1. 390 perpendicular: 3.2. 96 PERRON, 0 . : 6.3. 252254 Perron family: 6.3. 288 Jfamily: 6.3. 290 function: 6.3. 290 pface: 1.4. 21 PHRAGMEN, E . : 11.2. 389, 11.2. 444 PhragmenLindelof Theorem: 11.2. 444
An (S)



481
Glossary/Index P ICARD, E . : 9.1. 369, 9.4. 376 piecewise P: of an f in ffi.IR , that off a finite set
5.2. 206
S, f enjoys the property P,
���
differentiable: 4.8. 186 linear: 5.2. 206 ��� wellbehaved: 4.8. 186 PLANCHEREL, M . : 11.2. 441 Plancherel's Theorem: 11.2. 441 P OINCARE, H . : 8.6. 361, 10.2. 394 Poincare's Theorems: 8.4. 361, 10.2. 394 point function: 4.6. 164 POISSON, S. D . : 6.2. 273, 6.2. 275, 6.2. 277, 6.2. 244, PoissonJensen formula: 6.2. 277 Poisson kernel: 6.2. 273 ��� modification: 6.3. 286 ��� 's formula: 6.2. 275 ��� transform: 6.2. 275 polarization: 3.6. 129 pole: 5.4. 230 ��� at 00: 5.6. 252 ��� branched: 10.2. 399 of order n: 5.4. 232, 10.2. 399 polydisc: 10.1. 390 polygon: a union n of finitely many complex intervals for some curve ,,( , n = "(* : 1.7. 39 polygonally connected: 1.7. 39 polynomial function: for some N in N, a map ���
6.3. 249
���
f : en
'3
Z H
L
l o: l :S:N
co: zO: ,
{[ak, a k + d L < k < n  l ;  
3.2. 99
Polynomial Runge: 7.1. 306, 12.1. 448 polynomially convex: 12.1. 450 polyregion: 12. 1. 466 P OMPEIU, D . : 5.8. 263 Pompeiu's Theorem: 5.8. 263 P ONTRJAGIN, 1 . : 4.9. 189, 11.2. 440 Pontrjagin's Duality Theorem: 4.9. 189, 11.2. 440 poset: 1.2. 10 positive definite: for a vector space V, of a map
(, ) that
: V 2 '3
{x , y } H (x, y ) E e,
(x, x) 2: 0 and (x, x) = 0 iff x = 0, 3.6. 128
482
Glossary /Index

degree: of a polynomial
L
l o: l �N
co: z O: , that min { I n l }
>
0, 5.6. 252
set: 4.1. 138 potential function: for a vector field E, a scalar function <1> such that E grad <1>, 6.3. 287 precompact: of a set S in a topological space, that the closure S is compact, 
=
1.6. 38
preFubinate: 4.4. 134 preserves angles: 8.1. 336, 8.1. 339 Prime Number Theorem: [Schap] , 5.9. principal part: 5.4. 239 Principle of the Argument: 5.4. 242 PRINGSHEIM, A . : 10.1. 384 Pringsheim's Theorem: 10.1. 384 probability theory: [Kol] , 2.2. 66, 11.1. product: 3.5. 116 curve: 5.5. 245  measure: 4.4. 151  rule: 4.7. 184 topology: 1.2. 9 projection: 1.2. 10, 3.6. 129, 10.2. 391 properly discontinuous: 8.4. 358 punctured disc: 5.4. 234 Pvariation: 4.6. 169
269
431


Q QUOTIENT ALGEBRA: for an algebra A and an algebra homomorphism ¢, the image ¢(A) regarded as an algebra by virtue of ¢; ¢(A) is the quotient Aj¢ l (O), 3.5. 124  norm: 3.5. 124  set: 2.2. 62  space: for a vector space X and a subspace Y, { x + Y x E X },
3.5. 123
R
RADIUS OF CONVERGENCE: 5.2. 205 RADON, J . : 4.2. 144 RadonNikodym derivative: 4.2. 144  Theorem: 4.2. 144 range: for a map f : X H Y, f(X), 1.2. rational numbers: 1.1. 3 rearrangement of series: 10.1. 384 rectifiable: 2.4. 79, 4.8. 185
21
483
Glossary /Index
recursively: [Me] , 5.3. 223 refinement: 1.5. 32 refines: 1.5. 32 reflection: 8.2. 344 reflexive: 3.4. 112 region: 1.1. 3, 1.2. 11, 5.1. 204 regular ideal: 3.5.124  measure (space) : 2.3. 78 point: 10.1. 384, 10.2. 399 relative topology: 1.2. 6 relatively closed, compact, open: 1.2. 17 remainder formulre: 2.4. 80 removable singularity: 5.4. 230 reproducing kernel: 8.3. 353 residue: 5.4. 239 Residue Theorem: 5.4. 239 restriction: 1.2. 11 reversed inclusion: 1.2. 10 RrcKART, C . E . : 3.5. 121 RIEMANN, B.: 1.2. 12, 2.2. 73, 4.6. 170, 4.8. 185, 5.6. 251, 10.2. 405 Riemann integrable: 2.2. 73 partition: 2.2. 73  ' s Mapping Theorem: 8.1. 336  sphere: 5.6. 251  Stieltjes integral: 4.6. 170  ' s zeta function ( : 5.9. 269  Stieltjes sum: 4.6. 170  surface: 1.2. 12, 10.2. 405 RIEsz, F . : 2.3. 76, 3.6. 128, 3.7. 135, 4.3. 150, 11.2. 440, 11.2. 442 F. Riesz's Theorems: 2.3. 76, 3.6. 128, 4.3. 150, 1 1.2. 440 RIEsz, M . : 4.9. 191, 11.2. 436 M. Riesz's Convexity Theorem: 4.9. 191, 11.2. 436 right adverse: 3.5. 118  open: 1.1. 5 . . IDlIR , 11· m f(b)  f(x) , w hen It · . · m. C eXIsts,  h and denvatlve: an f In lor x
3.2. 102
bx
 ideal: 3.5. 124  inverse: 3.5. 117  open interval: 1.1. 5 ring: a set R and two maps
a + b E R; { a, b} H a . b E R;
+ : R2 '3 { a, b} . : R 2 '3
H
484
Glossary /Index both + and · are associative; + is commutative; . is +distributive,
3
of sets: 2.2. 58 rotation: in [lR.n l e ' a T such that 
162
IIT(x ) I
invariant: 4.5. 162 ROUCHE, E . : 5.5. 244 Rouche's Theorem: 5.4. 244 RUNGE, C . : 7.1. 301, 7.1. 305307, Runge domain: 12.1. 450 's Theorem: 7.1. 305 variant: 7.1. 307 Running Water Lemma: 4.6. 172 IRvalued functions: 1.2. 8 JR.vector space: 3.1. 89
==
Il x ll (T is a linear isometry) , 4.5

12.1. 450

s
SAKS, S . : 4.6. 169 saturated: 10.2. 403 SCHMIDT, E . : 3.2. 98 SCHOTTKY, F . : 9.1. 370, 9.4. 376 Schottky's Theorem: 9.4. 376 SCHWARZ, H . A . : 3.2. 96, 3.7. 135, 6.2. 280 Schwarzian derivative: 8.2. 347 Schwarz inequality: 3.2. 96 Reflection Principle: 6.2. 278 's formula: 6.2. 280  's Lemma: 7.2. 322, 12.1. 447 second (Baire) category: 1.3. 18 second countable: 1.2. 7 SELBERG, A . : 5.9. 269 selfadjoint: 3.6. 130 semihomogeneous: 2.1. 47 seminorm: 3.3. 105 separable: 1.2. 7 separated: 1.2. 11 separating algebra: 3.5. 126  elements in a dual space: 3.3. 108 shared: 7. 1. 303 sheaf: 10.2. 393 sheets: 10.2. 395 shrinks nicely: [Rud] , 4.6. 169 sides: 7.1. 302 aalgebra: 2.2. 58 

1.1
485
Glossary /Index
 compact: of a subset 5 of a topological space, that 5 is the union of count ably many compact sets, 4.2. 146  finite: 4.2. 146  ring: 2.2. 58 signed measure (space): 4.1. 138 simple function: 2.2. 63 pole: 5.4. 241  zero: 5.4. 241 simplex: 1.4. 21 simply connected: 5.5. 246, 10.2. 401 singleton: of a set 5, the #(5) 1 , 2.2. 67 singular: of a linear map T between finitedimensional vector spaces, that T 1 (O) =j:. {O}, 4.7. 178  component: 4.2. 144  point: 5.4. 232, 10.1. 384 singularity: 5.4. 230 SORGENFREY, R. H . : 1.2. 6 somewhere dense: of a subset 5 of a topological space X, that for some neighborhood N, 5 n N ::) N: 3.3. 109 Sorgenfrey topology: 1.2. 6 span: ( noun ) : for a subset 5 of a vector space, the set of all (finite) linear combinations of elements in 5: 1.4. 22  ( verb): a subset 5 of a vector space V spans V iff span (5) = V: =
3.2. 97
Spectral Theorem: 3.6. 130 spectral radius: 3.5. 123 spectrum of an algebra: 3.5. 124    element x: 3.5. 120    operator in [SJ]c: 3.6. 131 SPERNER, E . : 1.4. 27 Sperner map: 1.4. 27  simplex: 1.4. 29  ' s Lemma: 1.4. 27 stalk: 10.2. 394 . . . , ei n ) standard basis: for en , the set ei . of vectors such l �"� n that e ij Jij , 4.7. 182 starshaped: 5.3. 226 STEINHAUS , H . : 3.3. 110 stepfunction: an ffi. linear combination of characteristic functions of halfopen intervals of the form [a, b): 2.2. 75 stereo graphic projection: 5.6. 252 STIELTJES, T. J . : 4.6. 149 Stieltjes integral: 4.6. 49 STOKES, G . G . : 5.8. 235
{ �f (ei 1 ,
=
}
486
Glossary/Index
Stokes's Theorem: [Lan,Spi] , 5.8. 233 STONE, M. H . : 2.2. 64, 3.5. 126, 3.7. 136 StoneCech compactification: 3.7. 136 Stone's Theorem: 2.2. 64 StoneWeierstraB Theorem: 3.5. 126 straight line through a: 8.1. 338 strictly separating: 3.5. 126 strong form of FTA : 5.4. 244 stronger topology: 1.2. 6 strongest topology: 1.2. 6 subadditive: of a functional !, that !(I + g) � !(I) + !(g ) , 2.2. 68 subharmonic: 6.3. 285, 10.2. 404, 11.1. 432  in Q: 6.3. 285   the wide sense: 6.3. 293 subordinate: 1.2. 16 subspace: in a vector space V, a subset W that is also a vector space,
3.3.
105 summable: 3.3. 111 superadditive: 2.1. 48; 4.1. 141 superharmonic: 6.3. 287, 10.2. 404 support: 1.2. 15 supporting line: for the graph G of y = ¢( x ) and a point P G, a line L through P and such that near P, L is below surjection: 1.2. 7 surjective: 1.2. 7 SZEGO, G . : 8.3. 354 Szego's kernel: 8.3. 354
�f (p, f (p) ) on G; 3.2. 102
T
THE CANTOR SET: 1.2. 13 theory of games: [NeuM] , 11.1. 431 thick: for a measure space (X, S, f.l ) , of a subset
S, f.l * (E \ Y) = 0, 4.5. 160 TIETZE, H . : 1.7. 41 Tietze's Extension Theorem: 1.7. 41 THORIN, G . 0 . : 11.1. 431, 11.2. 434 Thorin's Theorem: 11.1. 431 TONELLI, 1 . : 4.4. 156 Tonelli's Theorem: 4.4. 156 topological algebra: 6.4. 296
Y such that for every E in
487
Glossary/Index 
field: a field
OC in which the maps OC2 '3 { a, b} H a + b, OC2 '3 {a, b} H ab, OC \ {O} '3 a H a  I ,
are continuous with respect to the topology of OC, 6.4. 296 group: 1.6. 34, 4.9. 187 space: 1.2. 6 vector space: 3.1. 89 topologist's sine curve: 10.2. 402 topology: 1.2. 6 total variation: 4.6. 169 totally disconnected: of a set S in a topological space, that the only components of S are points, 10.3. 414 finite: 2.5. 85, 4.2. 144 afinite: 2.5. 86, 4.2. 146 transcendental: 9.1. 36 9 transition map: 10.2. 402 transitive: of a partial order <, that 



{{a < b} !\ { b < e}} ::::} {a
translate: 2.5. 84, 4.5. 163, 9.2. 371 translation: 4.5. 162 invariant: 2.4. 78, 4.5. 159  / scaling: 3.3. 94 triangle inequality: 3.2. 96 trigonometric functions: 2.4. 79 trivial topology: 1.2. 6 TYCHONOV, A . : 1.2. 13 Tychonov's Theorem: 1.2. 13 type: 7.3. 324 u
UINDUCED NEIGHBORHOOD: 1.6. 35 Uinduced uniform topology: 1.6. 35 ultrafilter: 1.5. 32 unbounded near zero: 5.4. 230 uniform space: 1.6. 34 topology: 3.4. 111 uniformity: 1.6. 34 uniformization: 10.2. 398 Uniformization Theorem: 10.5. 422
<
e} , 1.2. 10
488
Glossary /Index
uniformly continuous: 1.6. 35 unit circle: 2.4. 81 unitary: 3.6. 131 univalent: of a map, that it is injective, universal covering space: 10.3. 411 unshared: 7.1. 303 upper semicontinuous: 1.2. 9 URYSOHN, P . : 1.2. 15 Urysohn's Lemma: 1.2. 15
8.1. 336
v
VALUES: 10.2. 392 vector: a map r '3 "I H x"! E X: 1.1. 5 space: an abelian group V over a field OC: 2.1. 44 homomorphism: 3.3. 103 vertex: 1.4. 21, 4.8. 186, 7.1. 304 vicinity: 1.6. 34 VITALI, G . : 2.3. 78, 4.6. 169, 5.3. 206 VitaliCaratheodory (Theorem) : 2.3. 78 Vitali's Covering Theorem: If I is a set of ndimensional intervals, and for each positive E and each x in a set E, there is in I an I such that x E I and ).. n (I) < E, then if 15 > 0 and )"�(E) < 00 , for some finite set 11 , . . . , !rn of pairwise disjoint elements of I, ).. � E \ < 15.


(
4.6. 169
 Theorem: 5.3. 228
U ;=Jj)
w
WEAK, WEAK': 3.4. 112 weaker, weakest: 1.2. 6 wedge product: 5.8. 261 WEIERSTRASS, K . : 3.5. 126, 5.4. 230, 7.1. 310, 7.1. 312, 8.6. 366 WeierstraB Approximation Theorem, 3.5. 126 elliptic function: 8.6. 366 product representation: 7.1. 312 WeierstraBCasorati Theorem: 5.4. 230 wellbehaved: 4.8. 186 ordered: of an ordered set X, that each subset has a unique min imal element, 1.1. 5 Wellordering Axiom: Every set X may be wellordered, i.e. , there is an order < such that for any two elements x and y of X, x < y or y < x and for any subset Z of X, there is in Z a (unique) z such that for any other element z' of Z, z < z' (every subset has a least element). 1.5.

32
489
Glossary/Index WEYL, H . : 10.2. 395 winding number: 5.2. 208 Wstructure: 10.2. 391 x
xFREE: independent (free) of x, 1.6. 37 Xholomorphic, Xmeromorphic: 10.2. 404 xneighborhood: 1.7. 39 xsection: 4.4. 153 y
YOUNG, G . C . : 4.6. 177 YOUNG, W . H . : 4. � . 190,
11.2. 386, 11.2. 442 z
ZAREMBA, S . : 8.1. 342 ZERMELO, E . : 4.5. 160 ZermeloF'raenkel system of axioms: [Me] , 4.5 160 zero: 5.4. 231 of order or multiplicity n o : 5.4. 232 Z ORN, M . : 1.2. 13, 1.5. 32 Zorn's Lemma: In a poset (r, < ) , if every ordered subset has an upper bound in r, for each "I in r, there is in r a maximal element f.l such that "I < f.l, 1.5. 32