Selected Papers

Proof. It is sufficient to prove the second assertion of the theorem, since this implies the first. Let the Banach space V consist of all continuous complexvalued functions h on e, with the norm II h II = sup (I h(z) II z £ e}, and let V* be the dual of V. Define VI and V~ similarly. It is well-known that V* consists of all finite complex-valued Borel measures on e. Let the meUf->ure J.l.n on e be defined by J.l.nCS) = Jsn~n g(z) dz, for all Borel subsets S of e. Then I h(z) dJ.l.n(z) = J~" h(z)g(z) dz - t J h(z) df..!(z) as n - t OJ, for every continuous function h on e. This means that (J.l.n} converges to f..! in the weak star topology of V*. It is known that this implies that II f..!n II is bounded. ~ ow define the measure Tn = rp(f..!n) on e, , so that J k(z) dTn(z) = J k(
e, .

59

THE STRUCTURE OF CERTAI:S MEASURES

ThEOREM

3.

The meaSllre

p.

285

is represented by g(z) dz.

Proof Let Zo be any point in U. Let 'I' be the conformal map of I z I < 1 .onto U for which '1'(0) = Zo and '1" (0) > O. Let {J n I be a sequence of simple dosed curves containing C ill their interiors and converging to B Let 'l'n be the conformal map of I z I < 1 onto the interior of J" for which '1'.,(0) = "'0 and ¥,~(O) > O. Then, by a theorem in Caratheodory [3), 'l'n converges to '1', as n - 7 co, uniformly 011 all compact subsets of I z I < 1. For each n we define the differential fn(z) dz = 'I'~'(g(Z) dz) = g('I',,(z» d'l'"(z) on 'I',~'(U). We also define fez) dz = ",-l(g(z)dz) = g('I'(z»d'l'(z) on I z I < 1, so that 'I'(f(z)dz) = g(z)dz It is clear that fn converges to f, as n -7 co, uniformly on all compact subsets of I z I < l. Let v" be the measure 'I'~'(p.) so that II 1'" il = II p. II We may therefore assume, by passing to a subsequence if necessary, that {v" I converges in the weak star topology to some measure v on I z I = 1. For each I z I < 1 and each n sufficiently large, we therefore have f (z) n

=

(

g '1'"

(z» d'l',,(z) = _1 dz 21T'i

J t -dp.Ct)'I',,(z) d'l',,(z) = -~ J dl'"(s) d'l'"Cz) . dz 21T'i 'l'nW - 'I'"(z) dz

Now the function aCt) = l/['I'"(S) - 'l'n(Z)] d'l',,(z)/dz - l/(t - z) of t is analytic in I r I < 1. Hence a('I'~l(S) is analytic interior to J" , and can therefore be uniformly approximated on C by polynomials Since p. is orthogonal to all polynomials, it follows that f a(S) dl'" (S) = f a ('I': I (S) dp.(S) = O. Combined with the abO\'e equality, this gives fn(z) = (l/21T'i) J dl',,(S)/(t - z). Taking the limit as n - 7 co, this gives fez) = (l/21T'i) f dv(S)/(t - z). Since, for each n, Pn = 'I'~' (p.) is orthogonal to all polynomials, the measure v is orthogonal to all polynomials. By the theorem of F. and 1\1. Riesz, it follows that fez) dz represents the measure v. By Theorem 2, it follows that the differential 'I'(f(z)dz) = g(z) dz represents some measure X on B. By the corollary to Theorem 1, X = ).I.. This completes the proof.

3 The structure of jJ.. Throughout this section, 'I' will denote a fixed conformal map of I z I < 1 onto U, and fez) dz will denote the differential 'I' -'(g(z) dz) = g('I'(z» d'l'(z). Since we have shown in Theorem ;) that g(z) dz represents jJ., it follows from Theorem 2 that f(z) dz represents some measure v on I z I = l. By the theorem of F. and M. Hiesz, fez) dz represents v in the sense of F. and M. Riesz, the measure v is continuous with respect to Lebesgue measure on I z I = I, and there exists an integrable Borel measurable function u on I z I = 1 such that properties (i), (ii), and (iii) of §1 obtain. The purpose of the present section is to investigate in more detail the relationship between the measures ).I. and P. LEMMA 1. There exists a Borel measurable function v on I z I = 1, with values in the set of essential boundary points of V, and a Borel set M consisting of almost all points of I z I = 1 relative to Lebesgue measure, such that'l'(z) converges to v(zo) whenever z converges non-tangentially to any point Zo of M. There exists a Borel

60

ERRETT BISHOP

measurable function w on B, with values in almost all z with I z I = 1.

Iz I =

1, such that w (v (z »

= z for

Proof. The bounded function tp assumes non-tangential boundary values on some Borel set M consisting of almost all points of I z I = 1, by a theorem of Fatou, to be found in Bieberbach [2]. The non-tangential boundary values of tp will define a Borel measurable function v on M, whose values obyiously lie in the set of accessible boundary points of U. If we define v on the rel'lt of I z I = 1 to have some fixed accessible boundary point of U as its value, then v will obviously have the required properties Also, v will be one-to-one on l~f, because distinct points of M correspond to distinct accessible boundary points (see Carath~odory [;)]). By Lusin's theorem, there exists a sequence I Tnl of closed subsets of M, such that v is continuous on T" and such that almost all points of I z I = 1 relative to Lebesgue measure are in U T". Since v is one-toTherefore 11 is a homeomorphism on one on 111, it is one-to-one on each Tn T. for each n. It follows that the function w defined on 11 (U Tn) so that w(v(z» = z for all z in U Tn is a homeomorphism on each veT,,). Therefore w is a Borel function. If we define w to have the value 1 on B - v (U Tn), then w will have the desired property. This completes the proof. Since the differential dz on I z I < 1 represents the measure Vo = dz on I z = 1, it follows by Theorem 2 that the differential tp(dz) = d
Proof of (i). The statement (i) is equivalent to the statement that dp.(z) = J h(v(z» dv(z) for all continuous functions h on C. Let Ir.} be any sequence of positive numbers smaller than 1 which converge to 1. Since fez) dz represents JI in the sense of F. and M. Riesz, it follows that 11<1- •• k(z)f(z) dz - J k(z) d/l(z) for all continuous functions k on I z I ~ 1. By Theorem 2, there exists a subsequence Is,,) of {rn) and a measure A on B such that f!. h(z)g(z) dz -+ J h(z) dA(Z) for eyery continuous function h on C, where On is the curve
J h(z)

f

h(z) dp.(z) =

J h(z) dA(Z) = lim Lh(z)g(z) dz = lim

r

h(tp(z»f(z) dz

= lim

1 2

0

Jlzl=sn

61

,..

h(cp(Sne;9»f(sne;9) dS,/9.

281

THE 5TRUCTCRE OF CERTAIN MEASURES

1\ow h(rp(s"e;"» is bounded uniformly in nand 0, and as n ---> co it converges for almo~t all 0 to h(v(e;8». Also f(sne;o) converges as n -> co in LI mean to u(e'o) Therefore h(rp(sne;6»f(s,/8) converges as n -> co in LI mean to hCvCe'"»u(e i6 ). It follows that the above limit equals

{r h(v(e,6»u(c;8) de;8 = L'-I h(v(z»u(z) dz = Jh(vCz»

d1l(z),

as \\a<; to he pro\'cd. Proof of (il) Since w 01' ie.; the idC'ntity function almost e\'erywhere on I z I = 1, we have weu) = w(\(1I» = (w 0/')(11) = II, as was to be proved Proof of (iii). Let 8 be any Borel subset of B. By (i) abO\'e, \ye have J.I.(8) = 11(1,-1(8» = I,-.(s) 11(Z) dz. :\o\\" (i) above applies to the measure J.l.o as well as to the nwac.;ure J.I., so that J.l.o = v(v o ) It follows that

r

u(w(z» d/-Lo(z)

=

f~,,,,

u(w(v(z») d1lo(z)

=

f~.(q

u(z) dz,

since w(u(z» = Z for almo~t all z of I z I = 1. Comparing the equations just obtained, we get jJ.(8) = Is u(w(z» djJ.u(z) , as was to be proved. 4. Applications. W{' no\\" use the theorems of the previolls seC'tions to gi\'e a solution to the Dirichlet problem on C. We continue to use the previous notation LEMMA

2.

If the measure

jJ.

considered above is real-valued, then J.I. = O.

Proof. Since jJ. is real-valued, the measure II = w(JI.) will be real-valued. Since II is orthogonal to all polynomials, it will be orthogonal to the real parts of all polynomials. Since any continuous function on I z I = 1, by Feier's theorem (see Zygmund [5]), is a uniform limit of real parts of polynomials, we see that II is orthogonal to all continuous functions, and therefore II = O. Since jJ. = V(II), it follO\,"s that J.I. = 0, as was to be proved. THEOREM 5 If h is any real-valued continuous function on B, there ext'sis a continuous extension of h 10 C, which is uniformly approximable on C by the real parts of polynomials, and therefore harmonic on U.

Proof L('t K denote the set of all continuous real-\'alucd functions on B which are uniform limits on B of the real parts of polynomials If h t K, then there exists a non-trivial real-valued measure J.I. on B such that I h dJl. ~ 0, and J1. is orthogonal to K It follo,,"s that jJ. is orthogonal to all polynomial;;;. By Lplllma 2, J1. = O. This contradiction shows that h t K, so that h is a uniform limit 011 B of a sequence of r(':11 parts of polynomials. Since the real part of a polynomial assumes its maximum for C at a point of B, the sequence \\'il! con\'erge ulliformly Oil C, and the limit function will give the desired extension of h. The next theorem is a genernlization of a theorem of Wenner [3J. The proof

62

J<;HHJ<;'l''l' IH:"HUl'

given here is modeled on a proof of Wenner's original theorem whiC'h is due to K. Hoffman. THEOREM 6. Let T be the Banach algebra of all comple.r contirmmlS functions on B. Let R be the subalgebra of T consisting of all functions in T which are uniform limits of polynomials. Let Q be any closed subalgebra of T with R C Q C T. Then either Q = R or Q = T.

Proof. It is rlear that every fum·tion ill R ran 1)(' extended to he continuous on C and analytic interior to C A"sume that this has lWE'1l dOlle Then for each w in U we have the homomorphism cp" of R into the complex Ilumiwrs defined by
o = (~<, + ~~2)(Zhl(Z) ax

ay

=

QtlJz2 + i ax

ahl(z}

ay

These are jUfit the Cauchy-Hiemann equations for hI' Therefore hI is analytiC' on U. It follows that h is in It, by a theorem of }Iergelyan [-!J Thus we see that h l: R for all h in Q, so that Q = R. We have left the case to consider in which there existsw in U for which ip", is not extendable to Q. Then the maximal ideal in R consisting of all h in R with hew) =
63

THE STRUCTl:RE OF CERTAIN MEASURES

289

structure of a measure JJ. on B which is orthogonal to all polynomials can be obtained as follows. We again define the function 9 on V by g(z) = (1/27ri) f dJL(n/(!; - z). Let Vi be any component of V, and choose Zo £ Vi . Let the curves I J n I converging to B and the mapping functions {CPn I be chosen as in the proof of Theorem:3 Then {CPn I will converge to the mapping fUllction cP of I z I < 1 onto Vi , uniformly on compact subsets of I z I < 1, by the theorem of Caratheodory quoted above. It then follows, exactly as in the proof of Theorem 3, that the differential fez) dz = cp-l(g(Z) dz) on I z I < 1 represents a measure v on I z I = 1. Thus, by Theorem 2, the differential g(z) dz representliJ a measure JLi on the boundary of Vi' It readily follows that the measure ).I 2:~~1 ).I, on B is orthogonal to all functions on B of the form h(z) = 1/(z - zo), where Zo is any point not in B. By applying the theorem of Mergelyan, ,ve see that therefore JL = L';:~l JL,. This fully describes the structure of JJ. in this case. In case U has an infinite sequence {Vi I of components, the same proof shows that the differential 9(z) dz on V, represents some measure JLi on the boundary of Vi , but it is not clear that the series 2:~~1 JLi need converge in any of the standard topologies for measures The fact that this series actually converges to ).I in the norm topology for measures requires a more refined analysis of the problem, which will be given in a later paper. REFERENCES

1. E BISHOP, On measures orthogonal to polynomials, Proceedings of the National Academy of Sciences, USA, vol 44(1958), No.3 2 L. BIEBERBACH, Lehrbuch der Funktionentheorie, vol. 2, Leipzig, 1930. 3. C. CARATHEODORY, Conformal Representation, Cambridge, 1952. 4. S. N. MEIWELYAN, On the repreoentation of functions by series of polynomials on closed sets, American :\lathematical Society Translation, No 85, 1953 5. J. WERMER, On algebras of continuous f1mctions, Proceedings of the American Mathematical Society, vol 4(1953), pp 866-869 6. A. ZYGMUND, 7'rigonometrical Series, ~ew York, 1955. UNIVERSITY OF CALIFORNIA BERKELEY

64

Volume 9, pp. 946-953, by permission of the American Mathematical SOciety.

APPROXIMATION BY A POLYNOMIAL AND ITS DERIVATIVES ON CERTAIN CLOSED SETS ERRETT BISHOP The work on the theory of approximations initiated by Weierstrass and continued by Walsh, Keldysh, and Lavrentiev, among others, has culminated in the following theorem of Mergelyan (See Mergelyan [3]): Given any compact subset C of the complex plane, which does not separate the plane, and given any continuous function] on C which is analytic interior to C, then] can be approximated uniformly on C by polynomials. This theorem leaves the following question unanswered: If ]0, ]1, ... , ]n are continuous functions on C, can a sequence {Pk of polynomials be found with the property that for each integer i with ~ i ~ n the sequence {p~ll, where p~1 denotes the ith derivative of h, converges uniformly on C to]i? If C is totally disconnected, it is easy to show that the answer to this question is always yes. We omit the simple proof, because a more general result will be given elsewhere. If C is a Jordan arc, the question becomes more complicated. It is clear that if C has a rectifiable sub-arc J, whose endpoints we call Zo and Zi, then for the approximation to be possible it is necessary that !d>+l(Z)dz=]i(Zl)-]i(ZO) for O~i~n-l. Thus, if the approximation is to be possible whatever the functions ]0, ]1, ... , in, it is necessary that C have no rectifiable sub-arcs. Conversely, if C is a Jordan arc having no rectifiable sub-arcs, we conjecture that the approximation is always possible. It is the purpose of this paper to prove this conjecture by means of an additional hypothesis, that C satisfy a Lipschitz condition of a fixed order c at a dense set of points. (This concept will be defined below.) The author has been unable to prove the conjecture without this restriction. If Sl and S2 are any subsets of the complex plane, define d(Sl, S2) =min {\Zl-Z2\\ZlES1, ZzEsd. DEFINITION 1. Let cf> be a homeomorphic map of [0, 1] into the complex plane, so that cf>[0, 1] is a Jordan arc C. We say that C satisfies a Lipschitz condition of order c at a point cf>(t) of C, tE [0, 1], ifthere exist A >0 and I> > 0 such that max {d(cf> [0, t], z), d(cf> [t, 1], z) } ~A\cf>(t)-z\·whenever \cf>(t)-z\ . Then we have

I

°

THEOREM 1. I] C has no rectifiable sub-arcs and if there exists c > 0 such that C satisfies a Lipschitz condition of order c at a dense set S oJ Received by the editors April 23, 1957, and, in revised form, May 3, 1958.

946

65

947

APPROXIMATION BY A POLYNOMIAL

points, then for any continuous functions fo, ... ,fn on C there exists a sequence {Pi} of polynomials for which P~tl-'>fk uniformly on C as i-'> 00 , for ~ k ~ n.

°

The proof of Theorem 1 will utilize the Riesz representation theorem (see Banach [1, p. 60 X is the set of all n+1 tuples (fo, . . . , fn) of continuous functions on C, topologized by the norm (fo, . . . , fn) = sup { Ifi(z) zE c, ~ i ~ n}, then by the Riesz theorem we see that to any bounded linear functional L on X correspond unique measures Mo,···, Mn on [0, 1] such that L(fo,···, fn) = ftfo((t»dMo(t) f~fn((t»dMn(t). Now let Y be the subset of X consisting of all (p, p[l), ... , p[nJ), where P is any polynomial. Then Theorem 1 states that Y is dense in X. By the Hahn-Banach theorem (see Banach [1 D, this is equivalent to saying that every bounded linear functional on X which vanishes on Y vanishes on X. If L is the bounded linear functional in question, then by the above representation of L we see that L(p, ... , p[n J) = f~P((t»dMo(t) f~p[nJ ((t) )dMn(t) = for all polynomials p. To prove Theorem 1 we must show that this implies that Mo=M1= . . . =Mn=O. The linear functional L and therefore Mo, . . . , Mn will be fixed during the discussion. As an abbreviation we set Lt(P) = f~P((t»dMo(t) f~p[nJ ((t) )dMn(t) for all t in [0, 1] and all functions p analytic in some neighborhood of C. Then L 1 (P) =0. Assume now that (t) ES. We proceed to obtain a new formula for Lt(p). To do this, take A and 0 as in Definition 1, and let E be positive and less than o. Let J be the circle z -(t) = E. Then by Definition 1, for z in J either d(z, U1»AE c or d(z, U2»AE c , where U1= [0, t] and U2 = [t, 1]. Thus we can write J = J 1U J 2, where J 1 and J 2 are disjoint Borel sets and d(z, Ui) > A eo for z in J;. Now if P is any polynomial, let K=max {lp(z)llzEJ}. Then

D.H

II

II

°

II

+ ... +

°

+ ... +

+ ... +

I

p(z) for

Iz -(t) I < E.

=

If for i

Ji(Z)

=

(l/27ri) =

I

1

p(f)df / (f - z)

1 and 2 we define

(1/27ri) l/(f)df/(f - z),

then we see that fi is analytic on the complement of the closure of J i, that If,ltl(z) I ~Kjld(z,Ji)-Ci+l), and that P(z)=h(z)+f2(Z) for Iz-(t) <e. Thus we see that IfJJI(z) [0, t] and O2 a neighborhood of [t, 1] such that 01 (\0 2 = {zll z -(t) < e} and 01(\J1 = 02(\J2 =. Then f1 +f2 = p on 01(\0 2 and fi is analytic in 0,. Therefore we may

I

I I

66

J<:RRJ<:'IT BISHUP

define an analytic function gl on 01U02 by specifying gl(Z) =!t(z) for z in 01 and gl(Z) = P(z) -j2(Z) for z in O2, Also define g2 on 0 1U02 by g2(Z) =h(z) for z in O2 and g2(Z) = p(z) - jl(Z) for z in 01, Then gl +g2 = P in 01U02. Also Ig~l(z)1 = Ijl1(z) I ~Kj![AEc]-(i+l) for z in Ui, where i = 1 or 2 and j is arbitrary. For i = 1 this inequality, in conjunction with the definition of L 1(gl), tells us that Lt(gl)j ~MIKE-c(n+l), where Ml is a constant. For i = 2 the inequality tells us that L(g2) -L t (g2) 1 ~ M 2KE- c(n+l). Thus we see that j L,(p) I = I L,(gt} +L 1(g2) j ~ I L1(gl) I I L t (g2) 1 = I L1(gl) I 1L 1(g2) - L(g2) I ~ MKE- m , where M = Ml +M2 and m is any integer larger than c(n+l). Taking p to be the polynomial (z-c/>(t))i, we see that K =Ei, so that Lt([z-c/>(t) ]i) <ME i - m for all E<8. If j>m, this implies that Lt([z-c/>(t) ]i) =0. Therefore L1(p) depends only on the first m 1 terms of the expansion of p in powers on z-c/>(t), so that

I

+

I

+

+

m

Lt(p) =

L: {3i(t)P !il (cfJ(t)) ,

where (3o(t),"', (3m(t)

;=0

are certain complex numbers. For the remainder of the proof, the only use which will be made of the fact that C satisfies a Lipschitz condition of order c at points of S will be to conclude that the expression just obtained for L1(p) is valid whenever c/>(t) is in S. Therefore, the conjecture of the introductory paragraphs can be proved whenever the expression just obtained for L1(p) can be shown to be valid for a set of values of t which is dense in [0,1]. We now obtain another formula for L1(p), where now t may be any point in (0, 1). For any polynomial p and any complex number z we have the Taylor's formula

p(z) =

t

p!i1(cfJ(t)) [z - .;(t)];

i=O

Thus

L1(p)

=

Ei E

=

t

1.

p!Hil(cfJ(t)) [cfJ(x)

I {

p!il(cfJ(t))

(± f ;=0

i=O

t

~ cfJ(t)]} d,uj(x)

[cfJ(X).- cfJ?;]H dJ.£i(X)), (1 - J).

0

where /J-j=O if j>n. If we define i

ai(t) =

L: ;_0

f

I

0

[cfJ(x) - cfJ(t)]H

..

(1 - J)!

67

dJ.£;(X),

APPROXIMATION BY A POLYNOMIAL

949

L:.o

for t in (0, 1), we therefore have L,(p) = pI.1(cf>(t))ai(t). We see that a. is continuous on the right. Comparing the two formulas ob~ tained for L,(P), we see that 'L':'..o(3.(t)pl i l(cf>(t») = 'L;:'o a.(t)plil(rfJ(t) for all t in (0, 1) with rfJ(t)ES, and for all polynomials p. It follows that ai(t) =0 for i>m, tECO, 1), and cf>(t)ES. Since S is dense and since a. is continuous on the right, it follows that a. = 0 for i > m. Be~ fore proceeding, we need a definition. DEFINITION 2. If f and g are two complex valued functions on [0,1], if tE [0,1), and if a is a complex number, then a is said to be a right conditional derivative at the point t of f with respect to g if there exists a sequence {t.}, with tiECt, 1), t.~t as i~oo, and

[((Ii) - f(t)] [get,) - get) ]-1 ----+ a as i ----+ 00. To get more information about ai, take 0
= J.'i [0,

{[ai(t) - Ai(t)] - [ai(to) - A,(to)] }[cf>(t) - cf>(to) ]-1

f'

i-I

=

{

L

;_0

i=O

10

i-lI'O +L f

J .

~ -

0

_EI - L i-O

[4>(x) - 4>(t)]H C ')1 dJJ.j(x) [cf>(X).(~

0

4>~to)]H dJJ.j(X)} [4>(t)

[4>(x) - cf>(t)];-; - [cf>(X) - 4>(to)]H

. J)! .

(~

0

..

(~

'0<"':5 1

-

[4>(t) - 4>(/0) ]-ldJJ.j(x)

-

[cf>(x) - cf>(t)]H

i-I

i-O

- 4>(tO)]-1

J)!

-

[cf>(t) - 4>(/0) ]-ldJJ.j(x).

J)!

The second of these summations less than i- 1

f

2· (2B)H-l

I dJJ.j(x) I in absolute value, if B is the bound of Icf>(x) I and if t is chosen so that IcJ>(t) -ep(to) I ~ 2- 1ep(x) -cJ>(t) I for all x in [to, t J. The latter condition will be satisfied if ep(t) -ep(to) I ~ Icf>(x) -cf>(lo) I for all x in [to, t], and L

i-O

..

'0<"':5 1

(~- J) I

1

values of t can be found arbitrarily close to to for which this will be true. Thus the second summation can be made arbitrarily small for certain values of t close to to. Due to uniform convergence under the integral signs, the other of the above summations approaches i-I

Li-O

f

10 0

[cJ>(x) - cf>(/o) Ji-i-l

(..J $ -

1)!

dILi(X) = - ai-l(to) as t ----+ to.

Thus we see that ai-l(to) is a right conditional derivative at to of cf>, for j ~ 1 and all to in (0, 1).

X/-al with respect to

68

Since aj=O for j>m and since Aj=O for j>n, if m>n we see that a".(t), which is a right conditional derivative at t of Am+l-a".+l =0 with respect to cP, must vanish for t in (0, 1). Thus a". =0. The argument can then be continued to show step by step that ai=O for i~n. Therefore an-let) is a right conditional derivative at the point t of A,,-an=An with respect to cp, for all t in (0, O. If an_l(t)r£O, this implies that [an-let) ]-1 is a right conditional derivative at t of cP with respect to An. Since an-l is continuous on the right, we can find u > t and r>O such that an-l (x) >r for x in [t, u). Therefore for x in [t, u) we see that lan_l(x)l-l
I

I

I

:'1

I

I
I


I

< ,-11 AnCW) -

An(X')

I.

Thus we have

I
-
<

I

~

\
-
,-1 \ AnCw) - AnCx') \

+ \
-
+ ,-1 IxX! I dAn \

I ~ ,-1 Ix I dAn \ • W

Therefore wE T so that [x, w] C T. Thus [x, w] is a neighborhood of x' in [x, y]. Hence T is both open and closed in [x, y]. Since xET, T= [x, y]. Therefore yET, so that IcP(y)-Cx)1 ~,-11~ldAnl for all x and y in [t, u). Therefore has bounded variation on [t, u], so that [t, u] is a rectifiable sub-arc of C. This contradicts the hypothesis. This contradiction shows that an_let) =0 for all t in CO, 1), so that an-l =0. Having proved this, we can use the same argument to show step by step that a n -2=a n -S= ... =ao=O. But ao(t) = f odJLo(t). Thus JLo vanishes on all subsets of [0, 1). Since there is inherent symmetry between the endpoints, JLo=O. Then

so that JLl = O. Thus we show step-by-step that JLo = JLl = . . . This completes the proof of Theorem 1.

69

= JLn = O.

951

APPROXIMATION BY A POLYNOMIAL

There exists Jordan arcs for which condition (2) is fulfilled. For instance, a Jordan arc which has no rectifiable sub-arcs and which has a tangent at a dense set of points will do, because the existence of a tangent implies that a Lipschitz condition of order 1 is fulfilled. To see this, assume that C has a tangent at ¢(to). By this we mean that the parameter t can be so chosen that ¢'(to) exists and is not zero. Now if C does not satisfy a Lipschitz condition of order 1 at ¢(to), then for each 0 >0 there exists z with ¢(to) <0 such that d(¢[O, to], z)
1 -zl z)
1¢(t

1)

z

-

1 < 1¢(to)

- zI

/4

and 1¢(t2)-zl <1¢(to)-zl/4, so that 1¢(t 1 )-¢(t 2)1 Also, 1

<1¢(to)-zl/2.

¢(to) - z 1

~

1¢(to)

- ¢(t 1)

1 + 1(t

1)

-

z

1< 1¢(to)

- ¢(/l)

1+ 1¢(to) -

z 1/4,

so that 31¢(to)-zl/4<1¢(to)-¢(t1 )1. Thus,

I ¢(tl)

- ¢(t2)

1 < 21 ¢(to)

- ¢(t 1)

1/3.

It follows that 'Y

=

1¢(/l)

1

- ¢(/ 2) [ \ 11 - t 2 \-I[ ¢(to) - ¢(t 1) \

1to -

t 1 1- 1]-1

< 2/3.

On the other hand, as 0-'>0 the quantity 'Y converges to 1 ¢' (to) \ [¢' (to) [-I = 1. This contradiction shows that C satisfies a Lipschitz condition of order 1 at ¢(Io). To construct a Jordan arc which has no rectifiable sub-arcs and which has a tangent at a dense set of points, let f be any continuous real function on [0, 1] such that l' exists at a dense set S of points and such that in any sub-interval the set of points where l' does not exist has positive measure. Let ¢Ct) =t+1f(t), so that ¢ [0, 1] = C is a Jordan arc having a tangent at the dense set of points ¢(S). Also C has no rectifiable sub-arcs, because if ¢ [t, u] were a rectifiable sub-arc then ¢ would be of bounded variation on [t, u], which would imply that l' would exist almost everywhere in [t, u], contrary to the condition on f. I t only remains to construct the function f. The standard techniques for the construction of nondifferentiable functions can be used (see Hobson [2]). Let {rn} =s be a sequence of irrational numbers which is dense in [0, 1]. It is easy to construct inductively a sequence {C n } of countable subsets of [0, 1], each consisting of rational numbers and each containing 0 and 1, such that the accumulation points of Cn are exactly rl, . . . ,rn> such that the distance

70

J:!..l\..l\.,c 1 1

IU;:,nVr

of two consecutive points t1 and t2 of Cn is not larger than 6-"d, where d is the distance between the sets {ttt t2} and {rl' ... , r n} , and such

that DnCC,,+I. Here Dn=CnUC n, where Cn consists of all points which lie midway between two consecutive points of C n • Thus D" consists of rational numbers, and therefore is disjoint from the sequence s. Define the function fn on [0, 1] as follows. Let fn(t) = if t E Cr. U {rI' ... , r,,}. For consecutive points tl and t2 in C", let fnCt) = 3 n(t - t1) if t1 ~ t ~ (i1 +t2) /2 and fn(t) = 3"(t2 - t) if (t1 +t2) /2 ~ t ~t2' Since any point of [0,1] which is not in c"uh, .. " rn} lies between two consecutive points of Cn, this defines fn uniquely at all points of [0, 1 J. The functionf" is clearly continuous, except possibly at the points r1, ... , rr" and If..' (t) exists and equals 3 n if t is not in the set DnU{rtt"" rn}. Also, if U1 and U2 are consecutive points of D", it is clear that If,,(u2) - fn(U1) (U2 -Ul)-l = 3". To investigate the behavior of fn at the point ri, where i~n, let t be any point of [0, 1] - {r1' ... , rn}. There exist consecutive points t1 and t2 of C" with t1 ~ t ~t2' By the definition offn' we have Ifn(t) I ~ 3 n(t2 -t1)' By the construction of Cn. we have t2-t1~6-nl t-riI2. Thus, Ifn(t) -fn(r;) =If,,U)1 ~2-nlt-riI2 for all t in [0,1]. In particular, we see thatf..' (r;) =0, so thatf,.' (t) exists if t is not in Dr., and thatf" is continuous at rio Therefore, fn is continuous on [0, 1]. We also see that ~fn(t) ~ 2- n for all t. The series L:=l fn therefore converges uniformly on [0, 1] to a continuous function J. It will be shown that 1'(t) exists if and only if t is a member of the sequence S. If t =r; is a member of s, letf=g+h, where g= L~:11fn and h= L:-dn. Then g'(r;) exists because r, is not in Dn for all n. Also,

°

I

I

I

°

00

00

Ih(u)-h(r.)1 ~Llfn(u)-fn(r;)1 ~L2-nlu-riI2~

lu-r.12

n=i

for all u, so that hi (r i) exists and is zero. Therefore l' (r ,) exists. Now assume that t is not an element of the sequence s. Let n be arbitrary. Since t is not an accumulation point of Cn, there exist consecutive points U1 and U2 in Dn with UI~t~U2. Thenf(uz) -f(Ul) = L~-lfk(UZ) -fk(Ul) because fk(U) =0 if uED nCC n+1 and k ;;;n+l. Thus

If(uz)

I (U2 -

U1)-1

- fn(U1)

I (U2 -

- f(U1)

;;; Ifn(Ul)

n-l

U1)-I - L k-I

n

;;; 3 n

-

L

3k

;;;

n.

k-I

71

Ifk(U2)

I

- A(UI) (U2 -

UI)-I

APPROXIMATION BY A POLYNOMIAL

953

are not bounded. It follows that f'(t) does not exist, as was to be proved. REFERENCES

1. S. Banach, Theorie des operations lineaires, Warsaw, 1932. 2. E. W. Hobson, The theory of functions of a real variable, vol 2, Cambridge, 1926. 3. S. N. Mergelyan, Uniform approximations to functions of a complex variable, Amer. Math Soc. Translation no. 101, Providence, 1954. UNIVERSITY OF CALIFORNIA, BERKELEY

,,)

A DUALITY THEOREM FOR AN ARBITRARY OPERATOR ERRETT BISHOP

L Introduction. This paper studies the spectral theory of an operator (or bounded linear transformation) on a Banach space B. Although it would be possible to modify the results to apply to operators on non-reflexive spaces, for simplicity only reflexive spaces are considered. The distinctive feature of the treatment is that we are primarily interested in the spectral properties of a completely arbitrary operator. That is, we seek a spectral theory which will be valid independently of any of the usual restrictions (such as normality or complete continuity). It is, of course, not to be expected, in view of many known counter examples, that such a theory will even approach in power the spectral theory of a Hermitian or normal operator on a Hilbert space. In fact, it is surprising that a spectral theory for an arbitrary operator exists at alL The results obtained here are incomplete, but it seems likely that any spectral theory which is valid for an arbitrary operator will be closely related to the theory developed here. It is interesting that certain known results in spectral theory, which imply the spectral theorem for a Hermitian or a unitary operator, can be obtained from the theory which we develop for an arbitrary operator. The principal ideas of this paper are developments of ideas which appeared in rudimentary form in unpublished portions of the author's University of Chicago doctoral dissertation. The program of the paper is as follows. Instead of considering one type of spectral theory, we consider four. Since the content of these theories is that a certain duality exists between an operator and its adjoint, and since the term "spectral theory" is somewhat deceptive, we shall employ the term "duality theory" in its stead. The duality theories are not all of comparable strength. The weakest, type 4, will be shown to be valid for an arbitrary operator. This will be the principal result. This having been shown, it will then be possible to show that certain of the stronger and more conventional duality theories are satisfied provided that the operator is subject to appropriate restrictions. The duality theories will concern the existence and properties of spectral manifolds. Two types of spectral manifold are needed. The first type of spectral manifold, the strong type, is defined as one might expect, but it is the introduction of the second or weak type of spectral manifold that permits the construction of a theory which will apply to an arbitrary operator. The two types of spectral manifold and the four types of duality Received July 18, 1958 379

73

380

ERRETT BISHOP

theory will be defined in § 2, and certain elementary consequences of the definitions will be obtained. In § 3 the principal result, that an arbitrary operator admits a duality theory of type 4, will be derived. The proof will entail a preliminary investigation of certain Banach spaces of analytic functions. In § 4, the concluding section, conditions will be given under which the stronger duality theories obtain. In particular, conditions on the rate of growth of the norm II (T - zI)-'11 of the resolvent will be given which imply the existence of a strong type of spectral theory. Such rate of growth conditions have been studied by Lorch [4] and Wolf [7], but the treatment given here is much more general and the methods are distinct. 2. Definitions and preliminaries. It can be seen that the definition given below of a strong spectral manifold coincides with the usual definition if the operator is Hermitian or if, more generally, the operator can be written in the form Dunford [2].

~ AdE(A), in the sense of Lorch [4] or

DEFINITION 1. Let T be an operator on a reflexive Banach space B. Let F be a closed subset of the complex plane. We define the strong spectral manifold M(F, T) to be the closure of the set of all vectors x in B which have the property that there exists an analytic function f on the complement -F of F with values in B such that (T - zI)f(z) = x for all z in - F. It will be seen that the strong spectral manifolds are not sufficient to give an adequate duality theory for an arbitrary operator. (What is meant by an adequate duality theory will become clear later.) We therefore define the second type of spectral manifold. DEFINITION 2. Let T be an operator on a reflexive Banach space B. Let F be a closed subset of the complex plane. We define the weak spectral manifold N(F, T) to be the set of all vectors x in B which have the property that for each c > 0 there exists an analytic function f on the complement - F of F with values in B such that II(T-zI)f(z)-xll<s for all z in -F. It is clear from the definition that N(F, T) is closed and that M(F, T) is a subspace of N(F, T). Examples can be given to show that M(F, T) can actually be a proper subspace of N(F, T); however, it can be show that M(F, T) = N(F, T) if T is a normal operator on a Hilbert space. It is also clear that M(F" T)cM(F" T) and N(F" T) cN(F" T) if F,eF.. Also, M(a(T), T) = N(a(T), T) = B, where aCT) denotes the spectrum of T. The task of investigating the spectral manifolds M(F, T) and N(F, T)

74

A DUALITY THEOREM FOR AN ARBITRARY OPERATOR

381

now presents itself. An obvious possibility is to attempt to prove, for instance, that M(F" T) and M(F" T) span M(FJ U F" T). Although this is the case if T acts on a finite-dimensional space or if T is a normal operator on a Hilbert space, it is not true in general. A much more promising possibility for investigation is suggested by the fact that if B is finite-dimensional and if F[ and F, are disjoint closed sets with F] U F;~a(T) then the subspace M(F" T~) of B"- is the annihilator M(Fu T)'L of the subspace M(FJ' T) of B. This is easily seen from the theory of operators on finite-dimensional spaces. By means of the theory of integration around subsets of a( T) which are both open and closed, which has been developed by Lorch and is given in Hille [3], it is not difficult to show that this statement remains valid \vithout the restriction that B be finite-dimensional. In case it is noL possible to decompose a( T) into disjoint closed sets, the essence of the statement can nevertheless be recaptured. This is done by means of the following definition. DEFINITION 3. An operator T on a reflexive Banach space B will be said to admit a d1lality theory of type 1 if JIJ(F" T)' ~1II(F" TX) whenever F[ and F, are compact and disjoint and if l~I(Gu T)'L c JJ(G2 , T"") whenever G1 and G, are the closures of open sets G J and G_ which together cover the complex plane. It is clear that T and T"" play symmetric roles m this definition because B is reflexive and because the inclusions M(F" T)'L ~M(F" T ') and 1I1(G[, T)" C 1IJ(G" T") are equivalent respectively to the inclusions

M(F2' T*-)'L ~ M(F[, T) and M(G21 T ")'L:= l\1(G" T) .

Later we shall see that not every operator admits a duality theory of type 1. More precisely, there exist an operator T and open sets G] and G 2 which cover the complex plane such that M(CI T)' is not a subset of M(G" T~). Since it will be seen that the inclusion JI(Fj) T)' ~M(F2' T*) is always valid, it is a case of the strong spectral manifolds being too small. This is what was meant above when it was asserted that the strong spectral manifolds are not sufficient to give an adequate duality theory for an arbitrary operator, and this is the reason for consideration of the weak spectral manifolds. Conditions on an operator which imply that it admits a duality theory of type 1 will be obtained in §4. Since the duality theory of type 1 does not have much surface resemblance to the usual spectral theories, we no,\, define duality theories of types 2 and 3 which bear a closer surface resemblance to the usual spectral theories, but which will be seen to be intimately related to the duality theory of type 1. Some of the ideas occurring in the development of these duality theories are closely related to ideas which occur in [1]. J •

75

382

ERRETT BISHOP

DEFINITION 4. The operator T on the reflexive Banach space B will be said to admit a duality theory of type 2 if M(G" T), ... , M(Gn , T) span B whenever Gil ... , Gn are open sets which cover the complex plane. If 'T is an operator and M is a subspace of B which is invariant under T, the restriction of T to M will be denoted by T / M. DEFINITION 5. Let T be an operator on a reflexive Banach space B such that for arbitrary open sets G" ... , Gn which cover the complex plane there exist subs paces M" ... , Mn which span B and are invariant under T and have the property that a(T/Mi)cG i • Then T is said to admit a duality theory of type 3. It was shown by Lorch [4] that T admits a duality theory of type 3, in our terminology, if (1) a(T)c (z: Izl = I} and (2) there exists a constant K such that II Tn II ~ K for all positive and negative integers n. This work has been generalized by various authors. In §4 general conditions will be obtained which imply that T admits a duality theory of type 3. We now define the fourth type of duality theory. DEFINITION 6. Let T be an operator on a reflexive Banach space such that the inclusions M(F" T)1.~N(F" T"), N(F)I T)1.~M(F" T'-), M(G II T)1. C N(G 2 , T ), and N(G" T)1.c M(G" Tl-) are valid for arbitrary disjoint compact sets F, and F, and arbitrary open sets G, and G 2 which cover the complex plane. Then T is said to admit a duality theory of type 4. It is clear that T and T* play symmetric roles in this definition. Although this definition seems awkeward, it is justified by the nontrivial theorem, to be proved below, that every operator on a reflexive Banach space admits a duality theory of type 4. This fact makes it plausible that there exists a third type of spectral manifold L(F, T), intermediate between M(F, T) and N(F, T), which is self-dual in the sense that L(F" T) 1 ~ L(F" T") and L(G" T)1. c L(G" T) whenever F, and F2 are disjoint and compact and G, and G, are open and cover the complex plane. The author has not succeeded in formulating a conjecture as to how such intermediate spectral manifolds might be obtained.

3. Duality theory of type 4. Since the definitions of the spectral manifold involve vector-valued analytic functions, it will be necessary to make a preliminary study of spaces composed of such functions. Let B be a Banach space with dual B*. Let U, be an unbounded open subset of the complex plane whose boundary C consists of a finite number of disjoint closed rectifiable Jordan curves. The set U, will be called an unbounded simple open set. Let U z be the complement

76

A DUALITY THEOREM FOR AN ARBITRARY OPERATOR

383

of U, U C, so that U, is a bounded open set whose boundary is C. The set U, will be called a bounded simple open set, and U, and U, will be called complementary simple open sets. Let W, be the set of all analytic functions from U, to B which vanish at infinity, and let W 2 consist of all analytic functions from U, to B*. We topologize W, and W, by the topology of uniform convergence on compact subsets of U, and U, respectively. It is clear that W, and W, are locally con vex linear topological spaces. Let V, consist of those functions in W, which can be extended to be continuous on U, U C, and let V, consist of those functions in W, which can be extended to be continuous on U, U C. If 1 E V, we define 11111 = sup {111(z) II: Z E U,}, and if 9 E V 2 we define Ilgll = sup {llg(z)ll: ZE U,}. The Banach space V, is called the Banach space of analytic functions from U, to B which have continuous boundary values, and V, is called the Banach space of analytic functions from U, to B* which have continuous boundary values. Before proceeding, we state the Cauchy integral formula in a form suitable to our purposes: there exists a unique orientation on C, called the canonical orientation, such that

whenever Zo E U 2 and 9 is an analytic function on some neighborhood of U, U C, and such that

whenever Zo E U, and 1 is an analytic function on some neighborhood of U, U C which vanishes at infinity. For Z in U" ZO in Un and x in B we define o(x, ZO, z) = (z - zo)-'x. For each x in B and each Zo in U" the element a(x, ZO, .) will be called an elementary element of V,. The Banach space V, called the restricted Banach space of analytic functions from U to B which have continuous boundary values, is defined to be that subspace of V, which is spanned by the elementary elements of V,. If 1 E V, and 9 E V" we define j

r ndc

<1, g) = - 12. <1(z), g(z»dz , where C is given the canonical orientation and where we have extended 9 to be continuous on C. It is clear that this defines a jointly continuous bilinear functional between V, and V,.

1 and

77

384

ERRETT BISHOP

LEMMA 1. Let V be the restricted Banach space of analytic functions from an unbounded simple open set U1 to a Banach space B which have continuous boundary values. Let V 2 and W 2 be defined as above. Then there exists a linear subset Y of W, and a norm on Y such that (1) Y is a Banach space. (2) V,c Y. (3) The inclusion mappings V 2 -~ Y and Y --"> W, are continuous. (4) The inner product between V and V 2 can be extended to an inner product between V and Y in such a way that Y becomes isometrically isomorphic to V*.

Proof. Let 'P be an arbitrary element of V -l- • For each x in B and Zo in U2 define 1'(x, zo) = r",(x, zo) = 'p(a(x, Z'l> • For Zo fixed, it follows that r(·, zo) is a continuous linear functional on B, because convergence of {x n } to x in B implies convergence of ~(l(Xn' zo, .)} to a(x, zo, .) in V" which in turn implies convergence of {(p(a(x", ZO, .»} ,= {r(xn,zo)} to (p(a(x,zo, = r(x,zo). Thus J'(·,zo)EB' for each Zo in U,. We now prove that l' E W,. By this we mean that the mapping Zo --"> r(., zo) is an analytic function from U, to B*. It is sufficient to show that this mapping is weakly analytic, i.e., to show that Zo ~ r(x, zo) is an analytic function on U 2 for each x in B (see Hille [3], page 53). To this end we consider the difference quotient

».

.»

[r(x, Z1) - r(x, ZO)](Z, - ZO)-1

= (p{[a(x, z" .) - a(x, ZO, • )](Z1- ZO)-1} .

As Z1 -+ ZO, the element [a(x, Zt> .) - a(x, ZO, • )](Z1 - ZO)-1 converges in V to flex, ZO, .), where (3(x, ZO, z) is defined by flex, zo' z) = (z - zo)-'x for x in B, Zo in U" and Z in U1 • The difference quotient in question therefore converges to 'p«(:J(x, zo, as Z1 --"> Zo' It follows that r E W 2 • We have thus defined a linear mapping 'I' --"> 1'

.»

.»

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A DUALITY THEOREM FOR AN ARBITRARY OPERATOR

385

limit, so that the inclusion map Y ~ W 2 is continuous. To prove V 2 c Y, consider 9 in V 2 and define the continuous linear functional 'P on V by 'p(f) == <1, g), where the inner product between V, and V 2 was defined above. Thus (P E V*, so that 1''1' E Y. Also,

1'",(x, zo) = rp(a(x, zo'

.» =


= _1_. \
~\ (z - zo)-'(x, g(z»dz 27r~

Jc

=',

<x, g(zo»

.

Thus I'", = g. It follows that V 2 c Y. We define the inner product between V and Y by <1, 1 ''I') = 'p(f), so that Y becomes isometrically isomorphic to V;. Since we have seen in the preceding paragraph that 'J!(f) = (f, g) for each 9 = l'.p in V" it follows that this inner product extends the inner product bet\'. een V and V,. It remains to show that the inclusion map V 2 --" Y is continuous, Let ~gn~ converge to 9 in V 2 • Define 'fJn in Vi' by 'fJn(f) = (t, gn) for all 1 in V, so that gn = l''''n by the above. To show that ~gn: converges to 9 in Y we must show that {'Pn} converges to 'jJ in V'. This follows from the joint continuity of the inner product between V and V 2 • This completes the proof of the lemma. We shall identify the space Y of Lemma 1 with V", so that Viwill be thought of as a subset of W 2 • Define the bounded linear mapping T of V into B by T 1 = lim =zl(z). Then the adjoint mapping ," of B* into V* is defined by taking ,"u to be that function of V, whose value at each point of U 2 is u, i.e., by setting (T'U)(Z) = u for all z in U 2 • This is seen by the following computation: H


'0')

Z

..

27r'L

=

oJ C

=~r <1(t), u)dt 2;:-t Je

.

The mapping T is onto, because nr(x, 2o, .) = x for all x III B and all in U" as a simple computation shows. In addition to the above objects, we now consider an operator T on B. We define the operator H on V by (Hf)(z) = (T - zI)l(z) + Tf for all z in U[. In order to obtain a formula for the adjoint operator H" on V*, consider an elementary element 1 = a(x, zo' .) of V and an element 9 of V*. We consider 9 as an element of W 2 • and let (1' be the element of V* which corresponds to g. Then <x, g(zo» = 'p(a(x, zo, = (p(f) , by the definition of g. Also, <1, g) = rp(f) when 9 is interpreted Zo

.»

79

386

ERRETT BISHOP

as the element T of V*. Thus
=

+X ZO)-' x + X

=


(T - zI)(z - ZO)-I X

= a(Tx, zo, z) - z(z -

Therefore Hf = a(Tx, zo, .) - zoa(x, zo, .). <x, (H*g)(zo»

=


=
=

= a(Tx, zo, z) - zo(z - ZO)-' x .

Thus


- zo<x, g(zo» = <x, (T-+ - zoI)g(zo»

.

Since this is true for all x in B, it follows that the adjoint operator is given by the formula (H*g)(zo) = (T* - zoI)g(zo) for all Zo in U2. Of the many possible norms for V x V, we consider only those norms obtained by defining

II (I,,!,) II

=

II (I" f,) II" =

(1) Ilf, 112

+ Ilf,II')'/2 ,

where r; is an arbitrary positive constant. Then the norm on V* x V* will be given by II(g" g,)11 = (~-Illg,II' + IIg,II')'/2. We define an operator K on V x V by K(I" f,) = (I" Hf, - f2)' A simple computation shows that K' = I and that K* is the operator on V* x V* defined by K*(g" g2) = (H*g, + g" -g,). Since K' = I it has an inverse and therefore the norm [ J on V x V defined by [fJ = II Kfll is equivalent to the norm II II. Similarly, the norm [ J on Vi' x V* defined by [gJ=IIK-+gll is equivalent to the norm II II. Also, for each 9 in V-+ x V* we have

[gJ

= IIK*g II = sup {I
g)l: IIfll = 1} g)l: IIK(Kf) II

Ilfll

= 1}

= 1} = sup {I
g)l:

[hJ

=

1]

Thus the norm [ J on V* x V* corresponds to the norm [ J on V x V. In the proof of Theorem 1 we shall need the following simple lemmas. They correspond to well-known results, but we sketch their proofs for the sake of completeness and in order to have them in exactly the form they are needed. LEMMA 2. Let A be a bounded linear transformation which maps a Banach space B, homeomorphically into a Banach space B,. Then the norm [ ] on Bi defined by [uJ = inf {II v II: v E B i, A -+v = u] corresponds to the norm [ ] on B, defined by [x] = II Ax II.

Proof. Consider u in Bi. By the Hahn-Banach theorem, there exists v in Bi such that <x, u) =
80

A DUALITY THEOREM FOR AN ARBITRARY OPERATOR

387

then
IIv,lI

~

sup {1
Thus

[u] = Ilvll = sup {I
~

1}

as was to be proved. LEMMA 3. Let A be a bounded linear transforrnation of a Banach space B2 onto a Banach space B,. Then the norrn on Bi defined by [u] = Ilkl-ull corresponds to the norrn on B, defined by [x] = inf {Ilyll: Ay = x}, and the norrns [ ] and II lion B, are equivalent.

It is easily seen that [ ] is a norm on B, and that II x II ~ Let [x,,} be a Cauchy sequence of elements in B relative to the norm []. We may assume, by passing to a subsequence if necessary, that [x n - xn+,l < 2- n. Thus we can inductively find Yn in B, with AYn = Xn and llYn - Yn+,11 < 2- n. Thus [Yn} converges to a vector Y in B,. Thus {xn} converges to x = Ay relative to the norm 1 on B Thus B, is complete under the norm []. By the closed graph theorem, it follows that the norms II II and [] on B, are equivalent. Also, if UE Bi then [u] = IIA-i'ull = sup U
Proof.

IIAII[x] for all x in B,. j

r

j •

THEOREM 1. Let T be an operator on a Banach space B. Let U, be an unbounded sirnple open set, and let V be the 1'estricted Banach space of analytic functions from U, to B. Let Yj be positive. For x in B define

{x} = [x}" = inf [("llfI1 2

+ IIHfll')!/2:

fE V, ;:f = x} .

For u in B* define

Then the function { } on B is a norrn which is equivalent to the norrn

II II and the function { } on B'" is the norrn on B* which corresponds to the norrn { } on B. Proof.

The map (u, g)

-->

(;:*u, g) of B"" x V* into V* x V* is the

81

ERRETT BISHOP

388

adjoint of the map (f" f2) ->- ('[ f" f') of V x V onto B x V. It follows from Lemma 3 that the norm on Bi- x V"- defined by [(u, g)] = [('["-u, g)] corresponds to the norm on B x V defined by [(x, f)] = inf ~[(fo, f)]: '[fa = x}, and that the norms [ ] and II lion B x V are equivalent. Now the map (u, g) ->- u of B* x V* onto B"- is the adjoint of the map x ->- (x, 0) of B into B x V. It follows from Lemma 2 that the norm defined on B* by [u] = inf {[(u, g)]: g E V"} corresponds to the norm defined on B by [x] = [(x, 0)], and that the norms [ ] and II II on B are equivalent. But [x]

=

inf {[(f, 0)]: '[f

=

x}

=

inf UIK(f, 0)11: '[ f

= inf {11(f, Hf)ll: '[I = x} = inf {(~IIII12 + IIHIII')'!2: '[I =

=

x}

x] = {x] .

Also, [u]

=

inf {[('[*u, g)]: g E V*}

=

inf {II K-1-(,[*u, g) II: g E V*}

inf {11(H*g + r-"-u, -g) II: g E V"} = inf UI (H*g - r-"u, g) II: g E V*} = inf {(~-'IIH"-g - r-"-uI1 2+ IlgI12)"2: gE V"-] = {u} . =

This proves Theorem 1. Of the two constituents, ~l '11111 and IIHIII, \vhich enter into the norm {xl = inf {(~IIIII' + IIHIII')'2: r- I=:c} on B, imprecisely speaking the second constituent

IIHIII = sup {II(T - zI)I(z) + x II: z E U

J}

measures the degree of approximation to which one can solve the resolvent equation (T - zI)I(z) = x on Up and the first 1}111II1 measures the size of the approximate solution. A similar comment can be made for the norm ~ ~ on B". Thus Theorem 1, which states that the norms correspond, establishes a relationship between approximate solutions to the resolvent equation for T on the set U and approximate solutions to the resolvent equation for T"- on the set U,. In view of the fact that the spectral manifolds were defined in terms of approximate solutions to the resolvent equation, it is not surprising that Theorem 1 should contain information pertinent to the duality theories defined above. Actually, the duality theory which we obtain as a consequence of Theorem 1 will be less precise than Theorem 1 itself. It therefore seems likely that a complete duality theory for an arbitrary operator will be based upon some version of Theorem 1 itself rather than upon the duality theory which we now proceed to derive from Theorem 1. It would have been possible to derive Theorem 1 in more generality by considering possibilities other than V for the Banach space of analytic functions from U l to J

82

A DUALITY THEOREM FOR AN ARBITRARY OPERATOR

389

B, but this was not necessary for the applications to be made. Let the subset N of B consist of all x such that for each c > 0 there exists f in V with IIHfl1 < c and 7:f = x. Let the subspace M of B* be the closure of the set Mo of all u in B" such that there exists g in V* with H"g = 7:"u. Then N is a subspace and M1. = N. COROLLARY.

We first show that N = [x: {xL --> 0 as r) --> 0]. If x EN f in V with 7:f = x and IIHfl1 < c. Thus {x}~ ~ (1J11fW + C')'/' --> C as IJ --> O. Since c is arbitrary it follows that {X}, --> 0 as 7J --> O. Conversely if {x}" --> 0 as ~'--> 0 then for each c > 0 there exists r; > 0 such that {xL < c, which implies that there exists f in V with Tf = x and (~IIJII' + IIHfll')" < c. Thus IIHfl1 < c, so that Proof.

for each c

> 0 there exists

xEN.

We next show that Mo = {u: {u; 1 is bounded for r; then there exists g in V* with H*g = 7:i-u. Therefore

> O}.

If u E Mo,

is bounded for r; > O. Conversely, assume now that {u}" < c for all positive ~'> Then for each positive integer n there exists gn in V" with (nIIH*gn - r*ull' + Ilgnll') < c'. Thus Ilgnll < c. Since the unit sphere of V" is compact in the weak star topology, there exists a cluster point g of {gn} in the weak star topology. Therefore H*g is a cluster point of {H*gn} in the weak star topology. Since IIH*gn - T*ull' ~ c'n- 1 , it follows that H*g = T*U. Therefore u E Mo. If xENand uEMo, then l<x,u)1 ~ {X}1{U},--> 0 as ~'-->O. Therefore Nand Mo are orthogonal. If x is not in N, then {X}1 does not converge to 0 as 7J --> 0, so that there exists a sequence {r;n) of positive constants converging to 0 and a constant c > 0 such that {xL > c for all n. It follows that there exists a sequence [un} of element; in B'" with {unL < 1 and <x, un) > c. We can thus find a sequence {gn} of elements i~ V* with

As before, there exists a cluster point (g, u) of the sequence Ugn, un)} in the weak star topology of V'" x B*. It follows, as before, that H*g = 7:"u. Thus u E Mo. Since also <x, u) ~ c, the vector x is not in MoL. Thus M if- = N. It follows that N is a subspace and that M 1. = N, as was to be proved. As an easy consequence of this corollary, we obtain the following fundamental theorem.

83

390

ERRETT BISHOP

THEOREM 2. Every operator T on a reflexive Banach space B admits a duality theory of type 4.

Proof. Let F, and F2 be disjoint compact sets. Then there exists an unbounded simple open set U" whose complementary simple open set will be called U., such that F,c U, and F,c U 2 • Define N as in the above corollary. To see that N(F" T)cN, take x in N(F" T) and take c: > O. Let Zo be any point in U2 and let inf {Iz - zol: ZE U,} = c > O. By the definition of N(F" T), there exists an analytic function f from -F, to B such that II(T - zl)f(z) - xll < c: for all z in -F" Since f is analytic on the neighborhood -F, of U, U C, Runge's theorem for vector-valued analytic functions (whose proof is exactly the same as the proof of Runge's theorem in the scalar case) implies that f can be uniformly approximated on U, U C by linear combinations of elementary elements of V,. Therefore f E V. Define f, in V by f, = - f + a(x

+ Tf, zo,

,) .

Then Tf, = -Tf + lim z(z - ZO)-'(X

+ Tf) =

X.

Also,

l1 = sup lIl(T - ZI)f,(Z) + Tf,ll: ZE U,} sup {II-(T - zI)f(z) + (z - zo)-'(T - zl)(x + T f) + T f,ll: sup {ll(T - zl)f(z) - x ll: Z E U1 } + sup {11(z - zo)-1(T - zI)(x + Tf)ll: c: + sup {1I(z - zo)-I(T - zl)(x + Tf)ll: ZE U,} .

IIHf,

= ~

~

E

U1 }

ZE

U,}

Z

Now

Ilx + rfll

= lim

II-[(T -

zI)f(z) - x] II ~ c: .

Thus lI(z - zo)-'(T - zl)(x

+ Tf)11

~ ~

11[(z - zo)-'(T - zol) - 1](x + Tf)11 [c-'II Tll + c-'Izol + 1]c: .

Thus

IIHf,l1

~

[c-'II Til + c-'Izol + 2]c: .

Since c: is arbitrary, x E N. Thus N(FH T)cN. We next show that M(F" T*)cM, where M is defined in the above corollary, It is sufficient to consider u in M(F" T*) such that there exists an analytic function g from -F2 to B* with (T* - zl)g(z) = u

84

A DUALITY THEOREM FOR AN ARBITRARY OPERATOR

391

for all z in -F" since the set of such u is dense in M(F" T*). Since g is analytic on the neighborhood - F2 of U 2 U C, g E V,c V*. By the definition of H* it follows that H*g = ,*u. Thus u E M, so that M(F" T*)cM. Since M 1- = N by the above corollary, it follows that

Since this holds for all operators T, we may replace T by T* and mterchange F, with F, to obtain

It remains to consider open sets G, and G2 which cover the complex plane. There exists an unbounded simple open set U, such that U,cG,U -a(T) and (J,cG" where U, is the bounded simple open set which is complementary to U,. Define V, M, and N as in the above corollary. We first show NcN(iJ" T). Consider x in N, so that for each c > 0 there exists f in V with ' f = x and IIHfl1 < c. This implies that

II(T - zI)f(z)

+ xII =

II(T - zI)(-f(z)) - xii


for all z in U,. Since -G,c U" it follows that x E N(G" T). We next show M c M(G" T). Consider u in M o, so that there exists g in V* with H'-g = ,'"u. Thus (T'- - zI)g(z) = u for all z in U,. If z E -a(T) let g(z) = (T* - zI)-'u. We thus obtain an analytic extension of g to U,U -a(T) which has the property that (T" - zI)g(z) = u for all z in U,U -a(T). Since -G, c U,U -a(T), it follows that u E M(G" T*). Thus MocM(G" T'-), and therefore McM(G" T*). Since M1- = Nand is reflexive, N 1- = M. Hence N(G" TV'- c M(G" T*). In this inclusion we may replace T by T'- and interchange G, and G, to obtain N(G" T'-)1-cM(G" T) or M(G" T)1-cN(G" T") .

This completes the proof of Theorem 2. We close this section by considering an example, which will show that M(F, T) N(F, T) in general and that not every operator admits a duality theory of type 1. Let {xn}:__ ~ be a complete orthonormal basis for a Hilbert space B, and let {an}:=-~ be a sequence of real numbers with 2- 1 < an < 2 and

'*

and lim inf

(a,a, . an)'!" = lim inf n-~

(a_la_, a_n)'i n = 2- 1 n_~

85

•

ERRETT BISHOP

392

Define the operator Ton B by T(Ea,x,) = Ea,a ,x , + 1. Then T", T-l, and (T ')-1 are operators of the same type as T. For each r with 2- 1 < r < 2 let Kr = {z: \z\ :;;; r} and Lr = {z: \z\ ~ r}. We shall show that M(KT' T) = M(LT' T) = to}, that N(K" T) = N(L" T) = B, and that corresponding equalities hold for Ti-. To show M(K" T) = {O}, let x be any vector in B such that there exists an analytic function f from -Kr to B with (T - z1)f(z) = x for all z in - K r • The Laurent expansion for f is therefore 00

fez) = -

E

z-n-1Tnx .

11,=0

Since this series con verges for Iz I > r we have lim

sUPn~=11

Tn x

wn :;;; r < 2 .

By the formula for T, this implies that x = O. Thus M(K" T) = [0;. Since T-l and T' are operators of the same type as T, we also have M(K" T-l) = M(Kn T*) = to}. Now consider x E M(L" T). Thus there exists f analytic for Iz I< r with values in B such that (T - z1)f(z) = x. Therefore

for Izl < r. Thus -zf(z) = g(Z-I) is an analytic function of Z-1 for Iz- 11 > r- 1 and (T-l - z-l1)g(Z-I) = T-I X • Therefore T-I X E M(Kr-t, T-l). By the above, this implies that T-I X = O. Therefore x = O. Therefore M(L" T) = {O}. Similarly, M(L" Ti-) = {O}. This already shows that T does not admit a duality theory of type 1, since by the above it is not true that

<s
M(K" T)l.cM(L" T*) if 2- 1

1 Then To show that N(K" T) = B, choose s N(K" T)~ cM(L" T"') = {OJ since T admits a duality theory of type 4. It follows that N(K" T) = B. The equality N(I-'n T) = B is proved similarly.

4.

The other duality theories.

DEFINITION 7. The operator T on the reflexive Banach space B is said to satisfy condition a if N(Fn T)cM(F" T) whenever Fl and F2 are closed and F, c interior F,. THEOREM 3. An operator T which satisfies condition a admits a duality theory of type 1. Proof. Assume that T satisfies condition a. joint compact sets. Then

86

Let Fl and F, be dis-

A DUALITy Tm:OREM FOR AN ARBITRARY OPERATOR

M(F" T)c N(F" T),

SO

~N(F"

that M(F" T)1-

393

T)1- ~M(F2> T*)

by Theorem 2. Consider now open sets G, and G, which cover the complex plane. Choose an open set G3 such that GJcG, and such that G2 and GJ cover the complex plane. Then N(G" T)cM(G" T) by condition a. Thus M(G" T)1- cN(G3 , T)1- C M(G 2 , T*) by Theorem 2. This completes the proof of Theorem 3. We now investigate conditions which are sufficient for T to admit duality theories of types 2 and 3. DEFINITION 8. The operator T on the reflexive Banach space B satisfies condition f:J if for every open set U and every sequence {In} of analytic functions from U to B such that (T - z1)ln(z) --+ 0 uniformly on U as n --+ = it follows that the sequence lin] is uniformly bounded on compact subsets of U. There are various reformulations of Definition 8. We shall need the following: T satisfies condition /'3 if and only if for every open set U and every sequence {Inl of analytic functions from U to B and every x in B such that (T - z1)ln(z) -> x uniformly on compact subsets of U as n --+ it follows that {In} is uniformly bounded on compact subsets of U. That this reformulation implies condition /3 is clear. To see that /'3 implies this reformulation, let {In} be a sequence of analytic functions from U to B such that (T - z1)ln(z) ---+ x uniformly on compact subsets of U, and let K be a compact subset of U. Then there exists a neighborhood U, of K such that (T - z1)ln(z) ----> x uniformly on U,. If Un} is not uniformly bounded on K, by passing to a subsequence if necessary we may assume that d n +, > n + d n for each n, where

=

d n = sup

~ll/n(z)

Thus the sequence {gn] defined by gn on K. On the other hand,

=

II:

Z

j~+ 1

E -

K} . In is not uniformly bounded

(T - z1)gn(z) = (T - z1)ln+,(z) -- (T - z1)ln(z)

converges uniformly to x - x = 0 on U,. This contradicts condition /3. Therefore {In} is uniformly bounded on K, as was to be proved. DEFINITION 9. The operator T on the reflexive Banach space B will be said to satisfy condition ~/ if for every open set U and every non-zero analytic f from U to B there exists z in U with (T - z1)f(z) O.

'*

THEOREM 4. If T satisfies condition /'3 and if F is an arbitrary closed subset of the complex plane, then (1) M(F, T) = N(F, T), (2) for

87

394

ERRETT BISHOP

each x ~n M(F, T) there exists an analytic function f from - F to B with (T - z1)f(z) = x for all z in -F, and (3) T satisfies conditions a and /. Proof. For each x in N(F, T) there exists a sequence {fn} of analytic functions from -F to B such that II (T - zI)fn(z) - x II < n -1 for all z in -F. By condition (3, {fn} is uniformly bounded on compact subsets of - F, so that there exists a subset, which we continue to call Un}, converging pointwise in the weak topology of B to an analytic function f from -F to B. Clearly (T - z1)f(z) = x for all z in -F. This proves assertions (1) and (2). If F, and F2 are closed and F, c interior F 2 , then N (F" T) = M(F" T)cM(F" T) by (1), so that T satisfies condition a. If T does not satisfy condition /, there exists an open set U and a non-zero analytic function f from U to B with (T - zI)f(z) = 0 for all z in U. Let fn = nf, so that (T - zI)fn(z) -'r 0 as n -'r = uniformly on U. Since T satisfies condition (:J, it follows that the sequence ~nf~ is pointwise bounded. This contradiction proves that T satisfies condition /'. The theorem just proved is related in part to results of Wermer [5]. As a corollary of this theorem, T admits a duality theory of type 1 if T satisfies condition /:J. THEOREM 5. If T+ sat'isfies condition (J, then T admits a duality theory of type 2. If T and T* satisfy condition (3, then T admits a duality theory of type 3.

Proof. Assume that T' satisfies condition {J. Then T", and therefore T, admits a duality theory of type 1. Also, T*- satisfies condition /'. Let G,,···, Gn be open sets covering the complex plane. If M(G" T), . ", M(G a , T) do not span B, there exists u * 0 in B' orthogonal to these subspaces. Since G,,···, Gn cover the complex plane, n7~1 -G, is void. There therefore exist open sets U,,"', Un such that -G,c U, and nr~1 U, is void. Since U, and G, cover the complex plane and since T admits a duality theory of type 1, we see that M((J., T)~ c M( U" T ') .

Thus u € M( Ui , T') for each i. Since T" satisfies condition fJ it follows from Theorem 4 that there exist an analytic function g, from - U, to B* with (T* - z1)g,(z) = u for all z in - U,. If z E - [i, n - U), we have (T* - zI)(g,(z)-g;(z» = O. Since g,-gj is analytic on -uin -(fj and since T* satisfies condition /, it follows that g,(z) = g,(z) for z in - U, n - (fj. Thus if we define the function 9 by g(z) = g,(z) for z in - U" then 9 is uniquely defined and is analytic on U~-l - (f,. Also

88

A DUALITY THEOREM FOR AN ARBITRARY OPERATOR

~95

(T" - zI)g(z) = u whenever 9 is defined. But the sets -l1; cover the plane because n~~l Ui is void. Also g(z) = (T' - zI)-J u -40 as Iz 1---> co. Thus 9 is everywhere analytic and vanishes at infinity, so that 9 = O. Thus u = (T > -- zI)g(z) = O. This contradiction implies that M(G]> T), "', M(G,,, T) span B. Therefore T admits a duality theory of type 2. Assume now that T also satisfies condition /3. We must show that T admits a duality theory of type 3. First consider a bounded open set H such that -H is connected. By Theorem 4 for each x in M=M(H, 1') there exists an analytic function f(x, .) from U =- -H to B such that (T - zI)f(x, z) = x for all z in U. Since l' satisfies condition /-], and therefore condition OJ, the function f(x, .) is unique. For each z" in U define the linear transformation T 0 and II Tzuxn II -4 co as n ---> CD. Since T satisfies condition ;3, it follows that U(x n, zo)} = [T'uxn} is bounded. This contradiction implies that Tzu is a bounded transformation for each Zo in U. If z in U is sufficiently large and if x E M, T,x = (T - zI)-lx EM. Since T,x is an analytic function on U and since U is connected, it follows that T,x EM for all x in M and all z in U. Thus for each z in U the operator T, on M is a right inverse of (T - zI)/A1. To show that T z is the inverse of (T - zI)/M it is sufficient to establish that x = 0 whenever x EM and (T-zI)x = O. To this end, consider x in M with (T-zI)x == O. Define the analytic function h from U to B by h(A) = x + (,\ - z)T"x. Then (T - AI)h(A) = (1' - zI)x

=

0

+ (z

+ (z

- A,)X

- A)X

+ (,\, -

+ (A z)x

=

- z)(T - AI)T"x

0 .

Since T satisfies condition (3, and therefore condition "j, this implies h = O. Thus x = h(z) = O. Therefore (T - zI)/M has an inverse for all z in U, so that (J(T/M(R, T»cH. Now let a covering G]O "', G n of the complex plane by open sets be given. There exists a family [H ik ] , 1 ;;;; i ;:;:: n, 1 ;:;:: k ;:;:: mil of open sets which cover the complex plane such that HikcG i and either Hi" is bounded and -Ri" is connected or Hikc -(J(T). From the preceding paragraphs we see that the M(H'k' T) span B and that (J(T / M(R" , T»cRik

if Hi" is bounded and -Hik is connected. Also a(T/M(H,,,, T»cH", if Rikc -(J(T) because M(Rit> T) consists only of the zero vector in this case. If this were not true, there would exist x -cf' 0 in M(R" , 1') and

89

396

ERRETT

BISHOP

-it.

an analytic function f from to B such that (T - zI)f(z) = x. If we extended the definition of f to -aCT) by defining fez) = (T - zI)-'x for z in -aCT), then f becomes an analytic function defined everywhere and vanishing at infinity. Thus f = O. Therefore x = (T - zI)f = O. Hence a(T/M(ii,., T»cii'k for all i and k. Define M! to be the subspace spanned by the sets M(H'H T), "', M (H""" T). Since -

-

M(Hie , T)cM(G" T) for each k, we have M,cM(G" T). Also M" "', Mn span B. To complete the proof of the theorem we must show that a(T/M,)cG,. As before, with the set H replaced by G" for each x in M,cM(G" T) there exists a unique analytic function f(x, .) from -Gi to B such that (T - zI)f(x, z) = x for all z in -Gi . As before, for each z in -G! define the linear transformation T z from Mi to B by Tzx = f(x, z). As before, T z is a bounded linear transformation for each z in -G t • If the set H,. is bounded and -iii. is connected, we see as before that the spaces M(iiild T) are invariant under T, for each z in -iiik~ -G,. Since otherwise H,. has the property that M(lt, , T) consists only of the vector 0, and since the M(iiik' T) span M, it follows that M, is invariant under Tz for all z in -G,. Thus the operator T z on M, is a right inverse of (T - zI)/ Mi' As before, it follows that T z is the inverse of (T - zI)/M, for each z in -G,. Thus a(T/M,)cGi , as was to be proved. As an immediate consequence of the preceding theorems, it follows that T and T* both admit duality theories of types 1, 2, 3, and 4 whenever T and T* both satisfy condition /3. DEFINITION 10. Let r be a function from the complex plane to the set [0, (;0]. The function r will be said to be a modulus of control for analytic functions if for every open set U and every sequence {fn} of complex-valued analytic functions on U such that Ifn(z) I ~ r(z) for all z in U the sequence {fn} is uniformly bounded on all compact subsets of U. THEOREM 6. Let T be an operator on a reflexive Banach space B. Let the function r defined by r(z) = II(T - zIt'l1 if Ze -aCT) and r(z) = (;0 if Z e aCT) be a modulus of control for analytic functions. Then both T and T'" satisfy condition /3 and therefore admit duality theories of types 1, 2, 3, and 4.

Proof. Since II(T - zIt'l1 = II(T* - zI)-'11 whenever Ze -aCT), it is sufficient to show that T satisfies condition /3. To do this, let U be open and let {fn} be a sequence of analytic functions from U to B such

90

A DUALITY THEOREM FOR AN ARBITRARY OPERATOR

397

that (T - zI)fn(z) -+ 0 uniformly on U as n -+ oo. It is then possible to choose a constant K such that Ilfn(z) II ;<::; KII(T - zI)-' II for all nand all z in U. Assume that there exists a compact subset C of U on which {fn} is not uniformly bounded. By the uniform boundedness theorem [3J, it is possible to choose u in B* such that the sequence {gn} of complex-valued analytic function on U defined by .9a(z) =
110g+ log+ sup {II(T - zIt'lI: I(z)

=

y}dy

is finite, then T admits duality theories of types 1, 2, 3, and 4.

Proof. By Wolf [6], the function r defined by r(z) = II(T- zI)-11l is a modulus of control for analytic functions, so that the result follows from Theorem 6. It follows that a Hermitian operator, for instance, admits duality theories of types 1, 2, 3, and 4. From this fact it is not difficult to derive the spectral theorem. REFERENCES 1. E Bishop, Spectral theory for operators on a Banach Space, Trans Amer Math Soc, 86 (1957), 414-445 2 N Dunford, Spectral operators, Pacific J Math, 4 (1954), 321-354 3. E Hille, Functional analysis and semi groups, Amer Math Soc Colloquium Publica tions, 31 (1948) 4 E. R Lorch, The integral representational of weakly almost periodic transformations in reflexive vector spaces, Trans. Amer. Math. Soc, 49 (1941), 18-40. 5 J Wermer, The existence of invariant subspaces, Duke Math J, 19 (1952), 615-622 6. F. Wolf, On majorants of subharmonic and analytic functions, Bull. Amer Math Soc, 48 (1942), 925-932. 7 - - - , Operators in a Banach space which admit a generalized spectral decom positirYn, Nederl Akad. Wetensch Indag. Math, 19 (1957), 302-311. UNIVERSITY OF CALIFORNIA BERKELEY

91

A MINIMAL BOUNDARY FOR FUNCTION ALGEBRAS ERRETT BISHOP

An algebra ~{ of continuous functions on a compact Hausdorff space C will be understood to be a set of complex-valued functions on C which is closed under the operations of addition, multiplication, and multiplication by complex numbers. The algebra ~{ is called separating if to any two distinct points of C there exists a function in ~{ which takes distinct values at the given points. The norm IIIII of a continuous function I on a compact space is defined to be the maximum absolute value of the function. The algebra 2l is thus a normed algebra. ~{ is called a Banach algebra if it is complete with respect to its norm, i.e., if the limit of every uniformly convergent sequence of elements of ~{ is in ~l. An important theorem of Silov (see [5], p. 80) asserts that if ~{ is a separating algebra of continuous functions on a compact Hausdorff space C then there is a smallest closed subset S of C having the property that every function of ~{ attains its maximum absolute value at some point of S. This set is called the Silov boundary of 91. A simple example is obtained by taking C to be a compact subset of the complex plane and ~{ to be the set of all continuous functions on C which are analytic at interior points; in this case the Silov boundary of ~{ coincides with the topological boundary of C. Given a separating normed algebra ~l of continuous functions on a compact space C, it seems natural to ask, in view of Silov's theorem, whether there exists a smallest subset M (not necessarily closed) of C having the property that every function in ~{ attains its maximum absolute value at some point of M. The answer in general is no. However, it will be shown (Theorem 1 below) that such a set M, called the minimal boundary of ~{, always exists if in addition it is assumed that ~{ is a Banach algebra and that there is a countable basis for the open sets of C, i.e., that C is metrizable. An example will be given to show that the metrizability of C is necessary. If the minimal boundary M exists, it is clear that the closure of M is the Silov boundary. An example will be given to show that M need not be closed, so that M in general is smaller than the Silov boundary. This raises the question of the topological structure of M, which is answered (Theorem 2) by showing that M is a G s, i.e., a countable intersection of open sets. 1.

Introduction.

Received December 1, 1958, in revised form February 23, 1959 Sloan Foundation Fellowship

629

92

The author holds a

630

ERRETT BISHOP

The next portion of the paper concerns the representation of bounded linear functionals on 21 by measures. It is an easy consequence of the classical Hahn-Banach theorem and the Riesz representation theorem that any bounded linear functional 'P on 21 of norm 1 can be represented by a (complex-valued, Borel) measure p of norm 1 on the Silov boundary S of

~(, in the sense that

'P(f)

= ~/dll for all f in ilL

It is natural

to conjecture that p can actually be taken to be a measure on the minimal boundary M of A. The author will devote a subsequent paper to a proof of this result and a consideration of related questions. Karl de Leeuw also has a proof of this result, based on work of Choquet [3]. In the present paper we prove a special case, which is needed to prove the general result and which will be sufficient for the applications considered here. This special case, Theorem 3 below, states that for any point x in C - M there exists a non-negative valued measure /1 of norm 1 on C - {x} such that f(x) =

~fdP for all f in ,)L

The final section is concerned with problems of approximation in one complex variable. Necessary and sufficient conditions are obtained on a compact set C without interior of the complex plane that every continuous function on C be uniformly approximable by rational functions whose poles lie in the complement of C. Mergelyan [6] has obtained sufficient conditions, of a different type, that the approximation be possible. A summary of the results of this paper was given in [1]. 2.

The minimal boundary.

1. Let f be a continuous function on a compact space Then S(f), the maximal set of f, consists of all points x in C such that If (x) I = llill. DEFINITION

C.

DEFINITION 2. Let ~l be a separating algebra of continuous function on a compact space C. A subset N of C is said to bound ':I( if N n S(f) is non-void for all f in ilL If the class of subsets of C which bound Il( contains a smallest set M, the set M will be called the minimal boundary of 21. THEOREM 1. Let ill be a separating Banach algebra of continuous functions on a compact metrizable Hausdorff space C. Then ')( has a minimal boundary M and M equals the subset Mo of C consisting of all x in C such that there exists f in ~( with S(f) = {x}

Proof.

Let N be an arbitrary subset of C which bounds 21.

93

For

A MINIMAL BOUNDARY FOR FUNCTION ALGEBRAS

631

each x in M o, there exists f in ~{ with S (f) = {x}. Thus {x} n N = S(f) n N is non-void. Hence x EN. Therefore MocN. To show that Mo is indeed the minimal boundary of ~, it remains to prove that Mo bounds~. It must therefore be shown that Mo n S(f) is non-void for each I in ~. Let f be given. Let I' be the class of all subsets "'I of G such that there exists Iy in ~r with S(fr) ="'1. By Zorn's lemma, there is a subclass 1'0 of I' which contains S(f), which has the finite intersection property, and which has the property that no larger subclass of r has the finite intersection property. Since G is ro"'l is non-void compact and since each cy in 1'0 is closed, the set D = and closed. Since there is a countable basis for the open sets of G, and since the family {G - cy: "'I E ro} of open sets covers G - D, there exists a sequence {"'In} from 1'0 such that {G - CYn} covers G - D, i.e., such that D = n"'ln. Fix a point Xo of D. Define

n

In = [fyn(xo)]-I/Y n

•

Clearly S(fn) = "'In and Illn II = In(xo) = 1. Thus the series ~n~~l 2- n l n converges uniformly on G to a function g in ~r with Ilg II = g(xo) = 1. If x E G - "'I" then Ig(x) I ~ ~ 2- n lfn(x) 1< 1 since Iln(x) I ~ 1 for all n and Ilk(x) I < 1. Therefore S(g)C"'l". Thus S(g)Cn"'lk = D. Assume that S(g) contains more than one point. Since ~{ separates points, there exists ho in ~ which is not constant on S(g). We may assume that the maximum of Ihol on S(g) is 1 and that ho takes the value 1 at some point of S(g). If we set h = ho + h~, it follows that the maximum of Ih I on S(g) is 2 and that this maximum is attained only where ho takes the value 1. Thus Ihl is not constant on S(g). Therefore the set E = {x: x E S(g) and Ih(x) I ;;c: Ih(y) I for all y in S(g)}

is a proper closed subset of S(g). Let x, be any point in E. Define the functions

and ho = fh(X1)]-lh .

Thus ligo II = go(x 1) = 1 and S(go) = S(g). Also ho(x 1) = 1, Iho(X) I ~ 1 if x E S(g), and IhoCx) I < 1 if x E S(g) - E. Let K = Ilho II. For each positive integer n, let Vn = {x: 1

+ 2- n(K -

1) ~ IhoCx) I ~ 1

+ 2- n+ (K 1

In .

Clearly UVn= {x: Iho(x)I>I}. Thus VnnS(g) = VnnS(go) is void for each n. Therefore IOo(x) 1< 1 for each x in V n • Since Vn is compact, it follows that there exists a positive integer Pn such that IOo(x)jPn :;;:; 1/2

94

632

ERRETT BISHOP

for all x in V n •

Since Ilgoll ;:;; 1, the series ho

+ 4(K -

1)2: 2- n g0'" T!=1

converges uniformly on G to a function k in k(x l )

=

1

+

4(K - 1)2: 2- n

=

1

~l.

+

We have

4(K - 1) .

n=l

If x E S(g) - E, then Iho(x) 1< 1 and Igo(x) I = 1, so that Ik(x) 1< H- 4(K -1). If xEG - UV", then IhoCx) I ;:;; 1 and Igo(x) I ;;::: 1, so Ik(x)l;:;; 1 + 4(K-1). If XE Vj' then Iho(x)l;:;; 1 + 2- 1 + I (K -1), Iglr)I"n;:;; 1 for all n, and Igo(x) IP j ;:;; 1(2, so that Ik(x) I;:;; 1 + 4(K - 1). Therefore k(x,) = 1 + 4(K - 1) = Ilkll. Thus Xl E S(k) and S(k) is disjoint from S(g)-E. Since Xl E S(g)cD= nro'Y, and since S(k) E 1', it follows from the maximality of 1'" with respect to the finite intersection property that S(k) E 1'0. Therefore S(g)cnr,,'/cS(k). Since S(g) - E is non-void, this contradicts the fact that S(g) - E is disjoint from S(k). Therefore the assumption that S(g) contains more than one point is false. Thus S(g) consists of a single point Xo. It follows that XoE Mo. Since S(g)cD = nro'YcS(f), it follows that Xu E S(f) n Mo. Thus S(f) n Mo is non-void, as was to be proved. We now give an example to show that Theorem 1 fails if G is not metrizable. Let I denote the unit interval [0, 1] with the usual topology. Let I' be an uncountable set. Let G consist of all families X = [X",}aE., with x", E I for each (l'. Thus G is the Cartesian product of an uncountable number of intervals, and is therefore compact. Let ~( consist of all continuous functions f on G which have the property that there exists / a countable subset J of l' such that f(x) = fCy) whenever x and y an:; points in G such that Xx = y" for all ()' in J. It is easy to see that ~( is a separating Banach algebra of continuous functions on G. By the Stone-Weierstrass theorem it follows that ~l consists of all continuous functions on G. Let Nl = {x: x" = 0 except for a countable set of a] and N2 = {x: x", = 1 except for a countable set of a}. It is easy to see that Nl and N2 bound \lL Since Nl n N, is void, it follows that ~( does not have a minimal boundary. For an example of a function algebra whose minimal boundary is distinct from its Silov boundary, let G be the subset {z: \z\ = I; of the complex plane and let ~( consist of all continuous functions f on G which have the property that there exists a continuous function j on [z: \ z \ ;;::: 1] such that fez) =f(z) for z in G, such that j is analytic on {z: Iz\ < I}, and such that j(l) = j(0). It is easy to see that ~l is a separating Banach algebra of continuous functions on G. It is also not difficult to show that the Silov boundary of ~ is G, whereas the minimal boundary

95

633

A MINIMAL BOUNDARY FOR FUNCTION ALGEERAS

of 91 is the set {z:

Izl = 1,z *- 1} = C- {1}.

THEOREM 2. Let 'ill be a separating Banach algebra of continuous functions on a compact metrizable Hausdorff space C. For each positive integer n, let Un consist of all points x in C such that there exists f in ~l with Ilfll ;;;;; 1, If(x) I > 3/4, and If(y) I < 1/4 for all y in Dn(x), where Dn(x) = {y: p(x, y) ;:::: n- ' } and fJ is a metric on C. Then Un is open and Un = M, where M is the minimal boundary of ~l.

n

Proof. If f is any function in 'ill, it is clear that the set a n(f) = {x: x e C, If(x) I > 3/4, If(y) I < 1/4 whenever ye D,,(x)} is open for each n. Since Un is the union of the sets belonging to the class {an(f):f e ~l,

Ilfll;;;;; I} ,

it follows that Un is open. If x e M, by Theorem 1 there exists f in ~( with S (f) = {x}. It is clearly no restriction to assume that Ilfll = 1. Hence If(x) I = 1. Since If(y) I < 1 when y is in the compact set Dn(x), it follows that there exists a positive integer Pn such that If(y) IP n < 1/4 when ye D,,(x). Thus x e (In(fPn). Therefore x e Un. Since this is true for each n, it follows that x e Un. Therefore Men Un. Now consider a fixed x in Un. We must prove that x eM. To this end, we construct by induction a sequence {gn} of functions in ~( having the following properties:

n

(i)

(ii) (iii) (iv)

n

IIgn+! - gnll ;;;;; 2- n + IIgnll;;;;; 3(1 - 2- n -

1

1)

gn(x) = 3(1 - 2- n ) Ign+l(y) - gn(y) I < 2- n - 1 if ye Dn(x) .

We first construct g,. Since x e U" there exists a function that Ilfll ;;;;; 1 and x e (J,(f). Let gl

f in

~{

such

= ~ [f(x)]-lf .

Since If(x) I > 3/4, we have IlgIII;;;;; 3/2 . 4/3 = 2 < 3(1 - 2- 2 ), so that g, satisfies (ii). Clearly g,(X) = 3(1 - 2- ' ), so that gt satisfies (iii). Hence g, satisfies all of the relevant conditions. Assume now that g" "', gk have been chosen to satisfy all of the relevant conditions. Since gk(X) = 3(1-2- k ), there exists an integer }>k such that Igk(y)I<3(1-2- k)+2- k- 2 for p(x, y) < }-" i.e., for y in C - D;(x). Since x e U j , there exists a function f in 'iJ{ such that Ilfll;;;;; 1 and x e a;(!). Define h = 3· 2- k - 1[f(x)]-lf. Thus hex) = 3 . 2- k - \ Since IIfll ;;;;; 1 and If(x) I > 3/4, we see that IIh II ;;;;; 2- k + l . Since also If(y) I < 1/4 for y in Dj(x), we see

96

ERRETT BISHOP

634

that Ih(y)1 < 2-~-1 for y in Dlx). mediately that

Let gUI

=

gk

+ h.

It

+3.

2- k -

follows Im-

that

and that (iii) If y

E

gUl(X)

+ hex) =

=

gk(X)

=

3(1 - 2- k -

1)

3(1 -- 2-<)

Dix), then

Igu/Y) I ;;:; IOk(Y) 1+ Ih(y) I < Ilg, \I + 2- k - 1 ;;:; 3(1 - 2-<-1) = 3 - 2- k < 3(1 - 2-<-') . If y

E

1

•

+ 2-

k - 1

C - Dix), then

Igk+l(y) I ;;:; Igk(y) I + Ih(y)1 ;;:; 3(1 - 2- k ) + 2-<-2 + Ilhl\ ;;:; 3(1 - 2- k ) + 2-!A + 2-<+1 = 3(1 - 2-~-2) . It follows that

Thus gk+l has the relevant properties. We have thus constructed the sequence {gn}. By condition (i), the sequence {gn} converges uniformly / on C to a function 9 in 1)(. By (ii), Ilg II ;;:; 3. By (iii), g(x) = 3. II' y E Da(x), then Ig(y) I ;;:; Ilgn II

=

+ 2..: Igk+l(y)

- gk(y) 1< 3(1 - 2- n - 1)

k=n

Thus S(g)

=

{x}.

Therefore

=

+ L. 2-"-1 <

3 .

k=n

XE

M, as was to be proved.

COROLLARY. If I){ is a separating Banach algebra of continuous functions on a compact metrizable Hausdorjf' space C, then the minimal boundary M of ~1 is a countable intersection of open sets.

3.

Representation by measures.

We now prove the fundamental result of this paper. THEOREM 3. Let 1)1 be a separating Banach algebra of continuous functions on a compact metrizable Hausdorff space C. Let I){ contain the function 1. Let x be a point of C - M, where M is the minimal

97

635

A MINIMAL BOUNDARY FOR FUNCTION ALGEBRAS

boundary of ';If.

Then there exists a non-negative Borel measure

norm 1 on C - {X) such that f(x)

= ~fd'Y

=

{y: y

E

of

for all f in ';/1.

Proof. We assume that a metric p on C is given. tive integer n, let Dn

'Y

C, p(x, y) ::;; n- I }

For each posi-

•

Let band c be real numbers such that 0 < b < 1/4 < 3/4 < e < 1. For each positive integer n and each positive integer m, let hr.m be a continuous function on C such that hn"Jy) = bUm for y ED", hnm(y) = e'lm for y € C - D'n, and bl / m ~ hnm(y) ~ e'lm for all y. Such a function exists because the closures of the sets Dn and C - D'n are disjoint. There are two cases to consider. Either there exists (Case 1) for each positive integer n a positive integer m and a function f in ';It such that If(x) I > (3/4)'/m and If(Y) I ~ hnmCY) for all y in C, or (Case 2) there exists a positive integer n such that for all positive integers m and for all f in ';It either If(x) I ~ (3/4)1/'" or If(Y) I > hnm(y) for some y in C. We shall show that Case 1 is impossible and that Case 2 implies the theorem to be proved. Assume now that Case 1 obtains. Let the positive integer n be given, and choose fin ';!l and a positive integer m such that If(x) I >(3/4)I/m and If(y) I ~ hnm(y) for all y. Write g =fm. Since If(y) I ~ hnm(y) ~ e' /m for all y, we have Ig(y)1 ~ e for all y . .Thus Iigll ~ e < 1. Since If(x)1 > (3/4)I/m we have Ig(x) I > 3/4. Since IfCy) I ~ hnm(y) = bl / m for y in D n, we have Ig(y) I ~ b < 1/4 for y in Dn. It follows that x E Un, where Un is the set defined Un = M, in Theorem 2. Since this is true for each n, we have x by Theorem 2. This contradicts the hypothesis of Theorem 3. Therefore Case 1 is impossible. We are therefore justified in assuming that Case 2 obtains. Thus there exists a positive integer n, henceforth fixed, such that for all positive integers m and allf in ~l either If(x) I ~ (3/4)I/m or I fey) I>hmn(y) for some y in C. Consider now a positive integer m. For each f in ';!( either If(x) I ~ (3/4)I/m or Ilfh- 1 11 > 1, where h = h nm . Thus If(x) I ~ (3/4)I/1n whenever f E ';II and Ilfh- 1 11 ~ 1. Let)S be the Banach space of all continuous functions on C, under the uniform norm, and let ~\ be the subspace Uh- I : f e ~{} of 5B. Define the linear functional cp on l.L~o by defining
en

98

636

ERRETT BISHOP

for all f in

~L

If we define the measure

=

/I",(S)

flm

by

Lh-ldJ)m

for all Borel subsets S of C, it follows that f(x) =

kdllm

for all f

in

~L In particular, 1 = Jdflm. Thus II/1m 11:;0; 1. Let the measure J):n, the restriction of J)", to the set Dw be defined by J)~,,(S) = J)m(S n D for each Borel set S. Let l.I~", the restriction of l.I mto C - D,,,, be defined 2n )

similarly.

Thus

II J)~n II +

IIl.I~"

II

=

II J) m II;::; (43)1/'" .

Similarly, let f/;" be the restriction of II,,, to D 2n , and let fl:" be the restriction of 11m to C - D2n • Thus Ilfl~11

+ 11f!~11 = IIp,,,II:;o;

1 .

Since [h(y)]-l = e- 1 / m for all y in C - D'n and since p:,.(S) =

L-n

h-ldJ)m, 2'1

for all Borel sets S, we see that /1;, = e-l/mJ)~" so that el/"'II P:" II = Ill.I:" II. Since Ih(y) 1-1;::; b- 1 / m for all y, and therefore for all y in D"" we see similarly that bl/mll,u~n II ;::; Ill.I~n II. Thus

Combined with the inequality

this gives

Thus

Since IIPml1 = Ilp~11

+ 11p~ll;::; b-l/mllJ)~,,11 + b-l/mlll.l:" II ;::; b-

1/ m

ll J)m II ;::; b-

1/

m ,

there exists a subsequence {Pm) of {Pm} which converges in the weak star topology for measures ou C to a measure ,u on C with 11/111 ;::; 1. Also,

99

637

A MINIMAL BOUNDARY FOR FUNCTION ALGEBRAS

for each f in ~. Since C - D 2n is open and since

for each m, we have

where p' is the restriction of p to C - D,n.

! rl/!~ ! )(! r !-

ll~ [(

bl')c l' - bh]-I

[(In

=

Now

= [In

(lnb)b h ][(lnc)c h

(lnb)bh]-I

-

lnb JElnc - lnb]-I = In(! b-I)[ln(cb-I)]-I

<1.

Thus, if p2 denotes the restriction of p to the set {x}, we have II p2J1 ~ II p'Jl < 1. Thus there exists a constant a with 1a 1< 1 such that 1fdp2 = af(x) for all t!ontinuous functions f on C.

tion of It to C - {x}, so that p2 Thus 1fd P = 1fd P2

for all f in

+

+ p3 =

k P3 d

~1. Therefore (1 - a)f(x)

IIp3J1~1-lIp211=1-lal,

111dPai ~ IIPaJl

~ 1-

Let p3 be the restric-

P and IIfl211

= af(x)

+ IIp 11 =11/111 ~ 1. 3

+ 1fdPa

= 1fdP3 for all f

in~.

Since

and since 1€~, we have 1-lal~11-al=

lal. Thus a is positive.

Define 7 = (1 - a)-'p3.

Thus f(x) = (1 - a)-'1fdP3 = 1fd7

for all f in~. 1 = 1d7.

Also, 117 II ~ (1 - a)-'lIp311 ~ 1.

Since 1 € ~{ we have

Therefore 7 is a non-negative valued measure.

This completes

the proof of the theorem. COROLLARY.

Let

~

be a separating Banach algebra containing the

100

638

~KKt;Tl

HI::'HUY

unit function of continuous functions on the compact metrizable Hausdorff space C. Let ~o consist of all functions on C which are real parts of functions in ~, and let ~ be the uniform closure of mo. Let Mo consist of all points x in C such that there exists f in ~H with If(x) I > If(Y) I for all y *- x in C. Then Mo equals the minimal boundary M of~. Proof. If x eM, there exists g in C such that Ig(x) I > Ig(y) I for all y *- x in C. It is no loss of generality to assume that g(x) = 1. If we let f be the real part of g, then f e lnoC~ and If(x)l = Ig(x)l > Ig(y) I ~ If(Y) I for all y *- x. Hence x e Mo. If x is not in M, then there exists a real-valued measure ry on M - {x} of norm 1 such that g(x) = ~gdry for all g in ~, by Theorem

3.

Since ry is real-valued, it follows that f(x) =

Thus f(x) =

~fd'Y for all f in

f in lH with 1

11 f d'Y 1< 1,

= f(x) >

m.

mo.

If x where in Mo, there would exist

If(y) I for all y

since If(y) I < 1 for y

~fdry for all f in

*- x.

Thus 1

*- x and since II?, II

= f(l) = ~fd'Y ~

= 1.

This contra-

diction shows that x is not Mo. Hence M = Mo, as was to be proved. DeLeeuw has found a proof of Theorem 3 which is somewhat simpler than the one given here. 4. Applications. We now apply the results of the previous sections to certain problems of approximation in one complex variable. DEFINITION 3. Let C be a compact subset of the complex plane. Then Ao(C) will consist of all continuous functions on C which are analytic at interior points of C, and A/C) will consist of all continuous functions on C which can be uniformly approximated arbitrarily closely by rational functions whose poles lie in the complement of C. It is clear that A l (C)c A 0(C), and that A 0(C) and A1(C) are separating Banach algebras of continuous functions on C. Mergelyan [2] has shown that A o(C) = A 1(C) in case the complement of C consists of only a finite number of components. No necessary and sufficient condition is known that A1(C) = A0(C). In case C has no interior, we shall obtain in Theorem 5 below a necessary and sufficient condition that every continuous function on C be uniformly approximable by rational functions with poles in - C. THEOREM 4. Let C be a compact subset of the complex plane with no interior and let M be the minimal boundary of A 1(C). Then either

101

A MINIMAL BOUNDARY FOR FUNCTION ALGEBRAS

A,cC)

= Au(C)

639

or C - M has positive 2-dimensional Lebesgue measure.

Proof. Assume that A,(C) *- Ao(C). We must show that C - M has positive 2-dimensional Lebesgue measure. Now A (C) is the Banach space of all continuous complex-valued functions on C, and A,(C) is a proper subspace, since A,(C) *- Ao(C). By the Hahn-Banach theorem, there exists a continuous linear functional 'P *- 0 on A (C) which vanishes " exists a finite on A1(C). By the Riesz representation theorem, there complex-valued Borel measure II on C which represents 'p. Thus It *- 0

and

~fdfL = 0 for all

f

in A,(C).

In particular,

~(Z

-1;)-'d,IL(t)

=0

whenever z is not in C, since the function (z - 1;)-' is a rational function of I; whose pole, z, is not in C. Since the function Z-I of z is integrable with respect to Lebesgue measure dxdy over any finite region of the plane, and since It is a finite measure on the compact set C, the integral

will exist for almost all values of z, and the function h(z) so defined, called the convolution of the measure p and the function Z-' and written h = z-I,,-p., will be integrable with respect to Lebesgue measure over any finite region of the plane. Since we have seen' above that h(z) = 0 if z is not in C, it f9llowS that h is integrable. Assume that the integrable function h vanishes almost everywhere, so that the integral h(z) = all z. tion

1

(z - I;)-'dp(t;) exists and vanishes for almost

To obtain a contradiction from this assumption, we use the equa-

aa. l. z z

=

no

([7] p. 49) from the theory of distributions. This means

that for any function g on the complex plane which vanishes in a neighborhood of co and which has continuous partial derivatives of all orders we have

- i r(z JJ

for all values of 1;.

-

t)-ll.(~ + i ~)g(z)dxdy 2

ax

ay

If we write gl(Z) =

~

(a:

= 7rg(t;)

+ i :y )g(Z) and integrate

both sides of the above equation with respect to p we obtain

HH (z - t;)-lgl(Z)dxdY}dp(t;) = - Hg,(Z)U (z - t;)-ldp.(t;) }dxdy = 0 ,

7r)g(t;)dP.(t) = -

102

ERRETT BISHOP

640

since h(z) vanishes almost everywhere. The use of Fubini's theorem is justified since z - t, and therefore (z - t)-', is measurable with respect to the product of the measures g,(z)dxdy and dp(O and since

is finite.

Now every continuous function go on C can be uniformly ap-

proximated by such functions

g,

that~go(t)d/l(O

so

=

O.

By the unique-

ness part of the Riesz representation theorem, it follows that II = O. This contradiction shows that there exists a set r of C of positive Lebesgue measure such that the integral h(z) exists and does not vanish for all z in r. We may clearly assume that at no point of r does /1 have point mass. Let Zo be any point in r, so that h(zo) = c exists and is not zero. Let f be any function in A,(C) such that f(zo) = O. Let Un} be a sequence of rational functions with poles in the complement of C converging uniformly to f on C. Since fn(zo) ~ f(zo) = 0 as n ~ oa, we see that {g",} converges uniformly to f on C, where Yn = fn - fn(z,,). Thus gn is a rational function with poles in - C which vanishes at zo, so that there exists a rational function g~ with poles in -C such that gn(z) = g~(z)(z - zo) for all z. Hence I

,!

for each n, since V;, E

AI(C).

Passing to the limit, we see that

~ f(z)(z

- zo)-ld/l(Z) = 0 .

Since this is true for all f in A,(C) with f(zo) = 0, it follows that for an arbitrary f in A,(C) we have

~ f(z)(z

- zo)-ldf1(z) =

+ ~f(Zo)(Z -

~ [fez)

zo)-ld/l(Z)

=

- f(zo)](z - zo)-ld/l(Z)

0

+ f(zo)( -h(zo» =

If we let Ou denote the measure of mass that J)

= (z - ZO)-'f1

+1

-cf(zo) .

at the point zo, it follows

+ COo

is a measure on C which annihilates A,(C). Now if Zo were in M, there would exist f in A,(C) with f(zo) = 1 and If(z) I < 1 for all z *- Zo in C. Since J)({zo}) = c

*- 0, it is clear that ~rdJ) *- 0 if n is sufficiently large.

103

l\ MINIMAL BOUNDARY FOl{ FUNCTION ALGEBRAS

641

This is a contradiction, since fn E AI(C). This shows that Zu is not in M. Since Zu was any point in r we have Fe C - M. Since /' has positive measure, C - M has pos-itive measure, as was to be proved. To restate the theorem, if for every point z on C, with the possible exception of a set of measure 0, there exists a continuous function f on C with If(z) I > If(n I for all I; cd=- z in C which can be uniformly approximated by rational functions with poles in - C, then every continuous function on C can be uniformly approximated by rational functions whose poles lie in - C. THEOREM 5. Let C be a compact subset of the complex plane without interior. Let Au(C) be the algebra of all continuous complex-valued functions on C and let A'(C) be those functions in AJC) which can be umformly approximated by rational functions with poles in - C, and let M be the minimal boundary of A I(C), Let Ju(C) consist of all continu01.ls real-'wlued functions on C, and let J,(C) consist of all continuous real-valued functions on C which are uniforrnly approximable by real parts of functions in A I(C), Let Mo consist of all points z in C such that there exists f in JI(C) with If(z) I > If(l;) I for all I; z in C. Then M = Mu and the following statements are equivalent:

"*

(i)

(ii) (iii) (iv)

AI(C) =-= Au(C) C - M has measure 0 M = C J,(C)

=

Ju(C) .

Proof. The fact that M = Mil is a special case of the corollary to Theorem 3. It is clear that (i) ~ (iii) ~ (ii). But (ii) ~ (i) by Theorem 4. Thus (i), (ii), and (iii) are equivalent. It is also clear that (i) ~ (iv). But (iv) implies that M" = C. Thus (iv) ~ (iii). This proves Theorem 5. Theorem 5 thus gives results concerning approximation on a nowhere dense subset of the complex plane by rational functions or by real parts of rational functions, and shows that the two problems are related. The results for approximation by the real parts of rational functions are similar in outward appearance to results of Brelot [2] and Deny [4], who consider approximation by functions harmonic in a neighborhood of C, but there does not seem to be an essential connection, due to the fact that a function harmonic in the neighborhood of C need not be the real part of an analytic function, since the conjugate harmonic function might be mUltiple-valued. REFERENCES

E Bishop, Some theOl'ems concerning function algeb7'as, Bull (1959), 77-78.

104

Amer Math. Soc 65

~KK~

1 1

Ol;:,nvr

2. M. Brelot, Criteres de refJUlarite et de stabilite, Acad Roy Belgique Bull Cl Sci 25 (1939), 125-137. 3. G. Choquet, Existence dC's npresentation8 int6graZ!'s au moyen des points ertremaur dans les cones converes, C R. Acad Sci. Paris 243 (1956), 699-702 4 J. Deny, Sur l'approrimation des fonctions harmoniques, Bull Soc Math France 73 (1945), 71-73 5 L. H. Loomis, Abstract harmonic analysis, New York, 1953 6 S. N Mergelyan, On the representation of functions by series of polynomials on closed sets, Amer. Math. Soc Translation no 85, 1953 7. L. Schwartz, Theorie des distnbutions vol 1, Actualities Scient et Indust, Paris, 1951 UNIVERSITY OF CALH'ORNIA BERKELEY, CALIFORNIA

105

the American Mathematical Society.

SOME THEOREMS CONCERNING FUNCTION ALGEBRAS BY ERRETT BISHOpl

Communicated by

J L Kelley, November 21, 1958

In what follows, C wIiI be a compact Hausdorff space, and m: will be a uniformly closed separating algebra of continuous complexvalued functions on C. That is, sums, products, and complex multiples of elements of mare in m, uniform limits of elements of mare in m, and for distinct points x and y in C there existsfin mwithf(x) y5.f(y). SiJov (see [1]) has shown that there exists a smallest closed subset B of C, called the Silov boundary of m:, such that for eachf in m: there exists in B with = where = max {IJ()') y E C}. The following theorem generalizes this result, in case C is metric.

If(x) I IVII '

x

I:

IVII

THEOREM 1. Let C be a compact metnc space, and let m: be a uniformly closed separating algebra oj continuous complex-valued Junctions on C. Then there exists a smallest subset M oj C, called the minimal boundary of m, such that for each fin 21: there exists x in M with If(x) = The set M 1'S equal to the set M 0 which is defined as follows:

I II! II .

Mo=

lx:xEc,

3finmwith If(x)

I>

If(y)lforallyy5.xincl.

The closure of M is the Silov boundary of 21:. The question of the topological structure of .M is answered by the following theorem. THEOREM

Un =

2. Lt't p be the metric on C. For each positive integer n, let

Ix: X E C,

3fin 21: with

IIfll

I f(x) I > 3/4,

~ 1,

and

I fey) I < 1/4 for

p(x, y) ~

,,-l}.

Then nUn=M.

Since it is easy to see that Un is open, it follows that },{ is a G•. lt is known that every bounded linear functional cf> on mcan be represented by a complex-valued Borel measure J.L with IIJ.LII = 11cf>1I on the Silov boundary B of 21:. I t is conjectured that J.L can be taken to be a measure on the minimal bounary of m:. The following theorem constitutes an important case of this conjecture. THEOREM 3. Let x be a point of C which is not in the minimal boundary M of m:. Let mcontain the constant functions. Then there exists a non-negative Borel measure J.L on B-1 x} of norm 1 such that f(x) = fdJ.L for all f in 21:.

J

1

This research was supported by the Sloan Foundation

77

106

,u

~AI\.C.l

I.

Dhll'lVl.

As an application of these concepts, we have the following theorem. THEOREM 4. Let C be a compact subset of the complex plane without interior. Let ~ be the algebra of all continuous functions on C which are uniform limits of rational functions with poles in - C. Let M be the minimal boundary of ~. Let 58 be the set of all continuous real functions on C which are umform limits of the real parts of rational junctions with poles in - C. Let N be the set

{x: x E C, 3f in 58 with If(x)

I>

IfCy)

I for ally ~

xin C}.

Then ]vI = N and the following conditions are equivalent: (i) ~ consists of all continuous functions on C, (ii) 58 consists of all continuous real functions on C, (iii) M = C, (iii i) C - .1£ has measure O. Theorem -l gives a necessary and sufficient condition on a compact subset C of the complex plane that every continuous function on C be uniformly approximable by rational functions Mergelyan [2] has given sufficient conditions. Added in proof. The conjecture preceding Theorem 3 has been proved. REFERENCES

1. L. H. Loomis, Abstract harmonic analysis, New York, 1953.

2. S. N. Mergelyan, On the representation of functions by series of polynomials on closet! sets, Amer. Math Soc. Translations, no 85, 1953. UNIVERSITY OF CALlrORNIA, BERKLEY

107

THE REPRESENTATIONS OF LINEAR FUNCTIONALS BY MEASURES ON SETS OF EXTREME POINTS by Errett BISHOP and Karel de LEEUW (I)

1. -

INTRODUCTION

Let X be a compact Hau'idorff space.

We shall denote by

C,(X) the real Banach space of continuous real valued functions on X, and by Cc(X) the complex Banach space of continuous complex valued functions on X, supplied with norms denoted by 11.11 and defined by

IIfII =

Sup i((x) I· XE'{

Suppose that B is some linear subspace of either Cr(X) or Cc(X) that distinguishes points of X and that contains the constant functions. We are concerned in this paper with the problem of representing the linear functionals in the dual space B* of B by measures on X. It is well known that such representations are always possible; if B is a linear subspace of Cr(X), by the Hahn-Banach theorem, any L in B* extends to a continuous linear functional of Cr(X) and thus by the Riesz representation theorem, there will be some signed Baire measure !J. on X so that

(1. 1)

L(f)

=

J(dp.,

all (in B.

If B is a linear subspace of Cc(X), a similar argument shows (1) The authors wish to thank the Sloan Foundation and the United States Air Force Office of Scientific Research, respectively, for the support of this research.

20

108

306

ERRETT BISHOP ET

K.~REL

DE

LEEUW

that each L in B* has a representation of the form (1. 1) for [J. a complex valued Baire measure. There are cases in which it is possible to find a. subset Y of X which is such that each L in B* has a representation of thp form (1. 1) for some [J. concentrated on Y. In this paper wp introduce such a subset, the Choquet boundary of B. The Choquet boundary of B is denoted by M(B) and consists of all points x in X having the following property: there is a unique positi()e Baire measure 1-'- that represent'>, in the sense of (1. 1), the linear functional L~ defined by (1. 2)

all fin B.

LAf) = f(x),

This unique 1-'- will of course be the unit point mass at x. In the case that B is a uniformly closed subalgebra of Cc(X) and X is metrizable, M(B) is the minimal boundary of [3] and [4]. We show in Section 4 that the extreme points of the subset

!L: L E B*,

L(1) = IILII = 1l

of B* are those Lc defined by (1. 2) for x in M(B). From this and the Krein-Milman theorem it follows easily that any L in B* has a representation of the form (1. 1) with [J. a measure concentrated on the closure of M(B). The question that concerns us is whether it is possible to choose the measure 1-'- so that it is concentrated on M(B) itself. / If X is metrizable, an application of the theorem of Choquet / in [7] shows that this is indeed possible. We proceed in the reverse direction, showing directly that the measure can be concentrated on M(B); this leads to a relatively simple proof of the Choquet theorem. In the case that X is not metrizable the situation is much more complicated. We give examples in the concluding section of the paper to show that M(B) need not even be a Borel set. We prove nevertheless that each L in B* has a representation of the form (1. 1) for a measure 1-'- that is ({ concentrated on M(B) » in following sense: it is a measure on the a-ring generated by M(B) and the Baire sets of X, and is zero on each set in this a-ring which is disjoint from M(B). Furthermore if L(1) = liLli, the measure 1-'- can be chosen to be non-negative. This leads to an extension of the theorem of Choquet to convex sets that are not metrizable.

109

THE

REPRESENTATIONS

OF

LINEAR

FUNCTIONALS

307

The existence of measures concentrated on the Choquet boundary is obtained roughly as follows. An ordering relation on the class of non-negative Baire measures on X is introduced. We say that fJ- is a B-cover of YJ if

J f dYJ J f dfJ-, I f2 dYJ < f f2 dp., =

and (1. 3)

allfinB, all fin B.

We say that fJ- is a proper B-col'er of Yj if fJ- is a B-cover of YJ and furthermore the inequality in (1. 3) is strict for some fin B. YJ is called B-maximal if it has no proper B-eover. A simple argument using Zorn's Lemma and weak* compactness assures for any given non-negative Bairc measure "1), the existence of a B-maximal fJ- that is a B-cover for "1). The crucial result now is theorem 5. 3 which shows that fJ-(S) = 0 if fJ- is B-maximal and S is disjoint from M(B). From this it follows simply that any B-maximal po can be extended to a measure « concentrated on M(B) » in the sense described above. Since each linear functional in B* has a representation of the form (1. 1) for some Baire measure fJ- on X, and such a p. is a linear combination of non-negative Baire measures, and each nonnegative Baire measure has a B-maximal B-cover that is concentrated on M(B), it follows that any linear functional in B* has a representation of form (1. 1) for some p. concentrated on M(B). Section 6 is devoted to uniformly closed subalgebras of C.(X). We give somewhat simpler proofs of some of the results of [3J and [4J and remove the hypothesis of metrizability of X imposed there. We show that if A is a uniformly closed subalgebra of Cc(X) that distinguishes points of X and contains the constant functions, the points x of M(A) can be characterized by either of the following conditions: I. For each neighborhood U of x there is a function f in A with Ilfll

< 1,

f(x)

> ~

and If(y) I <

{

for all y not in U.

I I. If S is a closed Go containing x, then there is some in A with If(x)1 = Ilfll

and

!y: If(y)1

=

f

Ilflll c S.

A subse.t Y of X is said to be a boundary for A if for each f

308

ERRETT BISHOP ET KAREL DE

LEEUW

in A there is some y in Y with If(y)1 = Ilfll. It is a simple consequence of the Krein-Milman theorem that the Choquet boundary of A is a boundary for A. ConditioR II shows that in addition, any Baire subset of X that is a boundary for A must contain M(A). Furthermore if X has the property that each point is a Go (in particular if X is metrizable), condition II shows that for each point x of M(A) there is some f in A that « peaks» at x, so that M(A) is the smallest boundary for A, that is, the minimal boundary (for X mctrizable, this was established in [3J and [4J). If not every point of X is a Go, there may bp no smallest boundary, even if A is all of Cc(X). Since each L in A* has a representation of the form (1. 1) for p. a measure concentrated on M(A), it is reasonable to inquire whether for any set Y that is a boundary for A, a measure can be found that represents L and is concentrated on Y. We show by example in Section 7 that thi"l cannot be done for linear subspaces of Cc(X) that are not subalgebras. Nevertheless we are able to show that for sub algebras such measures can always be found. The existence of these measures was suggested to us by Irving Glicksberg, who studied essentially the same problem for the case A = Cc(X) in [8J. In all that follows, by « subspace of Cr(X) » (or « of Cc(X) ») we shall mean a linear subspace containing the constant functions, but not necessarily closed or distinguishing points of X.' For applications it is useful to have results concerning not necessarily closed linear subspaccs; furthermore it is necessary for technical reasons for us to consider linear subspaces that do not distinguish points of X. By « measure » in the following, we shall always mean finite measure.

II. -- THE BASIC DEFINITIONS

In the following X is a fixed compact Hausdorff space. We shall denote by H the class of all non-negative Baire measures on X. This class H will be identified in the usual manner with a subset of the dual space of Cr(X). By the weak*

111

I

THE

309

I1EPRESENTATIONS OF LINEAR FUNCTIONALS

topology of H we shall mean the restriction to H of the weak* topology on the dual space of Cr(X). The basic fact concerning the weak* topology that we shall need is that bounded closed subsets of Hare weak* compact. The following allows us to reduce questions concerning linear functionals on sub5paces of Cr(X) or Cc(X) to questions about the Haire measures in H.

2. 1. - Let B be a subspace of Cr(X) or Cc(X). each L in B* has a representation of the form LEMMA

f fdfJ-,

L(f) =

(2. 1)

Then

all fin B;

where p. is a Baire measure which is of the form

(2. 2) ifB is a subspace of Cr(X), and of the form

p.! -

fJ-2

+ ifJ-J -

fJ-1l ••• , p.. in H,

ifJ-4I

if B is a subspace of Cc(X). Furthermore the fJ- in (2. 1) can be chosen to be in H if and only if L(1) = IILII. PROOF. We shall treat the case of B a subspace of Cr(X); the proof for Cc(X) is completely analogous. By the HahnBanach theorem, L can be extended to all of Cr(X) with preservation of norm; i.e. there is a linear functional L' on Cr(X) with L'(f) = L(f), all fin B,

and IIL'II = IILII. By the Riesz representation theorem, there is a signed Baire measure p. on X so that

L'(f)

=

.r

f dp.,

all fin Cr(X).

Each signed Baire measure on X is of the form (2. 2). L(1) = liLli, then

11fJ-11

=

IIL'il

=

IILII

=

L(1)

=

If

[J.(X),

so p. must be non-negative and thus in H. Conversely if p. is in H, the L defined by (2. 1) clearly satisfies L(1) = IILII. If B is a subspace of Cr(X) or Cr(X), and x is a point of X, we define H",(B) to be the subset of H consisting of all fJ- with (2. 3)

f f dfJ-

= f(x),

112

all fin B.

310

ERRETT BISHOP ET

K <\REL

DE

LEEUW

H.,(B) is always non-empty since it must contain at least the unit point mass at x. Since subs paces are assumed to contain the constant functions, by (2. 3) each [J. in H.,(B) satisfies p.(X) = 1. If S is any subset of X and B is any collection of functions on X, we define i8(S) to be the subset of X consisting of those points that cannot be distinguished from points of S by the functions in B; i.e. iH(S) is

1y : f(y) =

f(x)

for some

xeS and all

feB

l.

If P. is any Baire measure in H, we denote by p. its regular Borel extension. This is the unique regular Borel measure on X that agrees with p. on the Baire sets of X. It is defined by

P.(S)

=

Inf p.(U),

where U runs over all open Baire sets that contain S. If B is a subspace of either Cr(X) or Cc(X), the Choquet boundary of B, denoted by M(B), is defined to consist of those points x in X which are such that any p. in H..,(B) satisfies p.(in(x)) = 1. If B contains sufficiently many functions to distinguish a point x from all other points of X, ill(x) = fxl, so that x will be in the Choquet boundary of B if and only if the unit point mass at x is the only p. in Hc(B). If B is a subspace of Cc(X), we shall denote by Br the subspac(y/ of Cr(X) consisting of real parts of the functions in B. The following is immediate. LEMMA

H,,(B r )

=

2. 2. H,,(B).

M (Br)

=

M (B), and for each x in X,

We next introduce some ordering relations on H which are basic to the constructions that follow. If B is a subspace of Cr(X) and fI. and Yj are in H, we shall say that fI. is a B-co~er of Yj if all fin B, and (2. 4)

all fin B.

We shall say that p. is a proper B-cover of YJ if P. is a B-cover of YJ and furthermore the inequality (2. 4) is strict for some fin B.

113

THE

A measure

REPRESENTATIONS

OF

LINEAR

FUNCTIONALS

311

will be called B-maximal if it has no proper

Yj

B-co~er.

Much of the work done in the remainder of the paper goes into demonstrating that any B-maximal measure must be concentrated on the Choquet boundary of B. The following two lemmas will be applied later. LEMMA 2. 3. Let x be a point of X and Yj in H be the unit point mass at x. Let B be a subspace of Cr(X). Then any [.I. in Hx(B) is a B-cover of Yj.

By the Schwarz inequality, for each fin B,

PROOF. -

Furthermore for all Yj

f

in B.

LEMMA 2. 4. Let B be a subspace of Cr(X). in H has a B-co~er that is B-maximal.

Then each

PROOF. Consider subsets! [.I.a.! a.EJ of H indexed by totally ordered sets J, where the ordering is such that fAp is a B-cover of [.I.a. if IX <~. By Zorn's lemma there is a maximal such subset [.I.a.l a.EJ that contains Yj. Since B contains the constant functions, each [.I.a. is in

!

(2. 5)

!v:vEH,

v(X) = Yj(X) I.

(2. 5) is compact in the weak* topology and thus contains a weak* cluster point [.I. for the net ! [.I.a.l a.EJ. It is clear that [.I. is a B-cover of Yj and that [.I. is B-maximal.

III. -

REPRESENTATION OF LINEAR FUNCTIONALS IN THE SEPARABLE CASE.

The main result in this section is Theorem 3. 2. Using it we establish in Theorem 3. 4 the possibility of representing linear functionals on B by measures on the Choquet boundary of B in the case that B is separable, that is, has a countable

114

dense subset. Theorem 3. 2 will also be applied later in the non-separable case. We shall need the following well known lemma. LEMMA 3. 1. Let S be a closed subset of X. each positilJe real number c, the subset

f p.:

!J. e H,

!J.(X)

=

p.(T)

=

c,

for some finite

Then for T c S~

of H is dense in

!P.:!J.E H,

p.(X) = ~(S) =

c!

in the weak* topology. We can now prove THEOREM 3. 2. Let B be a subspace of Cr(X), and p. in H be a B-maximal measure. Let S be a closed subset of X which has the following property: there is a separable subspace D of B so that for each x in S there is some IJ' in HAB) with lJ'(in(x)) < 1. Then (l(S) = O. The statement of this theorem is necessarily complicated as it must be applied later to the situation where B is not separable. In that case D will necessarily be a proper subspace of B and will not distinguish points of X. In the application in this section however D = B, and in this case the hypothesis on S in the theorem becomes simply that it be disjoint from the Choquet boundary of B. /

!

PROOF OF THEOREM 3. 2. Let f fn: n = 1, 2, ... be a countable subset of D that is dense in D. For each pair of positive integers nand m define Lnm to be the subset of X consisting of all x for which there is some po in H",(B) with.

(3. 1)

For if !xa; ~ is a net of points in Lnm converging to a point x in X, and if for each or; a measure !J.a; is chosen in H",jB) so that

Lnm is closed.

by the weak'" compactness of

I!J.: fL e

po(X) =

H,

115

11

THE

REPRESENTATIONS

OF

LINEAR

313

FUNCTIONALS

the net j fJoa.l will have a weak· cluster point fJo in H...,(B), and fJo will satisfy (3. 1). We will show next that

S c U Lnm.

(3. 2)

Let x be a point of X that is in none of the L HAB). By the definition of the L nm ,

(3.3)

Jf!dcr

<

(fn(x))2

=

IJ{n dcr I 2

Il,".

Let cr be in

•

On the other hand, by the Schwarz inequality, (3. 4)

f f,. da J{~ dcr f 12 da > f {n dcr =

1

12 ,

with equality if and only if {n is constant a.e. with respect to cr. Comparing (3. 3) and (3. 4) we see that each (n must be constant a.e. with respect to cr. Since also fn(x) = (n da, it follows that fn is equal to (,,(x) a.e. with respect to cr for each n. Thus the set

J

in(x)

=

[y: (n(Y)

= (n(x),

n = 1,2, ... ]

has a measure 1. Since this holds for each cr in HAB), x cannot be in S. This completes the proof of the inclusion (3. 2). We now show that fl(S) > 0 contradicts the B-maximality of fJo. Suppose that P.(S) > O. Then by (3. 2), fl(Lnm) > 0 for some Lnm. Let v be the measure in H that is the restriction of fJo to Lnm : i.e., v(T) = fl(T n Lnm) for all Baire sets T. Now v =/= 0 since v(X) = fl(Lnm) -=1= O. By Lemma 3. 1, v is the limit in the weak* topology of a net [va.] of measures in H with va.(X) = v(X) and each Va. concentrated on a finite subset of Lnm. By Lemma 2. 3 and the definition of L llm it follows that there exists a corresponding net [YJa.] of measures in H such that each YJa. is a B-cover of Va. and in addition that

f f,.

dYJa. L

v~) + .J 'f,. dva..

If YJ is any weak· cluster point of the net [YJa.] , YJ l~ a B-cover of v and

J'r,. dYJ Lvi;) + f r. dv.

Thus YJ is a proper B-cover of v and it follows that (fJo -

116

v)

+ YJ

is a proper B-cover of fJ-. This contradicts the B-maximality of fJ- and shows that (1(5) must be O. This completes that proof of Theorem 3. 2.

COROLLARY 3. 3. - Let D be a separable subspace of C,(X). Then the Choquet boundary of D is a Gil. If YJ is any measure in H, then there is some measure fJ- in H with (3. 5)

J fdp. = J fdYJ,

all fin D,

and which is concentrated on M(D); i.e. satisfies f1(M(D)) = p.(X).

PROOF. - Let f fn: n = 1, 2, ... j be a countable subset of D that is dense in D. Define the closed subsets Lnm of X as in the proof of theorem 3.2. We shall show that M(D) is the complement of U Lnm and is thus a Ga. Let x be any point which is not in M(D). Then that part of the hypothesis of theorem 3. 2 which concerns B, D, and 5 is satisfied if B is taken to be D and S to be Ix j. The proof of Theorem 3. 2 shows that x is in U Lnm. Conversely, from the definition of the Lnm it is clear that no point in U Lnm can be in M(D). Thus M(D) is a Gil as claimed. Now let YJ be any measure in H. Let p. be aD-maximal measure in H that is a D-cover of YJ. The existence of such a p. is guaranteed by Lemma 2. 4. Equality (3. 5) holds since p. is a D-cover of YJ. To show that f1(M(D)) = fJ-(X), by regularity of f1 it suffices to show that (1(S) = 0 for each closed S that is disjoint from M(D). That this holds is a consequence of Theorem 3. 2 for the special case D = B. This completes the proof of Corollary 3. 3. The following is an immediate consequence of Lemma 2. 1 and Corollary 3. 3. THEOREM 3.4. - Let D be a separable subspace of C,(X) or C.(X). Then any linear functional L in D* has a representation of the form (3. 6)

L(f) =

Jf dp.,

all fin D,

for p. a Baire measure on X that is concentrated on M(D) in the sense that (1(S) = 0 for each Borel set S disjoint from M(D). Furthermore the p. in (3. 6) can be chosen to be in H if and only if L(1) = IILII.

117

THE

IV. -

REPRESENTATIONS

OF

LINEAR

FUNCTIONALS

315

EXTREME POINTS AND THE CHOQUET THEOREM

Let E be a real locally convex topological linear space with dual space E*. Let X be a compact convex subset of E and B the linear subspace of C,(X) consisting of all functions ( ha ving the form

((x) = F(x)

+ c,

all x in X,

for some F in E* and some real constant c. If p. is a real-valued measure on X whose domain includes the Baire sets of X and if S is a Baire subset of X, we shall denote by (4. 1)

.fs

the unique element y of E that satisfies F(y) = F dfJ- for each F in E*. We shall usc standard properties of the vector valued integral (4. 1) that are discussed for example in [6]. The following lemma was announced by Bauer in [2]. LEMMA 4. 1. Let x be a point of X. are equiyalent: 1. x is an extreme point of X. 2. x is in M(B).

Then the following

To show that 2 implies 1, let x be a point of X 1 that is not extreme. Then x = 2 (u y) for some u and y PROOF. -

+

=F y. Thus the measure fJ- in H that satisfies (l( f u}) = (l( f y}) = ~ and p.(X) = 1 is in Hx(B) by the defini-

in X with u

tion of B. Since the functions of B distinguish points of X, in(x) = Ix} so p.(ill(x)) = 0 and x is not in M(B). To show that 1 implies 2, suppose that x is an extreme point of X. Let fl. be in H,,(B). We shall show that fJ- is the unit point mass at x so that x is in the Choquet boundary of B. Since fJ- is in Hx(B), x = Y dfJ-(Y). If S is any Baire subset

f

118

ERRETT

BISHOP ET

KAREL

DE

of X with 0 < p.(S) < 1, and T = X tion as a convex combination

of points in X. (4.2)

p.(S)x

LEEUW

S, x has a representa-

Since:L is an extreme point of X, =

J:y dp.(y),

all Baire

Sc X.

J:F d;.,

all Baire

SeX, FE E*.

Equivalently, (4.3)

p.(S)F(x) =

(4. 3) is possible only if

(4. 4)

p.(j y : F(y)

=

F(x) l)

=

all

1,

FE E*.

If C(p.) is the carrier of p., that is, the smallest closed subset of X whose complement has p. measure 0, (4. 4) shows that C(p.)

c

!y: F(y)

=

F(x) J,

all FE E*.

But since the functions in E* distinguish points of X, C(p.) must be I x I and p. must be the unit point mass at x. Thus x is in M(B) and the proof of Lemma 4. 1 is completed. We can now establish the Choquet theorem by using our Theorem 3. 4. ,/ THEOREM 4. 2. - Let X be a compact convex metrizable subset of a locally convex topological linear space. E. Let X. be the set of extreme points of X. Then X. is a Go in X and every x in X has a representation of the form

(4. 5)

x =

Iy dp. (y)

for some non-negative Baire measure p. on X satisfying

p.(X.) = p.(X)

=

1.

PROOF. - Let B be the subspace of Cr(X) defined earlier. Since X is metrizable, B is separable, so X. = M(B) is a Go in X. Let x be a point in X and L the linear functional of B defined by all fin B. L(f) = f(x),

119

THE

REPRESENTATIONS OF

LINEAR

FUNCTIONALS

317

Then L(1) = IILII = 1, so by Theorem 3. 4 there is a non-negative Baire measure fJ. with fL(M(B)) = fJ.(X) = 1 and (4. 6)

L(F) = JF dfL,

all Fin E*.

But (4.6) is simply a restatement of (4.5). And fJ.(X.) = 1 since by Lemma 4. 1, X. = M(B). This completes the proof. LEMMA 4. 1. gives a characterization of the Choquet boundary in terms of extreme points in a very special situation. It will be necessary for our later work to have a suitable replacement of this in the general case. LEMMA 4. 3. - Let X be a compact Hausdorff space, B a subspace o{ Cr(X). Let Lo be a linear functional in B*. Then the following are equivalent: 1. Lo is an extreme point o{ (4. 7)

!L: L

E

B*,

L(1)

= liLl1 = 1j

2. There is a point x in M(B) so that

Lo(f) = {(x), PROOF. -

all (in B.

To show that 2 implies 1, suppose that Lo is not

an extreme point of (4.7).

Lo =

*

; (Ll

+ L 2),

with Ll and

'"

L2 in (4.7), Ll Lo. By Lemma 2.1 there are measures fJ.l and fJ.2 in H with fJ.l(X) = P.2(X) = 1 and Li (f) =

J{dfLi'

Since Lo =1= L 1 , P.l(iB(X))

<

all (in B,

1.

i = 1, 2. 1

Let p. = 2(fJ.1

+ fL2).

Then p.

is in H",(B) and fL( ill (x) ) < 1, so x is not in M(B). This completes the proof that 2 implies 1. To show that 1 implies 2, suppose that Lo is an extreme point of (4. 7). By Lemma 2. 1 there is a measure p. in H so that Lo(f) = {dp., all (in B.

J

Let SI be any Baire subset of X with 0 < :J.(SI) < 1, and S2 = X - SI. Then if the linear functionals Ll and L2 in (4. 7) are defined by 1 " Li(f) = :J.(SiL(/ dp., all {in B,

120

a:.nnr.J.J.

.Dl~HU.l'"

J:!.1

~A""''''''''''''

___

___ _

we have a representation of Lo as a convex combination

+

Lo = fJ-(SI) Ll ~J.(S2) L2 of points in (4. 7). Since Lo is an extreme point of (4. 7), this must be a trivial representation, so

fJ d'f =

'f(S).r f dfJ-,

for all fin B and all Baire Sin X. Thus each fin B is constant almost everywhere with respect to if, and if x is chosen to be in the carrier C(fJ-), of p., that constant value must be f(x). This shows that Lo(f) = f(x) for all fin B and that (4.8)

C(fJ-)

C

!y: f(y)

=

f(x),

all fin Bj = in(x).

Since fJ- could have been chosen to be any measure in HAB), (4.8) shows that x is in M(B). This completes the proof of the lemma. LEMMA 4. 3. allows us to draw a useful conclusion concerning the relation between M (B) and (CM) where B is a subspace of Cr(X) and C is a subspace of B. For this we need the following. LEMMA 4. 4. - Let El and E2 be real locally con~ex topological linear spaces and 9 : El ~ E2 a continuous linear transformation. Let Y be a compact con~ex subset of E 1 . Then for each extreme point ~ of
(4.9)

lL:LeB*,

L( 1)

= II LII =

1j

IL:LeC*,

L(1)

=

11·

under p is

(4. 10)

121

IILII =

I

THE

REPRESENTATIONS

OF

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FUNCTIONALS

319

Let x be a point in M(C). Then by Lemma 4.3 the linear functionnal Lx in C * defined by Lx(f)

=

{(x),

all (in C,

is an extreme point of (4.10). By Lemma 4.4, Lx is the image under l' of an extreme point of (4. 9), which must by Lemma 4. 3 be of the form Ly, all (in B, for some y in M(B); 9(Ly) = Lx means simply that Ly when restricted to C agrees with L.e, that is, (4. 11)

{(x) = {(y),

all (in C.

Since y is in M(B), (4.11) shows that x is in ic(M(B)). Since x was any point in M(C), it follows that M(C) c ic(M(B)), as was to be proved. It is worth noting that under the hypotheses of Corollary 4. 5, neither M(C) c M(B) nor M(B) c M(C) holds in general.

v. -

REPRESENTATION OP LINEAR PUNCTIONALS IN THE GENERAL CASE

The purpose of thIS section is to extend our Theorem 3. 4 on the representation of linear functionals to the case of subspaces that are not separable, and to use this result to remove the hypothesis of metrizability in the Choquet theorem. If B is a subspace of Cr(X) that distinguishes points of X, the fact that any linear functional in B* can be represented by a measure concentrated on the Choquet boundary of B follows simply from the following (which is our Theorem 5. 3) : If fJ. in His B-maximal and S is a Baire set disjoint from M(B), then fJ.(S) = O. This in turn is an immediate consequence of Theorem 3. 2 if it is possible to find for each x in S a measure rr in Hx(B) with rr(S) < 1. It is the establishment of the existence of these measures that is the main work of this section. This is accomplished by a reduction to the separable case and an application of Corollary 3.3.

122

ERRETT BISHOP ET

ur.

KAREL

Lr.r.U..,

For each subset S of X, we shall denote by "'/..s the characteristic function of S, ~ 1, x in S 'X.s(x) = ~ 0, x not in S. LEMMA 5. 1. Let B be a subspace of C.(X), S a closed subset of X and x a point of S. Then the following are equivalent: 1. Each po in Hx(B) has P.(S) = 1, 2. For each g in C.(X) with g ~ AS, Sup If(x): (e B, gl > 1.

«

Let fJ. be in Hx(B).

Suppose that 2 holds. for each g in C.(X) with g L Z.~, PROOF. -

Jg dfJ. 2 But

J

g!

(~ = «gl>1. g dfJ.: g e C.(Xl, gL"'/..sL

Sup) (dfJ.: (e B, Sup) ((x) : ( e B,

P.(S) = Infl

Then

J

so P.(S) L 1. For the converse suppose that 2 does not hold. a function in C.(X) with g > I.~ and Sup I((x) : (

e

«

B,

gl = 1 -

Let g be

Eo

Define the linear functional Lx on B by all (in B.

(5. 1)

L". is a positive linear functional on B (that is, non-negative on non-negative functions) and thus by a standard result (see [10J, p. 22) on the extension of such functionals, there is a positive linear functional L on C.(X) with L(f) = Lx(f) ,

(5. 2)

all (in B,

and L(g) = 1 - t:. By the Riesz representation theorem, there is a p. in H with (5. 3)

L(f) =

Jf dfJ.,

all (in C.(X).

Because of (5. 1), (5. 2) and (5. 3), po is in Hx(B) and P.(S) =

f AS dfJ. < Jg d[l.

=

1-

This completes the proof of Lemma 5. 1.

123

E

<

1.

321

THE REPRESENTATIONS OF LINEAR FUNCTIONALS

LEMMA 5. 2. - Let B be a subspace of Cr(X) and S a closed Gil in X. Let x be a point of s. Suppose that each ~ in H.,(B) satisfies fL(S) = 1. Then there is a separable subspace C of B which is such that each fL in H.,(C) satisfies ~(S) = 1. PROOF. -

Since S is a closed Go, there is a sequence

f gn : n

=

1, 2, ... J

in Cr(X) decreasing point wise to x.s. By « 1 implies 2 » of Lemma 5. 1, for each positive nand m it is possible to find some fnm in B with fnm < gn and fnm(x).2: 1 - m-l. Let C be the subspace of B generated by the fnm and the constant functions. C is a separable subspace. If fL is in H.,,(c), then :1.(S)

= I~f

Jgn dp. .2: I~f (S~p Jf'IlTI dp.) = I~f (S~p fnm(x)) L

1,

so p.(S) must be 1. THEOREM 5.3. - Let B be a subspace of Cr(X) that distin· guishes points of x. If S is a Baire set disjoint from M(B), and fL in H is B-maximal, then p.(S) =

o.

PROOF. - By regularity of p. we can assume S closed Baire and thus a Go. B distinguishes points of X so there will be a separable subspace D of B that distinguishes the points of S from those of X - S; i.e. that satisfies in(S) = S. We consider two cases. Case 1 : For each x ip S there is a a- in Hx(B) with a-(S) < 1. Case 2: For some x in S, and all a- in H",(B), a-(S) = 1. We shall use Theorem 3. 2 to show that p.(S) = 0 follows in case 1, and then use Corollories 3. 3 and 4. 5 and Lemma 5. 2 to show that case 2 cannot occur. Case 1: Sin('e iD(S) = S, for each x in S, in(x) c S. Thus for each x in S there is some '.T in H",(B) with a-(in(x)) < '.T(S) < 1, so by Theorem 3. 2, p.(S) = o. Case 2: If each a- in H",(B) satisfies a-(S) = 1, then by Lemma 5.2 there is a separable subspace C of B (which can be chosen to contain D) so that each a- in HAC) satisfies a-(S) = 1. We shall now use Corollary 3. 3 to contradict this by showing that there actually is a a- in Hx(C) with '.T(S) = o. Since D·c C and in(S) = S, ic(S) = S, and S will be disjoint 21

124

from ic(M(B)) since it is disjoint from M(B). But by Corollary 4. 5, M(C) c ic(M(B)), so S is disjoint from M(C). By Corollary 3. 3 applied to the unit mass Yj at x, there is a a in HAC) with &(M(C)) = 1 and thus with a(S) = O. This contradicts the earlier assertion that each a in Hx(C) must satisfy a(S) = 1. Thus Case 2 cannot occur, and Theorem 5. 3 is completely proved. COROLLARY 5.4. - Let B be a subspace of Cr(X) that distinguishes points of X. Let [J. in H be B-maximal. Then p. can be extended to a measure that is concentrated on M(B); to be precise, there is a measure ~ on the a-ring generated by M(B) and the Baire sets that satisfies fi(S) = p.(S), all Baire S, fi(M(B)) = p.(X) = 1.

and

PROOF. - Any set T in the a-ring generated by M(B) and the Baire sets has a representation of the form

(5.4)

T = \ Sl n M(B) I u [S2 n (X-M(B)) l,

Sl' S2 Baire.

It is simple to check, using Theorem 5. 3, that if p. is defined by with Sl as in (5.4),

il is well defined

and satisfies the conditions claimed. The following is now an immediate consequence of Lemmas 2. 1, 2. 4 and Corollary 5. 4.

THEOREM 5. 5. - Let B be a subspace of Cr(X) or C.(X) that distinguishes points of X. Let;J be the a-ring generated by M(B) and the Baire sets of X. Then any linear functional L in B* has a representation of the form (5.5)

L(f)

=

Jfdp.,

all fin B,

for p. a measure on j' that satisfies p.(T) = 0 for each T e ;Y disjoint from M(B). Furthermore the p. in (5. 5) can be chosen to be non-negative if and only if L(1) = IILII. We can now state the generalized Choquet theorem. Its proof is identical with that of Theorem 4. 2, except that Theorem 5. 5 is used instead of Theorem 3. 4.

125

THE

REPRESENTATIONS

OF

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323

FUNCTIONALS

THEOREM 5. 6. - Let X be a compact conlJex subset of a real locally conlJex topological linear space. Let X. be the set of extreme points of X and;1 the a-ring generated by X. and the Baire subsets of X. Then each x in X has a representation of the form Y d(J.(y)

f

for some non-negatilJe measure (J. on !l that satisfies

f1.(X.) = (J.(X) = 1.

VI. -

ALGEBRAS

If B is a subspace of C.(X) or C«X), a subset Y of X will be called a boundary for B if for each f in B there is some y in Y with If(y)i = Ilfll. The following lemma was announced by Bauer [2J. LEMMA 6.1. - If B is a subspace of C.(X) or C«X), M(B) is a boundary for B. PROOF. - If B is a subspace of C«X), and B. is the subspace of C.(X) consisting of the real parts of the functions in B, then M(B.) = M(B) and also any boundary for Br will be a boundary for B. Thus it suffices to consider the case of B a subspace of C.(X). The subset K of B* defined by

(6 1)

K =

jL: LeB*,

L(1) = IILI! = 1\

is convex and weak * compact. Choose any function h in B. Let Lo be a point of K with ILo(h)i= maxlIL(h)): L E K I, and Ko = I L: L e K, L(h) = Lo(h) I. By the Krein-Milman theorem, the compact convex set Ko has an extreme point. This extreme point must also be an extreme point of K. By Lemma 4. 3, sueh an extreme point will be of the form Ln LAf)=f(y),

for some y in M(B).

allfmB,

Since for each x in X, the Lx defined by

LAf) = f(x),

all fin B, 21.

126

&...0 ... , ... ,.&.1

IS In

..............

ua.........

__ ___ _

K, it follows from the choice of Ly that

Ilhll = =

max !lh(x)l: x e Xl max flL",(h)\ : x E Xl = iLy(h)1 = ih(Y)I·

Since y is a point of M(B), and h is an arbitrary function in B, M(B) is a boundary for B. Throughout the remainder of this section, A is a uniformly closed subalgebra of Cc(X) that distinguishes the points of X and contains the constant functions. It is well known that there is a smallest closed boundary for the algebra A, the Silov boundary (the Silov boundary has been related to extreme points in [1J, [2J and [5]). We are concerned here with boundaries that are smaller than the Silov boundary, and in particular with the question of whether if B = A, Lemma 6. 1 is the strongest result possible; i.e., whether any boundary for A must contain the Choquet boundary of A. We show that this is indeed so if each point of X is a Go, while we show in the general case that any boundary for A that is a Baire set must contain the Choquet boundary of A. In order to do this we must study two properties of points of X that are equivalent to being in M(A). \Ve shall say that a point x of X satisfies Condition I if for each open neighborhood U of X there is some f in A with

Ilfll< 1, If(x)1 > ~

and If(y)J<

~

for all y outside of U.

/

We shall say that a point x of X satisfies Condition II if for each closed set S containing x that is a Go, there is some function f III A with if(x)! = Ilfll and jy: If(Y)1

= IIfl15

c

S.

Note that if \ x I is a Gil, Condition II simply states that there is some f in A « peaking» at x. LEMMA

6. 2. -

PROOF. -

in A. Let U in Cr(X) with of U. Since ~(I xl) = 1.

If x is m M(A), x satisfies Condition I.

Let Ar be the subspace of real parts of functions be any neighborhood of x. Choose a function g 0 < g < 1, g(x) = 1 and g(y) = 0 for y outside M(A) = M(A,.), any [J. in H",(A r) must satisfy Thus by « 1 implies 2 » of Lemma 5. 1. applied

127

rHE

REPRESENTATIONS OF LINEAR

325

FUNCTIONALS

to S = !xj and B = Ar , there is some ho in Ar with ho < g log6 and ho(x) > logS' Let h= (logS) (ho -1). Since h is in A r ,

+

there is a k in Ar so that h ik is in A, and since A is a uniformly closed sub algebra, the function f defined by

f=

e(h+ik)

.is in A (the use of the exponential function at this point was suggested to us by H. Royden). It is simple to check that f satisfies the conditions wanted; i.e. Ilfll and

If(y)l~ ~ <

1

< 1,

If(x)l>

6

8 =

3

7;'

for y not in U.

LEMMA 6. 3. -- If x satisfies Condition I, it must satisfy

Condition I I. PROOF. - Let S be a closed Go containing x. Let r Vnl be a decreasing sequence of open sets with S = nV n' The construction of a function f in A with If(x)1 = Ilfll and

ly: If(y)1

= Ilflll

c

S

is identical with the construction in Theorem 2 of [4J, if the sets Dn(x) used in that construction are taken to be the X - V n' LEMMA 6. 4. -

If x satisfies Condition I I, it must be in M(A).

PROOF. - Let p. be in H,(A). Let S be any closed Go containing x. By Condition II there is an fin A with (6. 2)

,r

E

ly: If(y)1

= Ilfll!

c

S.

Since fJ.. is in Hx(A), f'f d:J. = f(x), which by (6. 2) is possible only if 1-'-(5) = 1. Thus by the regularity of p.., p..( Ixl) = 1, so x is in M(A). Thus we have established. THEOREM 6. 5. - Let x be a point of X. Then the following are equiIJalent: 10 x satisfies Condition I; 20 x satisfies Condition II; 30 X is in M(A). This equiIJalence for the case X metrizable is contained in [4J.

128

ERRETT

BISHOl' ET

KAHEL

COROLLARY 6. 6. - If each point smallest boundary for A.

DE

of

LEEUW

X is a Gil, M(A) is the

PROOF. - By Lemma 6. 1, M(A) is a boundary for A. By Condition II, at each point x of M(A) there is some f in A with

If(y) 1< If(x) \,

all y =1= x.

Thus any boundary for A must contain M(A). This result for X metrizable was established in [4J. COROLLARY 6. 7. - Let Y be any Baire subset of X that is a boundary for A. Then Y contains M(A). PROOF. - Suppa'll' on the contrary that there is some point x in M(A) that is 110t in Y. Then there is a closed set S containing x that is a G 0 and is disjoint from Y. By Condition I I there i~ an f in A with X

E

IY: [f(y)\ = Ilflll

c

S.

This f does not attain its maximum modulus on Y, contradicting the fact that Y is a boundary for A. It is however not true that M(A) is the intersection of all of the Baire boundaries for A, as can be seen from some of the examples in the next section. COROLLARY 6. 7. will now be used to show that if Y is any boundary for A, all linear functionals in A" can be represented as measures on Y. To establish this result we need first a lemma. LEMMA 6. 8. - Let Y be a boundary for A. Let fJ. be any Ar-maximal measure in H. If S is a Baire set disjoint from Y,

p.(S)

= o.

PROOF. - Since Y c X - S, the set X - S is a Baire boundary for A. By Corollary 6. 7, M(A) c X - S, so that S is disjoint from M(A). It follows from Theorem 5.3 that [1-(S) = O. THEOREM 6. 9. below now follows from Lemma 6. 8 in the same manner that Theorem 5. 5 follows from Theorem 5. 3. We omit the details.

129

~Hr; HEI'HESENIAl'IO:olS 01' LINCAH I'U'NCTIONALS

327

THEOREM 6. 9. Let Y be a boundary for A, and ~ the a-ring generated by Y and the Baire sets of X. Then each linear functional L in A'" has a representation of the form

(6. 3)

L(f)

=

J fdfJ-,

all

f

in A,

for fL a measure on :1 that satifies fJ-(T) = 0 for each T m :f that is disjoint from Y. Furthemwre the fJ- in (6. 3) can he chosen to be non-negatilJe if and only ~f L(l) = IILII·

VII. -

EXAMPLES

We present in this section a dass of examples showing that thr Choquet boundary, whieh must be a Go in the separable case, can be arbitrarilv bad in general. We al~o show that Theorems 5. J and 5.5 cannot be strengthened to assertiom about Borel sets rather than Baire sets. Finally there I'> a simple example which shows that the analogue of Theorem 6. 9 for subspaces rather than subalgebras is false. Let 1Y.c l.cEX be a family of disjoint non-empty topological spaces indexen by a topological space X. Let Y = yx

U

XE~

and 'it: Y -7 X be the projection map defined by 7t(y) = x if y is in Y". Let s: X ~ Y be a cross-section; i.e., 'its(.'r;) = :r for all x in X. We shall describe a topology \ .:alled the porcupine topology} for Y. Let 'tt be the class of all subsets U of Y that satisfy the following: there is some x in X so that U is an open subsrt of Y not containing s(x). Let:r be the class of all subsets S of Y that satisfy the following: there is some x in Y so that S is a closed subset of Y x not containing s(x). Let ttl be the class of all subsets of Y of the form 7t- l (V) - (SI U •.. uSn), where V is an open subset of X and the S, ar(~ in~. The collection 'U u qJ is closed under intersections and thus is the basis for a topology for Y. This is our porcupine topology. In this topology a net !ua.l of points in Y converges to a point y in Y:r - Is(x) I if and only if the net is ultimately in Y.J: and converges in the original topology of Y J: to y. If none of the Ua. are in Y x> ! ua.l converges to s(x) if and only if the net I

130

!n( Ua.) l converges to x in the topology of X. If X and the Y.z; are compact Hausdorff, then Y is compact Hausdorff. Suppose now that X and the Y x are compact Hausdorff. Let D be a subspace of Cr(X) and for each x in X, let Bx be a subspace of Cr(Y r)' Let B be the subspace of Cr(Y) consisting of all fin Cr(Y) such that f S is in D, and for each .1': in X, f restricted to Y x is in Bx. If D and all the 87: are closed subspaces and distinguish points, B will be a closed subspace and distinguish points. It is simple to check that the Choquet boundary M(B) of B is 0

(7. 1) We shall now consider a special cas(' of the above construction. Let X be an arbitrary compact Hausdorff sp8.ce and K an arbitrary subset of X. For each .'C in K, let Y x eomist of the one point S r, and for (,Heh .1' in X -K, let Y be the rliscrete topological space eonsisting of the three points rJ') Sec, tx;' Define s: X -?- Y by s (x) = S.c, all x III X. Let D be Cr(X), and if x is in K, let H.r = Cr(Y x)' If x is in X - K, let B be the suhspace of Cr(Y x) consisting of those f that satisfy 7

!

L

f(sJ')

1

= 2 (f(r,)

+ f(tx)).

The construction described above applied to D and the Bx yields a ('losed subspace B of Cr(Y) that distinguishes points. Its Choquet boundary is (7. 1) and is therefore easily seen to satisfy Y - M(B) = s(X - K). Since K was an arbitrary subset of X, this shows that the Choquet boundary can be arbitrarily bad. An example of a bad houndary has also been given by Choquet in [8J. Suppose now that in this example we take X to be the unit interval with the usual topology and K to be the void set. Let ') be Lebesgue measure on X, and let p. be the Baire measure on Y defined by

for all Baire subsets S of Y. Then fJ- is B-maximal. ~ever­ theless its regular Borel extension p. satisfies P.(M(B)) = O.

131

THE

REPRESENTATIONS

OF

LINEAR

FUNCflONALS

329

This is in contrast to Theorem 5. 3 which shows that a B-maximal measure must be (( concentrated on M(B) » In the sense that p.(S) = 0 for each Baire set S disjoint from M(B). The example shows that thc conclusion of the theorem cannot be strengthened to P.(S) being 0 for each Borel set S disjoint from M(B), even if M(B) itself is Borel. It also shows that the measures p. appearing in Theorem 5. 5 may not be regular. In order to obtain a morp stnkillg example, we take X to be the sub'let of the complex plane

X=rz:[z[<1j In the usual topology, and take K to be the set

K=;z:l z l<1l· For each x in K, let Y c consist of one point 8 c and for each 1: in X - K, let Y L be the discrete topological "pace consisting of the three points I r", 8 oX , and loX 1· Define 8: X ~ Y by s(x) = 8~, all x in X. Let D consist of all functions in Cr ( X) which are harmonic on K. If x is in K, let Bx = Cr (Y x). If x is in X - K, let Be be the sub'ipace of Cr( Y,,,) COIl'il!.tIng of those f that satisfy

The construction above applied to D and the Bx yields a closed subspace B of Cr(Y) that distinguishes points. Its Choquet boundary is easily seen by (7.1) to be M(B) = Y Let

Yo

be the point 8(0) =

So

8

(X).

of Y.

P.(M (B))

=

We shall prove that

0

for all p. in Hyo (B). To this end, WI' first note that the only compact subsets of M(B) are finite, by the definition of the topology on Y. It follows that in order to show that p. (M (B)) = 0, it will be sufficient to show that p. (I rx I) = P. t.e l) = 0 for each x in X K. :\ssllme tha t this is not the ca"e, so that

(r

132

330

ERRETT BISHOP ET KABEL DE LEEUW

p.( 1r:x, l) = C > 0, say, for some Xo in X - K. Let h be any non negative harmonic function on X with h(xo) > 1 and h (0) = C. Define the function ( in B by ((y)

Since p.

e

=

h(x), for all y in Y",.

H y " it follows that

Since also

((Yo)

=

this gIves a contradiction.

h(O)

=

C,

Thus

P.(M(B))

=

O.

This points up Theorem 5. 5, which states that there exists p. in H y , which can be extended to a measure on the cr-ring generated by M(B) and the Baire sets so as to have p.(M(B)) = 1. The point is that the extension of p. does not agree with the regular Borel extension p. of p.. The next example demonstrates that the analogue of Theorem 6. 9 for subs paces is false. Let X be the subset

Iz : Iz of the plane. the form

11

=

11

u Iz : Iz + 11 =

11

Let B be the space of all functions (on X of

((x, y)

=

ax

+ by + c,

(x, y) in X,

for a, band c real. Then each (in B that attains its maximum on X at p = (1, 1) also attains its maximum at (-1, 1), so X - Ip I is a boundary for B. However the linear functional L in B* defined by

L(f)

=

((p),

(

in B,

has no representation of the form

L(f)

= J(dp.,

(in B,

for p. a non-negative measure on X-) p l.

133

THE

REPRESENTATIONS

OF

LINEAR

FUNCTIONALS

331

BIBLIOGRAPHY [1 J R. ARENs and 1. SI"'IGER, Function values as boundary intcgrals, Proc. Amer. Math. Soc., vol. 5 (1954), 735-745. [2J II. B \UER, Un probleme de Dirichlet pour la frontierc de Silov d'un espace compact, C. fl. Acad. Sci., Paris, vol. 247 (19;)8), pp. 843-846. [3J E. BISHOP, Some theorems concerning function algebras, Bull. Amer. _'lifath. Soc, vol. 85 (1959), pp. 77-78. [4) E. BISHUP. A minimal boundary for function algebras, to appear in Pacific. JOltr. "Vlath. [5] II. BOH'II::;>O!BLUSI and S. K'\RLIl'if, Geometrical prnpertif's of the unit sphrre of Banaeh algpbras, Ann. of Math., vol. 62 (1955), pp. 217-229 [oj ~. BouRB \KI, Integratwn, Elements de Afathemat~ques, XIII, Book VI, Paris, 195:2. [7J G. CHOQUET, Existence des representations integrales au moyen des points extremaux dans les cones convexes, C. R. Acad. Sc~., Paris, vul. 243 (1956), pp. 699-702. [8] G. CHOQUET, Existence de" representatiom integrale~ dam leq cones convexes, C. R. Acad. Sci., Paris, Vol. 24.3 (1956), 736-737. [9J G. CHOQUET, Existence et unicite des representations integrales. Seminaire Bourbak~, Decembre 1956, 139-01 a 139-15. [10J 1. GLICKSBERG, The representation of functionals by integrals, Duke Math. Jour., vol. 19 (1952), pp 253-261. [l1J P. H\.LMOS, l'lifeasure Theory, Van Nostrand, 1950. [12J R. KA..DISON, A representation theory for commutativc topological algebra, Mem. Amer. 1I.1ath. Soc., no 7. [13J J. KELLEY, General topology, Van Nostrand, 1957. 1. The topologicdl terminology is that of [11], and the measurc thcoretic terminology is that of [9].

134

7 .. ", oy permission or tne American MatnemanCBI :'OCI"'Y.

741

SIMULTANEOUS APPROXIMATION BY A POLYNOMIAL AND ITS DERIVATIVES ERRETT BISHOp!

1. Introduction. \Ve shall prove the following theorem: THEOREM. If fo is a continuous function on a compact set C w1thout interior and with connected complement of the complex plane, and tf flo ... , fn are continuous functions on a compact totally disconnected subset E of C, then there exists a sequence {Pi} of polynomwls such that pi-+fo uniformly on Cas i-+ ~ , and Pi[l.l-+fl. uniformly on E as i-+ ~ , for 1 ~ k ~ n, where ppl is the kth derivative of Pi.

For the case in which E is void, this reduces to a well-known theorem of Lavrentiev [3]. A different case of the problem of simultaneous approximation by a polynomial and its derivatives-the problem of simultaneous approximation on a Jordan arc-has been considered in a previous paper [1], by methods distinct from the methods of the present paper. \Ve make some preliminary remarks concerning distributions. The distributions in question will all be defined on the complex plane and have complex values and have a compact support. If T is such a distribution and if f is a function defined on the complex plane and having partial derivatives of all orders, then (f, T) will denote the Received by the editors March 26, 1958 and, in revised form, December I, 1958. This work was partially supported by the Office of Naval Research through Contract ONR 041 157. 1

135

742

ERRETT BISHOP

[October

value of Ton f. Thus if T is a measure ,tL we have (f, ,tL) = ffd,tL, and if T is an integrable function h we have (f, h) = ff(z)h(z)dxdy. By Schwartz [4, p. 70], we have (a/az*) [(7rZ)-1 * T] = T. 2. Proof of the Theorem. Let 11-0 be any measure on C, and let 11-1, ... , ,tLn be measures on E such that jpdl1-o+ jp[1Jd,tL1 + ... + jp[nJdl1-n = 0 for all polynomials p, where by measure we mean finite, complex-valued, Borel measure. If we let T be the distribution T= L~~1 (-l)k [a/az]kl1-k' it follows that the support of T is a subset of E and that (p, T + ,(.La) = 0 for all polynomials p. Define the distribution R = (7rZ)-1 * (T +11-0), so that (a/az*)(R) = T +,tLo. Also, R agrees on the complement of C with the analytic function cp(z) = ([7r(z - n ]-t, (T +,tLo)(n) of z. Since the complement of C is connected, for each z in the complement of C the function (z - n- 1 of can be uniformly approximated on som~ neighborhood of C by polynomials, so that the derivatives of (z-n- 1 will be uniformly approximated on some neighborhood of C by the corresponding derivatives of the approximating polynomials. Since T +11-0 is orthogonal to all polynomials, we see that cp therefore vanishes on the complement of C. Thus the support of R is a subset of C. Now (7rZ)-1 is clearly the sum of an integrable function CPl and a continuous function CP2. Since the measure 11-0 has compact support, CPl * 11-0 is an integrable function and CP2 *,tLo is a continuous function. (See [4].) Therefore (7rZ)-1 * 11-0 is a locally integrable function. Since also (7rZ)-1 * T agrees with an analytic function on the complement of E, the distribution R = (7rZ)-1 * 11-0+ (7rZ)-1 * T will agree with a locally integrable function h on the complement of E. Since R vanishes on the complement of C, h will vanish on the complement of C. Now let be any point in C-E, and letf be an infinitely differentiable function with compact support, such that fez) = 1 for all z in some neighborhood U of and such that the support H of f does not intersect E Then the distribution (aJjaz*)R is equal to the integrable function caf/az*)h = g, so that a(fR)/az* = (aflaz*)R +JaR/az* = g+f(T +11-0) = g+JI1-O, since H is disjoint from E. Since the support of h is a subset of C and the support of af/az* is a subset of H - U, the support of g is a subset of HnC- U. Thus the distribution g+fl1-o is a measure II on HnC which agrees with,tLo on all subsets of U. For any polynomial p, we have (p, II) = (p, a (fR)/az*) = -(ap/az*, fR)= -(0, fR)=O. Thus II is orthogonal to all polynomials. By the hypothesis of the theorem, HnC will have no interior and will have connected complement. By the theorem of Lavrentiev, any

r

r

r

136

APPROXIMATION BY A POLYNOMIAL

continuous function on HnC can be uniformly approximated by polynomials. Thus v is orthogonal to any continuous function on HnC, so that v=O. It follows that J.Lo vanishes on all subsets of the neighborhood U of r. Since r was an arbitrary point in C-E, J.Lo is therefore a measure on E. If D is any open and closed subset of E, there will exist, by Runge's theorem, a sequence {Pi} of polynomials converging uniformly to 1 on some neighboruood of D and to 0 on some neighborhood of E-D. Thus all derivatives pIt! will converge uniformly to 0 on E as i~ co. Thus we see that O=jpidjJ.o+jpll!dJ.Ll+ .. + jp\n!dJ.Ln-->jJ.o (D) as i--> co, so that J.Lo vanishes on all open and closed subsets of E. Since E is compact and totally disconnected, the open and closed subsets of E form a basis for the open sets of E (see [2, p. 20 ]). Therefore J.Lo=O. Thus jpdJ.Ll+jp[lldJ.L2+ ... +jpln-l]dJ.Ln=O for all polynomials p. By repeating the above argument, we can therefore show successively that J.Lo = J.LI = ... = J.Ln = o. Let V denote the Banach space consisting of all n+ 1-tuples (fo, ii, ... ,iT.)' where 10 is a continuous function on C and ik for 1 ~ k ~ n is a continuous function on E, under the norm (fo, . . . = in where lIikll is defined in the usual fashIOn. The dual V* of V then consists of all n + 1-tuples of measures (J.Lo, J.Ll, ••• , J.Ln), where J.Lo is a measure on C and J.Lk for 1 ~ k ~ n is a measure on E. We have just shown that the only element of V* which vanishes on all elements of Vof the form (p, p[lJ, . . . , pIn]), where p is a polynomial, is the zero element From the Hahn-Banach theorem it follows that any element in V is a limit of elements of the form (p, pl1], ••• , pIn]), and this is just the theorem.

II

Ilioll + ... +II II ,

,in)ii

BIBLIOGRAPHY 1. E. Bishop, Approximation by a polynomial and its derivatives on certain closed

sets, Proc. Amer Math. Soc vol 9 (1958) pp 94Cr-953 2. \V. Hurewicz and H. \Val1man, Dimension theory, Princeton, 1948. 3. M. Lavrentiev, Sur les fonetions d'une variable complex representable par des ,eries de polynomes, Actualites Sci. Ind., Paris, 1936. 4. L. Schwartz, Theorie des distributions, vol 2, Actualites Sci. Ind , Paris, 1951. UNIVERSITY OF CALIFORNIA

137

~

I

..1,

pp. ,,;, 1 ~..1'+U

\..Opynglll

\bI

uUl\e UIlIVt=rsny

rr~bb

1

~OU.

BOUNDARY MEASURES OF ANALYTIC DIFFERENTIALS By ERRETT BISHOP

1. Introduction. We consider a compact set C in the complex plane with connected complement, and define a class H of analytic differentials on the interior U of C. In case C is the unit disk, for example, H will consist of all analytic differentials j(z)dz with f in the class HI (see Zygmund [7]). We also consider the class M of finite complex-valued Borel measures on the boundary B of C which are orthogonal to polynomials. Our main result, Theorem 3 below, will establish a natural one to one correspondence between Hand M. (Actually, Theorem 3 contains a somewhat more general result) If dw I: H, the associated Jl. in M is roughly describable as the boundary measure of the analytic dIfferential dw. The difficulty consists in showing that each Jl. in M is the boundary measure of some analytic differential dw in H. As a simple corollary of our main theorem, we shall derive Mergelyan's result (Theorem 4 below), that every continuous function on C which is analytic on U is uniformly approximable by polynomials. This paper is a sequel to a previous paper [1]. Many of the results needed here were established in [1] in somewhat less generality. Since the proofs of certain of these results carryover word for word to the more general case considered here, we do not give them in such detail as would otherwise be necessary.

2. The classes M and H. We now introduce the sets with which we shall be working, which are somewhat more general than the sets with connected complement mentioned above. DEFINITION 1. The compact set C in the complex plane will be called balanced if the boundary B of C equals the boundary of the unbounded component of - C. The open set U in the complex plane will be called balanced if U is bounded, if the closure C of U is balanced, and if U is the interior of C. The following lemma needs no proof. LEMMA 1. A compact balanced set has a balanced interior. An open balanced set is simply connected The union of any set of components of an open balanced set is an open balanced set.

By measure we always mean finite complex-valued Borel measure. We now define the class M. DEFINITION 2. Let C be a compact balanced set in the complex plane. The class M(C) is defined to be the class of all measures !J. on the boundary B of C such that

Received September 15, 1959. The author holds a Sloan Foundation Fellowship. 331

138

ERRETT BISHOP

J(t -

Z)-l dfJ.(t) = 0

for all z in - C. If U is an open balanced set whose closure is C, then M(U) is defined to be equal to M(C). It follows that f fdfJ. = whenever fJ. E M(C) and f is analytic in some neighborhood of C. The following lemma needs no proof.

°

LEMMA 2. If G is a compact balanced set in the complex plane and if V is the union of some set of components of the interz"or U of G, then MeV) C M(G). LEMMA 3. If G is a compact balanced set in the complex plane, if fJ. is in M(G), and if f is analytic in some neighborhood of G, then for each point z in the inter-ior U of C we have

J(t -

z)-lf(t) dfJ.(t)

= fez)

J(t -

Z)-l dfJ.(t)

Proof. The function g defined by get) = (t - Z)-l [f(t) - fez)] is analytic in a neighborhood of C. By the above remark we see that f gdp. = 0, from which the lemma follows. The following lemma is a slight generalization of Theorem 1 of [1]. LEMMA 4. Let G be a compact balanced set in the complex plane. be measured in M (C) such that

for all z in the

~·nterior

U of C. Then

fJ. =

Let

fJ.

and"

P.

Proof. By the definition of M(G), the measure X = p. - " in M(G) will be orthogonal to all rational functions whose poles lie in - B, where B IS the boundary of G. By a theorem of Mergelyan (see [4; 21]), every continuous function on B is uniformly approximable by such rational functions. (The hypotheses of this theorem of Mergelyan are satisfied because each point in B is in the boundary of the unbounded component of - B.) It follows that A is orthogonal to all continuous functions on B. Thus X = 0, as was to be proved. DEFINITION 3. Let U be a simply connected open set in the complex plane. A sequence hi I of compact subsets of U will be said to delimit U provided that (i) Each "Ii is the union of a finite number of disjoint rectifiable simple closed curves lying in U, no two of which belong to the same component of U, and (ii) If S is any compact subset of U, then for all i sufficiently large S will lie

in the union of the bounded components of - "Ii. We now define the class H (U), first remarking that an analytic differential dw on an open set U in the complex plane will mean a complex differential I-form on U of the type dw = f(z)dz, where f is an analytic function on U.

139

BOUNDARY MEASURES

333

DEFINITION 4. Let U be a bounded simply connected open set in the complex plane. The class lIeU) consists of all analytic differentials dw on U with the property that there exists a sequence bd of compact sets which delimit U and a positive constant K such that

Joy, I dw I <

K

for all values of i. If dw E lIeU), then II dw II will denote the greatest lower bound of all such constants K. If. U is connected, it is clear that lIeU) is conformally invariant. That is if II' is a one-to-one conformal map of U onto V, then lI(V) consists of all differentials of the form II' (dw) for dw E lIeU). It is also clear that then II dw II = 111P(dw) II. DEFINITION 5. Let U be a bounded simply connected open set in the complex plane, and let B be the boundary of U. A measure JI. on B is said to be a boundary measure of a differential dw in lIeU) if the sequence {'Y.} of Definition 4 can be chosen to have the additional property that

~,

h dw -)

J

h dJi.

as

t -)

<Xl

for all continuous functions h on U U B. The following lemmas need no proof. LEMMA 5. If JI. is a boundary measure of dw, then

II JI. II :::; II dw II.

LEMMA 6. Let U be a bounded simply connected open set and let r be the class of components of U. Let dw be an analytic dijJ·erential on U such that the restriction dwv of dw to V is m lI(V) for all V in r. Assume that Lv.r II dwv II < <Xl. Then dw

E

H(U)

and

II

dw

II :::;

L v.r

II dwv II·

The following theorem gives additional information about boundary measures. THEOREM 1. Let U be an open balanced set in the complex plane with closure C. Then any analytic dij)"erential dw m lIeU) has a boundary measure J.L, which is unique. The measure JI. belongs to M(U), and dw = f(z)dz is given in terms of JI. by the formula (*) for all z in U. Proof. Since dw E H(U), there exists a sequence bd which delimits U for which J"1' I dw I is bounded. By the weak * compactness of bounded closed sets in the space of measures on C, we see that there exists a measure JI. on C such that

1,

h dw -)

J

h dJi.

140

as

i -)

<Xl

334

ERRETT BISHOP

for all continuous functions h on G, where {a.} is some subsequence of {'Y.}. (This argument is given in detail in [1; 284].) It is clear that J.I. is actually a measure on the boundary B of U. By Definition 5, J.I. is therefore a boundary measure of dw. If z is any point in - G, then

J

(t - Z)-1 dJ.l.(t)

so that J.I.

(211"~rl

E

J

M(G)

=

M(U).

(t - Z)-1 dJ.l.(t)

= lim

fa. (t -

Z)-1 dw(t) =

0,

For all z in U we have

= lim (211"ifl

L

(t - Z)-1 dw(t)

=

lim (211"i)-1

L

(t - z)-lf(t) dt

= fez).

This proves the formula (*). N ow let 11 be another boundary measure for dw. Thcn formula (*) is valid with J.I. replaced by 11. Since both J.I. and 11 are in M(G), it follows from Lemma 4 that J.I. = 11 Thus J.I. is unique, as was to be proved. The following theorem is essentially Theorem 3 of [1], slightly generalized and supplemented by some additional remarks. THEOREM 2. Let G be a compact balanced set in the complex plane, and let J.I. be a measure ~'n M(G). Let the analytic dijJerential dw = f(z)dz on the interior U of G be defined by

(*)

for all z in U Let V be any component of U. is in H(V) and II dwv II :::; II J.I. II

Then the restriction dwv of dw to V

Proof. Let {J" I be a sequence of simple closed curvcf, surrounding G and converging to some compact set containing the boundary of C Let Zo be any point in V. Lct 'Pn be the Hiemann map of the interior of J n onto D = {w: I w i < I}, normalized by the conditions 'Pn(ZO) = 0 and 'P~(zo) > O. Let 'P be the Riemann map of V onto D, normalized in the same way. By a theorem of Caratheodory [3], {'Pn I converges uniformly to 'P on compact subf,ets of V. Since J.I. is in M(C) ann. 'Pn is analytic on a neighborhood of C, from (*) and Lemma 3 we have

for all z in V. Let lin be the measure 'P,,(J.I.) on D defined by lIn(S) = J.I.('P;:'(S)), so that II lin II :::; II !J. II. By substituting w = cp,,(z) in the cquation just derived, we obtain

141

BOUNDARY MEASURES

335

for all w in rp,,(V). Let v be any weak'" cluster point of the sequence I v" I, so that v is a measure on Iw: I wi = 11 with II v 11:$ 1111-11. It is clear that each v"' and therefore v, is orthogonal to polynomials in w. Passing to the limit in the last equation, we obtain

(271"i)-1

(**)

for

Iw I <

1, where g(w)

J

(t - W)-l dv(t) = g(w)

= t(rp-l(W)) [drp-l(W)/dw].

Since the measure v is orthogonal to all polynomials, it follows from the classical theory (which is quoted on page 283 of [1]) that v is the boundary measure of some analytic differential d{3 in H(D) and that II v II = II d{3 II By (*"'), ,ve see that d{3 = g(w)dw From the fact that H(D) is conform ally invariant, it follows thatrp-l(d{3) EH(V) Butrp-l(d{3) = rp-l(f(rp-l(W))drp-l(W)) = f(z)dz = dwv. Thus dwl' E H(V) and II dwv II = II d{3 II = II v II :$ II Jl. II, as was to be proved. 3. The main theorems. vVe come now to that part of the paper which extends the results of [1] in an essential manner. LEMMA 7. Let C be a compact balanced set 1'n the complex plane. Let U be the interior of C and let V be the union of some set r of components of U. Then there exists a sequence If d of functions analyt~c in some neighborhood of C such that

(a) I Mz) I :$ 1 for all Z 1'n C, (b) (fkl converges uniformly to 0 on compact subsets of V, and (c) Ifd converges uniformly to 1 on compact subsets of U - V. Proof· It is clearly sufficient to consider the case in which V consists of a single component of U, since the general case will then follow by taking products and using the diagonal process Let Zo be any point in V. As in the proof of Theorem 2, let I J"I be any sequence of simple closed curves surrOLlIlding C and converging to some compact set containing the boundary B of C. Let rpn be the Riemann map of the interior of J" onto D = f w I w I < 11 with rpn(zo) = 0 and rp~(zo) >. O. Let rp be the Riemann map of V onto D, normalized in the same way, so that {cp,,1 converges uniformly to rp on compact subsets of V. Since the functions {CPn} are uniformly bounded on each component TV of U, we may assume, by passing to a subsequence if necessary, that {cp,.} converges uniformly on compart subsets of lV, for each component TV of U. Since cp,,(TV) is disjoint from cp,,( V), for TV ~ V, and since cp,,( V) e\"entually fills all of D, it follows that the limit of {rp,,} 011 TV has modulus everywhere 1. Thus f cp" I converges uniformly on compact subsets of W to a constant IX" of modulus I. N OIV for any finite set r of components of U, not containing V, and any f > 0,

142

.J!,j11.1'tl'.lr.L D10nvr

and any positive number K, we can find a positive integer N such that N > K and l(aw)N - 1 I < E for all Win r. (see [5; 132]). If S is any compact subset of U, this means that we can choose L such that

I (CPn(Z»N

- 1

I<

E

for all n > L and all Z in S (\ Uw.rW. Since sup {I cp(z)l: z we can also choose K and n so large that

I (cp'(Z)N I <

E

S (\ V}

<

1,

E

for all z in S (\ V. It follows that a sequence If k I can be constructed having properties (a), (b), and (Cl') where (cr) is the property (er) ltd converges uniformly to 1 on compart subsets of U WEl W. Since this is true for each r, and since U has a denumerable number of components, it follows by the diagonal process that a sequence ''lith properties (a), (b), and (c) can be constructed. LEMMA 8. Let C be a compact balanced set and let /Jo be a measure in M(C). Define the analytic differential dw = f(z)dz on the interior U of C by

(*)

for all z in U. For each component W of U let /Jow be the boundary measure of the difJerential dww obtained by restricting dw to W. The existence of /Jow is guaranteed by Theorems 1 and 2. Then II Lr/Jow II ::; II /Jo II, where the summation is taken over any finite subset r of components of U. Proof. Let V be the union of those components of U which are not in r. By Lemma 7, there exists a sequence It.d of functions analytic in some neighborhood of C and having properties (a), (b), and (c) above. Let Ilk be the measure fk/Jo, defined by IIk(S) = Is fkd/Jo. Thus Ilk t M(C) and II Ilk II ~ II )Jo11. Let II be any weak* limit point of the sequence {Vk} so that II II II ~ II /Jo II. For z in U. the quantity (**) is a limit point of the sequence whose terms are (21rtl- 1

J

(t - Z)-1 dVk(t) = (2mrl

J(t -

z)-IMt) dp,(t)

= Mz)f(z), by Lemma 3, so that a(z) = 0 if z E V and a(z) = fez) if z E U - V. Since p,w is a measure in M(W), we have, on the other hand,

143

337

BOUNDARY MEASURES

depending on whether that

Z E

War

(271"~")-1

(***)

Z E

f

U - W. If we write u

=

Lrlo'w , it follows

(t - Z)-I duet) = a(z)

for all z in U. Now Io'w E M(G) by Lemma 2. It follows that u E M(G). By Lemma 4 it follows from (**) and (***) that u = JI. Hence II u II = II JI II ~ II JJ. II, as was to be proved. LEMMA 9. Let U be a connected balanced open set, and let JJ. be in M(U). Then {i) JJ. is a measure on the set P of accessible boundary points of U, in the sense that there exists a Borel subset G of P such that JJ.(S) = JJ.(G S) for all Borel sets S and (ii) JJ. vanishes on points.

n

Proof. Assume first that the complement of the closure G of U is connected. It follows from Lemma 1 and Theorem 4 of [1] that there exists a measure JI .on the set F = {w: I w I = I} which is absolutely continuous with respect to Lebesgue measure, a Borel function von F with values in P such that JJ. = v(JI), where the measure v(JI) is defined by v(JI)(S) = JI(v-I(S)) for all Borel sets S. The proof of Lemma 1 of [1] shows that there also exists a sequence {Tn! of dosed subsets of F such that T = UTn contains almost all points of F, and such that v is one-to-one on T and continuous on each Tn. Thus v(Tn) is compact, so that G = veT) is a Borel subset of P. It follows that JJ.(S) = JI(v-I(S)) = JI(v-I(S) n V-I (G)) ,

since T C v-leG) and since T contains almost all points of F relative to Lebesgue measure, and therefore relative to JI. But

JI(v-I(S) n v-leG)) = JI(v-I(S n G)) = JJ.(S n G),

n

so that JJ.(S) = JJ.(S G) for all Borel sets S. Thus JJ. is a measure on P. If z is any point, the set V-I ( {z J) T consists of at most one point, because v is

n

n

.one-to-one on T. We therefore have JJ.({z)) = JI(v-1({z)) T) = 0, since JI is absolutely continuous Thus JJ. vanishes on points. In the general case, when - G is not necessarily connected, the proof proceeds as follows. Lemma 1 and Theorem 4 of [1] are first proved for the general case. These proofs are word for word the same as the proofs given in [1]. This having been done, the remainder of the proof is exactly the same as the proof just given of the special case. This completes the proof of Lemma 9. LEMMA 10. Let U be a balanced open set and let V and W be components of U. Let dw be £n H(V) with boundary measure JJ. and let da be in H(W) with boundary measure JI. Then JJ. and II are mutually singular.

Proof. Assume that JJ. and JI are not mutually singular. Since JJ. is a measure .on the set P of essential boundary points of V and JI is a measure on the set Q of essential boundary points of W, and since both JJ. and II vanish on points, it

144

uuu

.l!.iIUt.r...J..l. luOnut

follows that P n Q is infinite. Let Zo and z, be distinct points of P n Q. There exists a simple arc J, whose interior lies in V and whose endpoints are Zo and z, , because Zo and z, are accessible. There similarly exists a simple arc J. whose interior lies in Wand whose endpoints are Zo and Z,. Thus J = J] U J~ is a simple closed curve lying in C Let L be the interior of J. Since no point in L is in the unbounded component of - C, and since C is balanced, we see that no point in L is in the boundary B of C It follows that L lies entirely in C. Since L is conneeted and clearly contains poilJt~ of both U and V, this is a contradiction. Therefore fJ. and II are mutually singular, as was to be proved. We now prove the main theorem. ThEOREM 3. Let C be a compaci balanced set with interior U. Define the mapping from H(U) to AJ(C) by u(dw) = fJ., where fJ. is the boundary measure for d:.J. Then u is a one-to-one map of H(U) onto JlJ(C) and II dw II = [I u(dw) [I for each dw in H(U). (J"

Proof. We first remark that u(dw) is actually in JI(C) for each dw in H(U), because M(U) C M(C) by Lemma 2 and u(dw) £ 111(U) by Theorem 1. By Theorem I, we see that the map (J" i~ one-to-om' To finish the proof, we must show that any fJ. in M(C) is the image of some dw in H(e) and that [I dw II = II!J. [t To this end, consider an arbitrary !J. in M(C) Define the analytie differential dw = j(z)dz on U by t(z) = (211'i) -]

(*)

J

(t - z) -1 dfJ.(t).

We shall prove that dw £ lIeU), that u(dw) = fJ., and that I! dw [[ = [[ fJ. [I, thus completing the demonstration of the theorem. Let r denote the set of components of U, and for each V in r let dwv be the restriction of dw to V. By Theorem 2, dwv £ H(V). Let fJ.v be the boundary measure of dwv. By Theorem 1, we see that}Lv £ M(V) and (**)

for all z in V If we apply Theorem 2 to the measure fJ.v in AI(V) C M(C), \\e therefore have [[ dw y II ~ II }Lv II· Sinee the measures!J.V are mutually singular, by Lemma 10, we have

II L

!J.v

Vdl

for every finite subset n of

L

II

dwv

r.

II ~

II

=

L II !J.v II YEn

It follows from Lemma 8 that

II

L

all sums being taken over V in

L

ILv

II

=

II

L}Lv

n. Thus we have

v.r

1\

dwv

II ~ IIIL II·

145

II ~ I[ IL II,

339

BOUNDARY MEASURES

It follows from Lemma 6 that dw Thus II £ M(C) and

(***)

fez)

=

£

R(U) and

J

(2nr 1

(t -

II dw II Z)-l

~

II Jl. I.

Write

II

= u(dw).

dll(t)

for all z in U, by Theorem 1. Because of (*) and (***), it follm,s from Lemma 3 that II = Jl.. Thus Jl. = u(dw) and II dw II ~ II f.l II· But II Jl. il ~ II dw II by Lemma 5. This completes the proof We now derive Mergelyan's approximation theorem for sets which do not separate the plane. THEOREM 4 (Mergelyan). Let C be a compact set with connected complement in the complex plane Let f be a contmuous function on C whlch is analytw on the interior U of C Then f can be uniformly approximated on C by polynomials Proof. Assume the contrary. It follows that f cannot be uniformly appro>..imated on the boundary B of C by polynomials There therefore exists a measure Jl. on B which is orthogonal to all polynomials, but not orthogonal to f. Since

Jl. is orthogonal to polynomials and since - C is connected, we see that M(C). By Theorem 3, there exists dw in H(U) such that Jl. is the boundary measure of dw. By Definition .5, it follows that there exists a bequence (-y,} Jl. £

of compact subsets of U ,,,hich delimits U such that

I.

h dw

---7

J

h dp.

as i

---7 00

for all continuous functionb h on C. Applied to the function f, this gives

1f

dp. = lim

1'"1' f dw =

0,

since fdw is an analytic differential on U. This eontradietb the fact that not orthogonal to f, thereby proving the theorem

Jl. IS

4. Remarks. 1. We have relied heavily, in proving Lemma 4, on Mergelyan's approxi-

mation theorem ([4, 21]) for approximation by rational functiom, on compact sets without iuterior Lemma 4 could be proved equally ,yell by appeal to the general approximation theorem announced ab Theorem 4 of [2] and to a cIabsical theorem of Wah;h [6] on approximation by harmonic functionb. The author wishes to take thib opportunity to point out that Theorem 5 of [1] ib a special case of this theorem of Walsh II. ''Ye have derived Theorem 4 as a bimple consequence of Theorem 3 This raises the question of whether, for a compact set C with connected complement, Theorem 3 can be derived from Theorem 4 That is, it might be suspected that Theorem 3 and Theorem 4 are equivalent, Theorem 3 being a reinterpretation

146

O":l:U

ERKE'rI' HII:!J::lUr

in the dual space of Theorem 4, This does not seem to be the case, so that Theorem 3 represents an essential strengthening of Theorem 4. III. It is natural to ask whether the analog of Theorem 3 holds for more general sets C, provided that H(U) and M(C) are defined appropriately. This seems to be a difficult question. It is clear that some hypotheses on Care necessary. REFERENCES

1. E. BISHOP, The structure of certain measures, this Journal, vol. 25(1958), pp. 283-289. 2. E. BISHOP, Some theorems concerning function algebras, Bulletin of the American Mathematical Society, vol. 65(1959), pp. 77-78. 3. C. CARATHEODORY, Conformal Repre8entation, Cambridge, 1952. 4. S. N. MERGELYAN, Uniform approximation to functions of a complex variable, American MathamaticaI Society Translation no. 101, 1954. 5. O. PERRON, Irrationalzahlen, New York, 1951. 6. J. L. WALSH, The approximation of harmonic functions by harmonic polynomials and by harmonic rational functions, Bulletin of the American Mathematical Society, vol. 35(1929), pp. 499-544. 8. A. ZYGMUND, Trigonometrical Series, New York, 1955. UNIVERSITY OF CALIFORNIA BERKELEY

147

A GENERALIZATION OF THE STONE-WEIERSTRASS THEOREM ERRETT BISHOP

1. Introduction. Consider a compact Hausdorff space X and the set C(X) of all continuous complex-valued functions on X. Consider also a subset ~ of C(X) which is an algebra, which is closed in the uniform topology of C(X), which contains the constant functions, and which contains sufficiently many functions to distinguish points of X. Such an algebra ~ is called self-adJ'oint if the complex conjugate of each function in %( is in~, The classical Stone-Weierstrass Theorem states that if 21 is self-adjoint then ~l = C(X). If ~ has the property that the only functions in ~I which are real at every point of X are the constant functions then 21 is called anti-symmetric. Clearly antisymmetry and self-adjointness are opposite properties, in the sense that if ~ has both properties then X must consist of a single point. Hoffman and Singer [2] have studied these two properties and given several interesting examples. The present paper was inspired by their work but it more directly relates to a previous paper of Silov [3]. The purpose of the present paper is to prove the following decomposition theorem for a general algebra ~ of the type defined above. THEOREM. There exists a partition P of X into disjoint closed sets such that ( i) for each S in P the restriction ~s of '2{ to S is anti-symmetric, (ii) if a function f in C(X) has, for each S in P, a restriction to S which belongs to 2l s , then f is in ~, (iii) for each S in P, each closed subset T of X - S, and each c > 0 there exists g in ~l with II g II ;£ 1, with Ig(x) -- 11 < c for x in S, and with Ig(x) I < c for x in T.

Property (ii) of this theorem is the essential new fact of this paper. ,!,he construction given below which leads to the partition P is due to Silov [3], who in essence proved (i) and (iii). Silov proved a weaker property than (ii). Our proofs are different from those of Silov, although the construction is the same. The fact that the Stone-Weierstrass theorem is a special case of the theorem to be proved here is clear. If 21 is self-adjoint then each ~s is self-adjoint. Since ~s is also anti-symmetric, each set S in P consists of a single point. Therefore ~s = C(S). By the theorem to Received June 15, 1960,

777

148

778

ERRETT BISHOP

be proved, it follows that each function in C (X) is in Ill. Thus III = C (X), which is the conclusion of the Stone-Weierstrass Theorem. 2.

Proof of the theorem.

The key step in the proof will be the

following lemma. LEMMA. Let Y be a compact Hausdor.tf space and lB be a subalgebra of C (Y) which contains the constant functions. Let ~ be all real functions in lB. Define Yl 0= y" for Y, and Y2 in Y, to mean that f(Y,) = f(Y2) for all fin !H. Let Q be the set of all equivalence classes for this equivalence relation, so that Q is a partition of Y into disjoint closed sets. Let p. be a finite complex-valued Baire measure on Y and f a function in C (Y) such that

(a)

11p.11~1,

(b)

}gdP.

=

0

(c)

} fdp.

*-

O.

for all g in )S,

Then there exists So in Q and a finite complex-valued Baire measure

lJ on So such that (a,) IllJ II ~ 1, (b,) ~ gdlJ = 0 for all g in ~\ (c,)

In

\

~fdlJ \ ~ \ ~fdP. \.

Proof. It is clearly no loss of generality to assume that)S is closed C (Y). Let 'Y = {gil be a finite set of functions in ~ such that gi ~ 0 for all i,

Xg i = 1 . Let r denote the class of all such 'Y. by writing

Define a partial ordering on

r

{g;} ~ {gil

if there exists a mapping rp of the set of indices j onto the set of indices i such that

for all i. To see that r is a directed set relative to this partial ordering, let {gil and {gil be any two elements of r. Then the set {giga is clearly a common successor of {gil and {gil. Consider 'Y = {gil in r. For each index i let P.i be the measure

149

A GENERALIZATION OF THE STONE WEIERSTRASS THEOREM

779

defined by

for each Baire subset H of S.

Clearly

and kdfl

=

~ ~fdfli .

Thus for at least one value of i with

II fl. II*-O we have

Choose such a value of i and write

It follows that

(a) and (fj)

By the compactness in the weak* topology of the unit sphere of the set of Baire measures on X, it follows that the net {fly} has a cluster point ); in the weak" topology. Let Xo be any point in the support of the measure );, and let So be that member of the partition P which contains Xo. Let X, be any point in X - So. Thus there exists ho in :R with hoexo) *- hoCx,). Let

If the real constants

C,

and

C2

are chosen properly then

It follows that there exists a neighborhood Uo of Xo and a neighborhood U, of X, such that h,(x)

< t,

and

150

X €

Uo

Let A. be continuous real-valued function on the range of h, with 0 ~ A, ~ 1, ;>.,(t) = 0 for t ~ t, A.(t) = 1 for t ~ t. By the Weierstrass approximation theorem, ;>.,(t) is a uniform limit of polynomials in t. Therefore the function h. = A.oh, IS III m. Clearly 0 ~ h. ~ 1, h,(x) = 0 for x in U o, and h,(x) = 1 for x in U,. Define g, = h. and g, = 1 - h.. so that {gJ E r. If 'Y = {gj} is an element of r which follows {g;}, then each vanishes on either Uo or U,. Therefore the support of f1~ is either a subset of X - Uo or of X - U,. Thus the support of v is either a subset of X - Uo or X - U,. By the choice of x o, it follows that the support of v cannot be a subset of X - Uo and is therefore a subset of X - U,. Therefore x, is not in the support of v. Since X, was any point in X - So, it follows that the support of v is a subset of So. Thus v is a Baire measure on So. It is clear from the definition of v and from (0:) and (,8) that (a I) and (e,) are valid. It only remains to prove (b,). Now for each g in )B and each 'Y in r we have

g:

by (b) and the fact that ggi E)B. Passing to the limit gives (b,). This completes the proof of the lemma. Let Q be the class of ordinal numbers whose cardinal numbers are less than or equal to 2~, where f3 is the cardinal number of X. For each (J in Q we define ):)y transfinite induction a partition Pa- of Q into disjoint closed sets. This is to be done in such a way that P u is a refinement of P, for (J > T. The definition is started by defining P, = {Xl'. so that the first partition P, consists of the set X alone. Assume that P, has been defined for all ordinals T < (J. If (J has a predecessor (Jo, let S be any element of P Uo ' and let l! be the set of all functions in ~l which are real on S. Partition S by defining X, == x, for X, and x. in S to mean that !(x,) = !(x,) for all ! in~. Clearly S is partitioned into disjoint closed sets by this equivalence relation. The totality of all sets into which the elements S of PUo are partitioned in this way is defined to be the class P u • If (J has no predecessor, define X, == X 2 , for X, and x. in X, to mean that X, and x. belong to the same element of P, for all T < (J. The equivalence classes of this equivalence relation clearly form a partition P u of X into disjoint closed sets. Thus the classes P u are defined for-

151

A GENERALIZATION OF THE STONE WEIERSTRASS THEOREM

781

all a in Q, and it is clear that P u is a refinement of P whenever a > r. Assume that PU + 1 is a proper refinement of P u for all a in Q, i.e., that P U - 1 -=1= P u • Then P u +! contains a set not in any p. for r < a + 1. Therefore the cardinal number of subsets of X is at least equal to the cardinal number of the set Q. This contradicts the choice of Q. Therefore there exists an ordinal in p such that Pp+! = Pp. We shall show that the partition P = P r satisfies all requirements of the theorem. The fact (i) that 'lls is anti-symmetric for each S in P = Pp is a consequence of the fact Pp = Pr+!. We next prove (ii). Consider to this end f in C(X) such that the restriction of f to S belongs to 'lls for all S in P. Assume that f is not in '2l. By the Hahn-Banach theorem, there exists a bounded linear functional on C(X) which vanishes on m and does not vanish at f. By the Riesz representation thaorem, this functional can be realized as a measure I'- on X. Thus T

~gdl'- = Vdl'-

0 ,

gem

*0.

we may clearly assume that II I'- II ~ 1. We now construct, by transfinite induction, a set Su in P u for each a in Q and a finite complex-valued Baire measure I'-u on Su with


(1)

Su c S-

(2)

I kdl'-u I ~ I kdl'-i

(3)

Ii I'-u II ~ 1 ,

(4 )

~gdl'-" =

for

0,

T

,

'

all g in

~r.

Clearly the measure 1'-, = fl does the trick for a = 1. Assume therefore that the sets S. and the measures p. have been constructed for all r < a. If a is a limit ordinal, let

and let I'-er be any cluster point in the weak* topology of the net {I'-.}-<er. Clearly this set Ser and the measure I'-u do the trick. If a is not a limit ordinal, then there exists r with a = r + 1. Let Y be S. and let )J3 be the restriction to Y of the algebra Ilt. By the above lemma, applied to the measure 1'-" there exists a finite complexvalued Baire measure fl" on some Su e P u, with ST c Y = Sq, such that

(3)

II I'-er [I

(4)

~gdl'-er=O,

~

1, all g in

~

152

782

ERRETT BISHOP

(2)

I ~fdp" I ~ ! VdPr I ~ If d Pl

.

Thus the set S" and the measure P" do the trick. This completes the construction of the sets S" and the measures fl.". For a = p we have

and ~gdPp = 0 for all g in~. Therefore the restriction of f to Sp is not in ~sp. This contradicts the assumption of (ii), thereby proving that f is in 21. This proves (ii). It remains to prove (iii). To do this we prove by induction on a that for each a in Q, each S in P", each closed subset T of X - S, and each c > 0 there exists g in ~ with II g II ~ 1, I g(x) - 11 < c for all x in S, and I g(x) I < c for all x in T. Once this is done (iii) is obviously obtained by letting a equal p. Since the existence of g is clear if a = 1, consider a > 1 and assume that all smaller values of a have been disposed of. If a is a limit ordinal, there exists T < a and R in P. with S c R and T c X - R. By the induction hypothesis, there exists g in I2l: with II g II ~ 1, I g(x) - 11 < c for all x in R, and I g(x) I < c for all x in T. Since S c R this function g has the required properties. It remains to consider the case of an ordinal a which has a predecessor T, so that a = T + 1. Let R be that element of P, for which S cR. Let ~ be the closure in C(R) of I2l: R • Let x, be any point in Sand Xo any point in Tn R. By the definition of the partition P" there exists h, in I2l: such that the restriction of h, to lR is real and such that h,(x o) *- h,(x,). Let A. be a function on h,(R) with 0 ~ A. ~ 1, A.(h,(xo)) = 0, A.(h,(x,)) = 1. Define the function h. in C(R) by Since A. is a uniform limit of polynomials, h. E~. Clearly II h211 ~ 1, h 2(x O) = 0, and h 2(x,) = 1. Thus h 2(x) = 1 for all x in S. By the compactness of Tn R, the product of a certain finite number of such functions h2 will be a function hg in ~ with II hgll ~ 1, hg(x) = 1 for x in S, I hg(x) I < S for x in Tn R, where S is an arbitrarily small positive number. By the definition of ~, there exists h. in I2l: with

for all x m R.

so that h5

E

12l:.

Define

Also

153

A GENERALIZATION OF THE STONE WEIERSTRASS THEOREM

783

for all x in R. Thus there exists an open set Uo c X with R c Uo such that I h.(x) I < 1 for all x in Uo• If 15 is sufficiently small it is clear that

I h.(x)

-

11 < ~ 2

for all x in 8 and

I h.(x) I < e for all x in T n R. Thus there exists an open set UI in X with Tn R c UI such that I h.(x) I < e for all x in UI • By the induction hypothesis at stage T there exists h6 in ~ with II h611 ~ 1, I h 6(x) - 11 < e/2 for x in R,

for all x in X - Uo, and

I h6(X) I < min {e I h5(X) I-I:

W

x e T -

UI }

,

for all x in T - UI • Define g = h5h6' so that g e~. For x in Uo we have I h.(x) I ~ 1, so that I g(x) I ~ 1 since II h611 ~ 1. For x in X - Uo we have I g(x) I ~ 1 because of H. Thus II g II ~ 1. For x in 8 we have I h6 (x) - 11 < e/2 and I h.(x) - 11 < e/2, so that

I g(x)

-

11

~

I h (x) II h.(x) 6

-

11 + I ho(x)

-

11 < e •

For x in U1 we have I h 5(x) I < e, so that I g(x) I < e since II hall ~ 1. For x in T - U1 we have W, so that I g(x) I < e. Thus I g(x) I < e for all x in T. Thus g has all of the required properties. This completes the proof of (iii) and thereby the proof of the theorem. We note in conclusion that property (iii) and some results to be found in [1] imply that ~s is closed in C(8) for each 8 in P. REFERENCES 1 H. S. Bear, Some boundary properties of function algebras, Proc Amer Math Soc, 11 (1960), 1-4. 2 K Hoffman and 1. Singer, On some conjectures of Gelfand (in Russian), Uspehi Math Nauk (N S), 14 (1959), 99-114 3. G E. Silov, On rings of functions with uniform convergence (in Russian), Ukrainian Math Journal, 3 (1951), 404-411

154

by permission of the American Mathematical Society

A PROOF THAT EVERY BANACH SPACE IS SUBREFLEXIVE BY ERRETT BISHOP AND R. R. PHELPS

Communicated by Mahlon M. Day, August 19, 1960

A real or complex normed space is subrefiexive if those functionals which attain their supremum on the unit sphere S of E are normdense in E*, i.e., if for each f in E* and each E > 0 there exist g in E* and x in S such that I g(x) I = Ilgll and Ilf - gil < E. There exist incomplete normed spaces which are not sub reflexive [1] 1 as well as incomplete spaces which are subreflexive (e.g, a dense subspace of a Hilbert space). It is evident that every reflexive Banach space is subreflexive. The theorem mentioned in the title will be proved for real Banach spaces; the result for complex spaces follows from this by considering the spaces over the real field and using the known isometry between complex functionals and the real functionals defined by their real parts. We first cite a lemma which states, roughly, that if the hyperplanes determined by two functionals f and g (of norm one) are nearly parallel, then one of IIf-gll. IIf+gll must be small.

LEMMA. Suppose E is a normed space and E>O. If f, gEE*, Ilfll = 1 =llgll, are such that Ig(x)1 ~E/2 whenever f(x) =0 and Ilxll ~1, then Ilf-gil ~E or Ilf+gll ~E.

A proof of the lemma may be found in [2, Lemma 3.1]. To prove the theorem suppose fEE* and E>O. We may assume that Ilfll = 1; by the lemma, we want to find gin E* such that I g(x) I ~ 1 for all x in T= {x:f(x) =0 and Ilxll ~2cq, and for which there exists x in S such that g(x) = 1 = Ilgll. Let C be the convex hull of the union ofI the sets T and U = {x: Ixii ~ 1}, and suppose there exists Xo in U which is also in the boundary of C. Since C has nonempty interior, by the support theorem there exists g in E*, Ilgll = 1, such that 1 An easily described example has been suggested by Y. Katznelson: Let E be the space of all polynomials on [0, I), with the supremum norm. (The example in [1] shows clearly how the method of proof given below fails without the assumption of completeness.)

97

155

98

ERRETT BISHOP AND R R. PHELPS

sup(g(x):xEcl =g(xo). It follows that g(xo)=l=l!xoll, so that the lemma applies and the theorem is proved. Thus, it remains to show that Unbdry Cis nonempty. To this end, choose z in U such that f(z»O and let K=(f(Z»-1(1+2c 1 ). Define a partial ordering on the set Z= {xE U:f(x) ~f(z) 1 as follows' Say that x>y if

(i) (ii)

f(x)

IIx - yll

> fey)

and

~ K[j(x) - fey)].

Suppose that W is a totally ordered subset of Z; by (i), the net of real numbers IJCx): xE is (bounded and) monotone, and hence converges to its supremum. From (ii) it follows that W is a Cauchy net; by the completeness of E, W converges to a point y in U. By the continuity of f and the norm it follows that y is an upper bound for W. Thus, by Zorn's lemma, there eXIsts a maximal element Xo of Z; since xoE uec, we need only show that xoEbdry C. If not, then Xo is in the interior of C, and there exists a> 0 such that xo+azE C. From the definition of C we see that there exist y in U, x in T and A in [0,1] such that xo+az=Ay+(1-A)X. Thenf(z) ~f(xo) Xo, a contradiction which completes the proof. A possible generalization of this theorem remains open: Suppose E and F are Banach spaces, and let £(E, F) be the Banach space of all continuous linear transformations from E into F, with the usual norm. For which E and F are those such that = II (for some x in E, II xii = 1) dense in £(E, F)? This is true for arbitrary E if F is an ideal in m(A) (the space of bounded functions on the set A, with the supremum norm). Added in proof: If C is a bounded closed convex set, let C' = IJEE* :f(x) = sup lJ(y): yE for some x in C A slight modification of the above argument shows that C' is dense in E*. This solves a problem proposed by Klee in Math. Z. vol. 69 (1958) p. 98.

wl

Ily-xoll (1-A)lly -xii

(l-A)(lIyll +llxll)

T

cl

(1-A)(1

II Til

Txll

I.

BIBLIOGRAPHY

1. R. R. Phelps, Subreflexive normed linear spaces, Arch. Math. vol. 8 (1957)

pp 444--450 2. - - , A representation theorem for bounded convex sets, Proc Amer Math. Soc., to appear. UNIVERSITY OF C~LIFORNIA, BERKELEY

156

MAPPINGS OF PARTIALLY ANALYTIC SPACES.· By

ERRETT

BISHOP.

1. Introduction. We study a mapping f of a partially analytic space K into n-dimensional complex affine space Cn. A partially analytic space K is a Hausdorff topological space which is a countable union of conditionally compact open sets, which is locally connected, and which is endowed, at least partially, with an analytic structure. This means that a countable family {Ma} of subsets of K is given, each with the structure of a complex analytic manifold. An algebra ~ of continuous functions on K, called the analytic functions on K, is also given, with the property that each function in ~ is analytic on each "Ma. The algebra ~r is closed in the topology of uniform convergence on compact subsets of K. The coordinates of the mapping fare required to lie in~. The purpose of this paper is to show that for certain spaces K the mapping f can be chosen to have certain properties. For instance, sufficient conditions are given for there to exist an f which is proper, or which separates points, or which has a certain rank at certain points. Since the complex spaces of Cartan [4J or Behnke and Stein [2J can be thought of as partially analytic spaces, these results apply to complex spaces. 1 Theorem 1 is concerned with the dimensions of the level sets of the mapping f. The statement of Theorem 1 is somewhat complicated; a special case reads as follows. If the level sets of ~ intersect each Ma in a countable set, then f can be chosen so that the level sets of f intersect each Ma in a set of (complex) dimension at most max {d a - n, O}, where da - dim M a. Theorem 1 is closely related to a theorem of Grauert ([6], Satz 12), but the proof given here differs from Grauert's proof and a more gpneral theorem is obtained. The next part of the paper is concerned with analytic polyhedra. With f as above, write

Po-{p: pEUM... lfc(p)1
1

209

157

210

ERRETT BISHOP.

Theorem 3 states that if S is a compact subset of P then there exists an analytic mapping g of K into Cd which defines an analytic polyhedron Q with seQ C P, where d is the maximum of the dimensions of the Ma (which is assumed to be finite). Theorem 3 (in a more general form) is basic to the theorems which follow. To prove Theorem 3, one first modifies the mapping t = (f1, . ,tn) slightly so that for each 0: the sets of simultaneous constancy of the functions tdt1'· .. , tn/l1 on

have the same dimension at each point of Ifa as the sets of simultaneous constancy of 2f. That this can be done follows from Theorem 1 if n > d, and if n < d there is nothing to prove. This modification of having been made, choose the constant r> 1 such that I (p) I < r 1 for all p in Sand 1 < i < n, and write

ti

t

Then if N is sufficiently large the mapping g = (g2' . . ,gn) of K into Cn-1 defines an analytic polyhedron pI with S C P' c P. After n - d such constructions one obtains the analytic ployhedron Q of Theorem 3. Next it is assumed that K is weakly holomorphically convex, which means that K = U Ma and that for each compact S C K each component of the set

s= {p: pE K, I l(p)1 <maxq€S I l(q)1

for all I in 2f}

is compact. Since weak holomorphic convexity implies that K is the umon of an increasing sequence of analytic polyhedra, one is able to apply Theorem 3 to show (Theorem 4) that K admits an almost proper analytic map I into Cd, where d is the maximum of the dimensions of the Ma and where almost proper means that each component of the inverse image of a compact set is compact. There follows a section of the paper devoted to the study of certain special analytic polyhedra (Theorem 5). As a result of this study we obtain some integral formulas (Theorems 6 and 7) which are an attempt to generalize the usual one dimensional planar formula to several dimensions and to manifolds. These formulas recall the several dimensional integral formulas of Weil [13] and Bergmann [3], although they seem to be fundamentally different from those formulas. Next we return to the study of mappings and show (Theorem 8) that it K is weakly holomorphically convex, if 2f separates points of K, and if certain other minor conditions are satisfied, there exists a proper mapping

158

211

PARTIALLY ANALYTIC SPACES.

of K into Crl+" with d as above, where proper means that the inverse images of compact sets are compact. The proof of this theorem goes as follows. Let f be an almost proper mapping of K into Crl. Let {Sd be an increasing sequence of compact sets the union of whose interiors is K. Let

J k = {z:

Z

= (Z,,"

,Zrl) E Crl, I Zi I
For each positive integer k let HI. be the union of those components of f- ' (J k) which intersect Sk. Let Gk be the set [f-1(JI.;) n HI.+1J -H k • The sets G" and HI. are disjoint compact sets. If ~k denotes the set of all uniform limits on Gk U H k of functions in ~, it can be shown that the maximal ideal space of the Banach algebra ~h is just the set GI. U Ih. It follows from the general theory of Banach algebras that there exists WI< in ~ arbitrarily large on G" and arbitrarily small on HI.. Since {H d is an increasing sequence of compact sets the union of whose interiors is K, it follows that the functions WI. can be

w" converges uniformly on ~r with I f rl+1 (p) I > k for all p in

chosen so that the series

~

compact subsets of K

k=1

to a function f ,hI in GI. and all k. that the mapping f X frl+1 of K into Crl+1 is proper, write

F,,= {p: pE

K.I f.(p) I
To see

1
and consider p in F k • Let j be the smallest integer not less than k such that p E H j • If j> k then p E Gj-l- Therefore k > I frl+1 (p) I > j -1. This contradiction shows that j = k, so that p E HI<.' Thus Flo C H k , so that Fie is compact and the mapping f X frl+1 is proper. It is now relatively simple (Theorem 9) in case K, for instance, is a Stein manifold to choose functions f1O+2' . ,f2n+1 in ~r such that (f,,' . " f2n) has rank n at pach point of K and (f,,' . " f2n+1) distinguishes points of K. This is done by choosing the functions successively to decrease the dimensions o~ certains singular sets associated with the mapping. This last result is a special case of as yet unpublished results 2 on the problem by R. Narasimhan, who used other methods. The first result of this type is due to Remmert [8]. If one looks at the proofs morp closely, it can be seen that all of the mappings considered are dense in the topology of uniform convergence on compact subsets of K. It is intended to devote a subsequent paper to applications of the results given here. I

• Since this was written Narasimhan's paper has appeared in Amerioan Journal of Mathematios, vol. 82 (1961), pp. 917-934.

159

212

ERRETT BISHOP.

2. Dimensions of level sets of mappings. Rather then begin by considering what is known in the literature as analytic space we introduce a notion which is somewhat more general and which is sufficient for the purposes of this section.

Definition 1. A partially analytic space K consists of a Hausdorff space K, a countable family {Ma} of subsets of K, each of which is a (complex) analytic manifold, and a set ~ of continuous (complex-valued) functions on K. The relative topologies of the subsets Ma are the same as the topologie<; which they possess as analytic manifolds. The space K is a countable union of open sets whose closures are compact, and each component of an open set in K is open. Each function in ~ is analytic on each Ma. ~(is closed ill the topology of uniform convergence on compact subsets of /(. Also ~[ is an algebra in the sense that sums, products, and complex multiples of elements in Ware in W. In the abo,·e definition and in the sequel an analytic manifold will be understood to be connected and coverable by a countable number of coordinate neighborhoods. Thus each Ma has a (complex) dimension da , which is the same as the dimension of Ma at each of its points. As an abbreviation, we speak of a partially analytic space K, the rest of the structure-the manifolds M a , called the structure manifolds of K, and the algebra W, called the set of analytic functions on K-being understood. The set of continuous complex-valued functions on K is written C (K), and the subset of C (K) consisting of those functions analytic on each )1 a is written A (K). The set of all analytic functions on an analytic manifold M is written A (M). The topology of uniform convergence on compact subsets of K is easily seen to be a metrizable topology for C(K). To see this, let {F .. } be a sequence of compact sets whose interiors cover K, and let 11/11,,==sup{lf(p)l: pEF.. } for each

1 in

C (K).

Let

..

p(f,g) = ~2-"1I I-g 11=1

11 .. (1

+

II

I-g 11 .. )-1

for all 1 and gin C(K). Then p is a metric for C(K) and C(K) is complete in the metric p. As a closed subset of C (K), W is also a complete metric space with metric p. We now recall some standard material from the theory of several complex variables which will be of use later. See in this connection the notes [4], [5],

160

and [10J, and the paper [12]. A set A in an analytic manifold },f is analytic if to each point p of M there exists a neighborhood U of p and a finite set of analytic functions on U such that A n U is the set of common zeros in U of the functions. A set A is locally analytic in M if to each p in A there exists a neighborhood U of p and a finite set of analytic functions on U such that A n U is the set of common zeros in U of the functions. A locally analytic set is analytic if and only if it is closed. A point p of a locally analytic set A in a manifold jl is called regular if there exists a coordinate neighborhood U of p and coordinate functions f 1,· . ., f n on U such that A n U

=

{q:

fi (q)

=

0, 1 < i < le}

for some positiye integer le. The set AR of regular points of a locally analytic set A is a locally analytic set which is open and dense in A; if A = A R , then A is called a regular locally analytic set. A connected regular locally analytic set is called a submanifold of 111. Then AR has a countable number of components, each open in A, each of ",hich is a submanifold of Jl and each of which inherits from .11 in a natural way the structure of an analytic manifold. The (complex) dimension of A, which equals one-half of the topological dimension of A, is the maximum of the dimensions of the components of A R . If p is a point in A, the local dimension of A at p is the minimum of the dimensions of the locally analytic sets A n U, where the sets U are neighborhoods of p. The dimension of A is the maximum of the local dimensions of its points. The set As = A - AR is a locally analytic set whose dimension is strictly less than that of A; it is an analytic set if A is an analytic set. This gives rise to a sequence

of locall~ analytic se~s, each a closed subse~ of the preceding: such t~atl A' = (A H) S for 1 < '/, < le and such that A k 1S empty. Thus A 1S the umor! of the disjoint regular locally analytic sets (A ih,
°

• The referee has pointed out that this fact, while true, is not trivial if ~ consists of all functions on an analytic space K which are analytic in the narrow sense, that is,

161

214

ERRETT BISHOP.

Definition 2. A set f 1 , ' • " f d of analytic functions of ad-dimensional analytic manifold M will be called a coordinate set of functions at a point p in M if there exists a neighborhood U of p and a constant r > 0 such that

maps U homeomorphically onto the set E={Z=(Z"

. ',Zd): !Z,-fi(P)!
in d-dimensional complex affine space Cd. '1'he set U will then be called a coordinate neighborhood of p for the functions f 1,' . " f d· LEMMA 1. Let K be a partially analytic space with a single structure manifold M. Let m be a positive integer not larger than the dimension d of JIJ. Let f 1, . ,fd in A (1V1) be a caordinate set of functions at a point p of M, and let U be an associated coordinate neigh borhood,

Let fm be in ~{ and h in A (M). Then the set G of functions g in m wch that the level sets V (sets of simultaneous constancy) of the functions

+

f 1,'

. " f m-1, g h on U are all of dimension d - m or less is dense in ~r and is a countabZe intersection of open subsets of ~r.

Proof. Let go be any element of ~{ and € a positive number. assume the value b at some point in U. Define d+l

g

=

go

+ ~j=l aj(fm-b)j -

Let fm

d+l

~ aj(-b)j,

j=l

where the aj are constants to be chosen. Thus gEm and p (g, go) < € if the aj are sufficiently small. Choose the aj to be that small and in addition to have the property that the functions

(*) have no common zero in U. This is possible because the range of the mapping 0':

P~

(c 1,' . " Cd+1),Ci= (i!)-l ai(go+h)/(a!m)i] p

of U into Cd+l has topological dimension at most 2d and therefore contains which arise from functions analytic in open subsets of the analytic manifolds in which K is considered as locally imbedded. The difficulty in this case is to show that W is closed in elK). This was proved by Grauert and Remmert in Mathematische Annalen, vol. 136 (1958), pp. 245-318.

162

PARTIALLY ANALYTIC SPACES.

no neighborhood of the origin. To see that 9 is in G, it is enough to show that g h is not constant on each of the coordinate planes

+

L

=

{p: p E U,fi(p)

=

Ci, 1
since the sets V are then proper analytic subsets of such planes L and therefore have dimensions less than the dimensions d - (m - 1) of the planes. h is constant on some plane L. Thus Assume therefore that 9

+

8i (g

+ h)/(8fm)i

d+1

=

8i (go

+ h)/8(fm) + ~;=1(].;(j!j(j -i)!) (fm- b)H i

vanishes everywhere on L. If po denotes a point of L for which fm(Po) = b, it follows that the functions (*) have a common zero at po. This contradiction to the choice of the constants (].j means 9 E G. Since go and £ were arbitrary, G is dense in W. To finish the proof of the lemma, define Fie, for each positive integer k, to consist of all 9 in W such that there exists a coordinate plane L with ICil
Definition 3. If M is an analytic manifold, a subset S of A (M) is said to have dimension m over a subset T of A PI) at a point p of M, if the level set of T containing P includes the level set of S containing p and if the difference in the local dimensions at P of these two leyel sets is m. The set S is said to have dimension mo over T on 111 if each level set of S on M is contained in a level set of T on }rf and if rno is the maximum of m for p in M. If T is void, then one speaks simply of the dimension m of S at a point p or of the dimension rno of S on J1. The rank of S at p is the rank at p of the Jacobian matrix of S with respect to some roordinate set of functions at p. LEMMA 2. Let K be a partially analytic space with a single structure manifold 111 of dimension d. Let B be a subset of A (1II) and h an element of A (1If). Let P consist of all points p in 111 such that the dimension of WU B U {h} over B at p is at least 1. Let G be the set of all 9 in W such that the dimension of B U {g h} over B at p is at least 1 for all p in P. Then W- G is of the first category in ~L

+

Proof. The proof proceeds by induction on d. The lemma is clearly true if d = 0, since in that case the set P is void. Assume therefore that the lemma is true for all values of d less than the given value. At each point p in M let J p be the (possibly infinite) Jacobian matrix of the functions in B with respect to a set of coordinates at p, and let J p have rank mp. Let m be

163

216

ERRETT BISHOP.

the maximum of the mp. Choose Po with mpo = m, and let f1" . " fm be elements of B whose Jacobian matrix at po has rank m. Let V be the set of points in M where the Jacobian matrix of these functions has rank m, i. e., where these functions are part of a coordinate set of functions. Thus V is open in M. The level sets of B have dimension d - rn at each point of V. Either the level sets of WU B have dimension d - m at each point of V (case I) or there exists go in W such that the Jacobian matrix of f1" . ',fm, go has rank m 1 at some point of V (case 1). In case 1, let lV be the set of all points in M where the Jacobian matrix of fl" . " fm, go has rank m 1. In case I, either the Jacobian matrix of f 1, " , f m, h has rank m at every point of V (case 3) or has rank m 1 at some point of V (case 2). In case 3 define W = V. In case 2 let lV consist of those points of M at which the Jacobian matrix of f 1,' . " f m, h has rank rn 1. Thus we see that in all three cases W is a non-void open subset of lrl and M - W is an analytic subset of .'vI. Thus M - W is the union of a countable family {Mo:} of disjoint submanifolds of M. The dimension da of Ma is less than d. Let the partially analytic space Ka consist of the Hausdorff space K with the single structure manifold M a and with the set of analytic functions W. Let P a consist of all points p in Ma such that WU B U {h} has dimension at least lover B at the point p on the manifold Ma. Since do: < d, it follows from the induction hypothesis that the set Gex of all g in W such that B U {h g} has dimension at least lover B on the manifold Mex at all points of P ex has the property that W- Go: is of the first category in W. Consider now case 1. For each point p in W, let Up be a coordinate neighborhood of p for some coordinate set of functions containing the functions f1" . " fm, go· Since the topology of M has a countable base, a countable family {U j } of the Up covers 111. From Lemma 1 it follows that W- Gj is of first category in W, where Gj consists of all g in W such that the level sets of f1'" ,fm' g h on U j have dimension at most d - m -1. Define

+

+

+

+

+

+

In cases 2 and 3, define

Thus, in any case, H is of first category in W. We shall complete the proof by showing that W- G c H. To this end, consider g in W- G. Thus there exists p in P such that the dimension of B U {g h} at p equals the dimension of B at p. Let D, E, and F respectively be the level sets of B, of

+

217

PARTIALLY ANALYTIC SPACES.

+

h}, and of ~ U B U {h} which contain p. Let their respective dimensions at p be A, p., and v. Thus A = p.. Since pEP, we have v < A. Let 0 1 be a neighborhood of p such that D n 0 1 , EnO l , and F n 0 1 have dimensions A, p., and v respectively. Since the level set F n 0 1 of ~ U B U {h} on 0 1 which contains p has dimension v, there exists a neighborhood 0 C 0 1 of P such that every level set of ~ U B U {h} on 0 has dimension at most v (see [5], Exp. XIV). Now for each IX the set En 0 n Ma. is a countable union of compact sets; so is the set En 0 n w. ~\lso, En 0 is the union of En 0 n Wand the sets En 0 n Ma.. By a standard theorem of dimension theory (see [7], p. 30), it follows that A = dim(E n 0) is the maximum of the dimensions of En 0 n Wand the sets En 0 n !lIa.. Thus

B U {g

dim(E nOn Q) =A, where Q is either lV or one of the sets Ma.. Thus there exists a point po in E nOn Q such that the dimension of E n Q at Po is A. By the choice of 0, the dimension at po of the level set Fo of ~ n B n {h} containing po is at most v. Also, the dimension of the level set D n Q of B at Po on Q is at least A, since it contains the set E nOn Q which has dimension A at po; on the other hand, the dimension of D n Q at po is at most p. = A, and therefore this dimension equals "-. Consider first the case in which Q = Ma for some IX. Since v < A the dimension of Fo n Jfa at po is less than the dimension of D n Q = D n lila at po; therefore po EPa. Also the dimension of D n JIa. at Po is A and the dimension of En]Ia at po is A. By the definition of Ga, it follows that g E ~ - Ga. Thus g E H. It remains to consider the case Q = lV and to show that g E H in this case also. To do this, cases 1, 2, and 3 must be considered separately. consider! first case 1. Let U j be one of the sets of the family {U j } which contains po. By the choice of f1' . ,fm any function f in B must be constant on each component of each level set of these functions on W: otherwise, there would exist f in B such that the Jacobian of fl' .. ,fm' f with respect to a coordinate set of functions would be of rank m 1 at some point of W. Thus the dimension A at Po of the level set D n Q = D n W of B equals d - m. Since also A is the dimension of the level set E n W = E n Q of B U {g h} at po, it follows that g E ~ - Gj • Thus g E H. Next consider case 2. Since the function g in ~ is locally constant on the level sets in W of the coordinate functions f 1, . . , fm, the level sets of f 1, . . , f m, g h on Ware locally the same as those of the functions

+

+

+

165

218

ERRETT BISHOP.

f " . , f m,

h and therefore have dimension d - m - 1. Thus the dimensions at po of the level sets of Band B U {g h} are d - m and d-m - 1 respectively. This contradicts the fact, derived above, that both these dimensions are.\.. Thus case 2 does not arise. N ext consider case 3. In this case it is clear that locally the level sets of B, B U {g h}, and ~ U B U {h} are the same as the level sets of f" ,fm. This contradicts the fact that the level sets of B and ~ U B U {h} containing po have respective dimensions.\. and v < .\. at Po. Thus case 3 also does not arise. This completes the proof of Lemma 2.

+

+

Definition 4 (by induction). Let X be the Cartesian product of complete metric spares X,,· ,Xn • If n = 1, then Fe X is said to be of third category if it IS of first category. If n> 1, then F is said to be of third ClLtegory if the set of all points p in X, X XX 11-1 for which the set of points q in Xn such that (p, q) E X is not of first category is of third category. It is clear that a countable union of sets of third category i" of third category, and that the complement of a set of third category is uen'<e THEOREM 1. Let K be a partially analytic space. For each at, let Ba be a subset of A (.11",) and let hI"" .. , hk", be elements of A (JI",), where le is a fixed positwe integer. Let pir>., 1 . we have

(*)

The dimension on the manifold Ma at p of B", U {g,

+ hI""

.. , g ..

+h

k ,,}

over Ba is at least i. Then k~( - G is of third category tn the le-fold cartesian pi"oduct k~r of ~( with itself.

We show by induction on le that the theorem is true for all positive integers le. We aSbume that the theorem is true for all values of le smaller than the one being considered. The proof of the theorem when le = 1 is the same as the proof of the general inductive step, and so need not be considered separately Having fixed le, let fI,· . ,h be any elements of~. Let Qia, 1
166

PARTIALLY ANALYTIC SPACES.

+

+

of B U {g, h' "" . ,g1v--1 hk-1",} over B", is at least i has the property that k- 1m: - Go is of third category. Fix some (g" .. , g1v--1) in Go. Let Qa be the set of points p in Ma for which the dimension at p of 8 1=

m: U

B", U {g,

+ h ' a,

.

,g"-1

+h

k - 1 a}

U {hka}

over

is at least 1. By Lemma 2, the set Z of gIG in ~r such that for each ex and each p in Qa the dimension at p of Ba U {g, h ' a, . , gk hka } over Ba U {g, h1a, .. , gl.--1 P-1",} is at least 1 has a complement of first ., gl.--1 category. We first show that 9 = (g1, .. , gkJ is in G for each 91, in Go and each g.. in Z. Afterward, we show that this implies that km: - G is of third category. We therefore now prove that 9 E G. Consider a point p in some Pia. Let A be the dimension at p of

+

+

+

T1 =

~{U

Ba U {h 1 a,

+

,h".,}

and p. that of Be., so that A- p. :> i because p E pia. at p of

Let v be the dimension

+

Then either A = v or A = v 1 (see [10], p. 43), since T 1 contains one more function than T 2. Consider first the case v = A and i < le. Then v - JJ- :> i, so that p E Qi a. Therefore the dimension at p of Ba U {g1 h1a, .. , gk-1 hk - 1a } over Ba is at least i. Therefore 9 satisfies (*) at the point p. N ext consider the cabe in which either v = A-I or i = le. Thus v-p.:>i-l, so that pE Qi- 1 a if i#l. Therefore the dimension at p of Ba U {g1 h' ", . ,g1v--1 hle- 1a} over Ba is at least i - I ; this clearly also holds if i = 1. Thus to show that 9 satisfies (*) at the point p it is enough to show that the dimension at p of Ba U {g, h 1a,·· ,g .. hka } over Ba U {g, h 1a,· .. , gk-1 hk-1a } is at least 1. Therefore it is enough to show that p E Qa, or that the dimension of 8 1 over 8 2 is at least 1 at p. Consider first the case v = A-1. N ow the level sets of 8 , and 8 2 U m: are the same as the level sets of T1 and T2 respectively, because g1,· .. , g1v--1 are in m:. Thus the dimension at p of 8 1 over 8 2 is at least A- v = 1, so that p E Qa and 9 therefore satisfies (*) at the point p. Consider next the case A = v and i = le. Then A :> p. i = JL le, and as above we see that A is the dimension of 8 1 at p. Since 8 2 contains lc - 1 more functions than Ba and since B", has dimension JL at p, the dimension

+

+

+

+

+

+

+

+

+

167

+

220

ERRETT BISHOP.

of 8 2 at p is at most p,+ (k-l). Therefore the dimension of 8 1 over 8 2 at p is at least p, k - (p, k-l) = 1. Thus P E Qa so that g satisfies condition (*) at the point p in this case and therefore in all cases. It follows that (gl,' . " gk) E G. To see now that "& - G is of third category in k&, note that 1..-1& - Go is of third category in k-l& and for each (gl, " , gk-l) in Go the set of gk in & with (gl,' . " gk) in k& - G is a subset of & - Z and therefore of first category in &. Therefore k& - G is of third category, as was to be proved.

+

+

3. Analytic polyhedra and weak holomorphic convexity. In this section we investigate mappings of a d-dimensional partially analytic space into Cd. Theorem 1 of the last section of course has applications to such mappings, but the purpose of this section is to obtain stronger results, under the assumption that K has certain analytic convexity properties. The basic notion is that of an analytic polyhedron. Definition 5. A subset L of a partially analytic space K is called central if it is the union of some subfamily of the family of structure manifolds of K and if each component of an open set in L is open. Let (fl, " , fn) be functions in & and let L be a central subset of K. Define Po= {p: pE L, I fi(p)

1< 1,1
Let P be a union of certain components of Po. If the closure P of P is a compact subset of L, then P is called an analytic polyhedron defined by the functions (f1," , f,,), and L is called the frame of P. The dimension of P is the maximum of the dimensions of & on the manifolds },fa n P for},fa C L. P is called reduced if n < dim P; otherwise U is called unreduced. An unreduced analytic polyhedron P is called prepared if for each structure manifold },fa of K which is a subset of L and intersects P the set & has dimension 0 on thE' manifold },fan P n {p: f 1 (p) =1= O} over the set {f2{fl)-'" . ·,f.. (fl)-l}. BdryP denotes the boundary of P when P is considered as a subset of its frame L. Note that an analytic polyhedron P is an open subset of its frame L, since each component of the open set Po in L is open in L. Therefore

bdryP=P-P. 3. If P is an analytic polyhedron defined by fl' . ea'Ch pin bdry P there exi5ts k, 1
, f n, then for

Proof. Let Po be defined as above. Assume the lemma is not true. Then I fi(p) I < 1, 1
168

221

PARTIALLY ANALYTIC SPACES.

in Po, and since p E bdry PCP, it follows that pEP. shows that the lemma is true.

This contradiction

THEOREM 2. Let K be a partially analytic space and let fl,· . ,fn be functions in m: defining an analytic polyhedron P with frame L. Let S be a compact subset of P and r > 1 a constant such that

For each positive integer N let Q be the union of those components of Qo = {p: pEL, I (rf, (p))N

I < 1, I (rf,;(p) )N -

(rfl (p) )N

I < 1,2 < i <

n}

which contain points of S. Let R be the union of those components of

which contain points of s. If N is sufficiently large, then Q is an analytic polyhedron with frame Land SeQ C P. If P is prepared and if N is sufficiently large, then R is an analytic polyhedron with frame Land S eRe P. Proof. Clearly seQ and S C R if N is chosen so large that I f,;(p) I 1 < i < n, pES. Clearly, also, the fact that Q and R are analytic polyhedra with frame L will follow from the inclusions Q C P and Rep. Consider first the general case, so that P need not be prepared. We wish to prove Q C P. Now if p is any point in bdry P, then I f,;(p) 1= 1 for some i. If I fl(p) I >r" then by the definition of Qo it is clear that p is not in Q.

< irN,

If

I fl(P) 1<

r"

then

I (rf.(p))N -

(rfl (p))N I >rN -

(rrl)N

=

r N -1,

so that p is not in Q is N is sufficiently large. Thus if N is sufficiently large Q does not intersect bdry P. Thus each component of Q is eitHer a subset of P or disjoint from P. Since each component of Q intersects S C P, it follows that Q C P, as was to be proved. Now assume that P is prepared. We must show that ReP if N is sufficiently large. To this end it suffices as above to show that R does not intersect bdry P. Assume therefore that (bdry P) n R is non-void for a sequence {Nm} of values of N, where Nm->;oo as m->;oo. Choose a constant c with r l < c < 1 and write where

v=

{p:

I I,; (p) I < c, 1 < i < n}.

169

222

ERRETT BISHOP.

Since (bdry P) n R is non-void, there exists a component G of R - V which intersects bdry P. Now G contains a point qo of the boundary (in L) of V n L, since otherwise G would be a component of R containing no points of S. Let T m be the component of TOm containing qo, so that T meG. Now T m intersects bdry P, for otherwise T m would be a component of R - V and would therefore equal G, contradicting the fact that G contains a point of bdry P. For each p in T m there exists le, 1 < k < n, with \ flc (p) \ :> c. For this value of k we have

r N \ f,(P) \N:> \ (rh(p))N where N

=

N m'

1-\ (rh(p))N f, (p) oF 0

Hence, for pETm,

\ (f.(p) )N(f,(P) )-N -1\

<

(rf1(p))N \ :> (cr)N -1, and

«cr)N _1)-1

< 2(cr)-N

if N = N m is sufficiently large. Let Dm consist of all complex numbers z such that I zN-ll < 2(cr)-N. For z in D m ,

1-2(cr)-N

< \ z \N < 1 + 2(cr)-N,

so that 1-

2 (cr) -N

<

Iz \

< 1 + 2 (cr) -N,

or \1-\ z \ I < 2 (cr) -N. Also, if arg ZN is chosen between -.". and ."., then I zN - 1 I :> i \ arg ZN \, so that I arg ZN I < 4 (cr) -N. There therefore exists an integer j, 0 < j < N -1, such that

I arg z - 2j.".N-1 \ <4N-1 (cr)-N < 4(cr)-N if arg z is chosen properly.

\ z - Ci I <

Thus, if Ci

=

I I z I - I Ci II + I arg z -

exp (2.". (-1) IljN-1), we have

arg Ci I < 2(cr)-N 4(cr)-N = 6 (cr)-N.

+

Therefore Dm is contained in the union of the circular neighborhoods of radii 6(cr)-N about the Nm-th roots of unity. These neighborhoods are clearly disjoint if N m is sufficiently large. Thus the functions f. (f1) -" 2 < i < n, each map T m into the union of these disjoint neighborhoods. Since T m is connected, fdfd -1 maps T m into a particular such neighborhood F m" Let the corresponding N m-th root of unity be Cm.i' Thus

for all p in T m' By passing to a subsequence of {Nm} if necessary, we may assume that {Cm.'} converges for each i to a number C. as m ~ 00. The sequence {Tm} of compact subsets of the compact set P may also be assumed

170

PARTIALLY ANALYTIC SPACES.

to converge to a compact subset T of P, in the usual topology for compact subsets of a compact space. f'm is connected because T m is connected, so that T is connected. Clearly T is a closed subset of P. Since T m contains pointR in bdry P and bdry V, T also contains points in bdry P and bdry V. Also for all p in T. Partition T into equivalence classes by defining p == q, for p and q in T, to mean fi (q) = fi (p), 1 < i < n. Since T is connected and contains points in bdry P and bdry V, since each f; is bounded in absolute value by c < 1 on bdry V, and since to each point in bdry P there exists fi having absolute value 1 at that point, it follows that some fi assumes an uncountable number of values on the set Tn P. Thereforc there are an uncountable number of partition classes of Tn P. Thus there exist an Ma C L such that Ma intersects an uncountabll:> number of partition classes of T n P. Let S be that level set of the functions f 2(f1) -".. , f n (fd -ion PnMan {p: fI(p) ~O} which contains the set Tnp, so that S contains an uncountable number of level sets of f1' ,fn on "Ala. The locally analytic set 8 in Ma splits into a countable family {Sd of irreducible analytic sets, none of which is a subset of the union of the others There exists ko such that f,,' . ,fn are not all constant on S"". Since Sho is irreducible the sets of constancy of f1' ", fn on Ska arc all of dimension less than the dimension of 8 ko ' If po is a point in 8 4 not in any of the other 8/.., it follows that the dimension on the manifold M a of f 1,' • " f n oycr f 2 (f d -" . , f n (f1) -1 at Po is at least 1, contradicting the fact that P is prepared. This completes the proof. LEMMA 4. Let {Lj} be a finite family of centred subsets of a partiaUy analytic space K, and let f,,' . " f n be functions in ~r such that for each j there exists an integer y = y (j), 1 < y (j) < n, such that f 1, . ,f"t define an analytic polyhedron P j with frame L j • For each j let 8 j be a compact subset of Pj. Then there exist functions 1'" ",1'.. in ~r such that for each j the functions /'1" . ,f'"t define an analytic polyhedron P'j with frame L j with

Sj

c

pi; C P j

such that P'j is unreduced and prepared if P j Proof.

i~

unreduced.

Choose t:> 0 such that

SjCPjn{p:lfi(p)1

<

1-2t:,1
for all values of j. Consider a value of j for which P j is unreduced and let Ma be any structure manifold which intersects P j and is a subset of L j • Let

Nja=PjnMan {p: f,(p)

171

~O}.

ERRETT BISHOP.

The dimension of W on NjO/ is less than 'I, since P j is unreduced. level sets of Ware the same as the level sets of

Since the

the dimension of H on NjO/ is at most 'I -1. Let Gi be the set of all (g2,' . " gn) in n-lW such that for all a the set H has dimension 0 on the manifold NjO/. over the set

and let G =

nG

j•

By Theorem 1, with EO/. the void set,

n-l~r

-

Gj is of third

j

category in n-IW.

Thus the set n-lW -

G = U (n-lW -

Gj )

j

is of third category in n-lW, where the union is taken over all j for which P j is unreduced. Thus there exists (g2" . " gn) in G such that for all j we have I gi (p) I < £ whenever 2 < i < 'I (j) and p E Pj, whether or not Pi IS unreduced. Define

and 1'1 = fl' Since the level sets of Ware the same as the level sets of H, we see by the above that ~r has dimension 0 over the set

on the manifolds Njcx for each j for which P j is unreduced. Pi be the union of those components of

For each j let

P"j={P: pELj,ll'i(P)1
For each p in Sj and for 2 < i < 'I, we have

Clearly also 11'1 (p) I < 1 for p in Si' Therefore S} C P"j and hence 8 j C P l f • Consider now p in bdry Pi' By Lemma 3, I fi (p) I = 1 for some i, 1 < i < y. If i = 1, then clearly p is not in P'j. If 2
that p ¢ p} in this case also. Thus p l j does not intersect bdry Pj' Since each component of P'j contains a point of 8 j C Ph it follows that P'j C Pj. Thus P'j is an analytic polyhedron with frame L j defined by /'1" . " /''Y' If P j is unreduced then p l j is also unreduced since p l j C P j implies dim P'j IiO

172

225

PARTIALLY ANALYTIC SPACES.

-< dim P J ; p'J is also prepared since we have seen that /' 2(f/l) -1,.. , /' "Y ( / ' l) -1 on each of the manifolds

~

has dimension 0 over

This completes the proof. THEOREM 3. Let {L J} be a finite family of central subsets of the partially analytic space K. Let the functions fl,· .. , fn in ~ define for each j an analytic polyhedron P J with frame L J. Let 'Y = 'Y (j) = min {n, dim 1\}. Let 8 J be a compact subset of PJ. Then there exist functions 1'1, ,f~ in ~ such that for each j the functions /'1'· .. , r"Y define an analytic pulyhedron p'J with frame L; such that 8; C P'; C PJ.

Proof. Let T be the class of all positive integers m such that there ,h5 define exist hl' .., hn in ~ such that for each j the functions hl' an analytic polyhedron Q; with frame L; with 8 J C Qj C Pi> where 1l=1l(]) =max{y(j),m}. Clearly the set T is non-void, since by hypothesis nE T. Clearly also the lemma will be proved if it is shown that 1 E T. To this end, assume 1 ¢ T. Let m be the least integer in T, and choose hl' .., h" as above. We will prove that m - 1 E T and thus obtain a contradiction. The polyhedra Q; with 'Y (j) < m are unreduced, since dim Q; -< dim P J = 'Y (j). Therefore, by applying Lemma 4, we may assume that the polyhedra Q; with 'Y (j) < m are prepared. Let r> ] be such that I hi (p) I < 1 -< i -< 'Y (j), p E 8j, for all values of j. We define

r"

h'm-l= (rhm)N- (rhl)N, h'm = (rh.)N, h'm+l = (rhm+1)N - (rh.)N

h'" =

(rh,,)N -

(rh1)N.

If N is sufficiently large it follows from Theorem 2 that h /1 , · • . , h' fl defi ne an analytic polyhedron Q'; with frame L J such that 8 J C Q'; C Q; C Pi> where .B = max{y(j), m - I } . This shows that m - 1 E T. From this contradiction we conclude that 1 E T, as was to be proved. Definition 6.

A partially analytic space K is called weakly holomor-

2

173

226

ERRETT BISHOP.

phically convex if it is the union of its structure manifolds and if for each compact 8 C K each component of the set S

=

{p: p E K, [ f (p) [ < max q E 8

[

f (q) [, all f in m}

is compact. Definition 7. A continuous mapping of a topological space T, into a topological space T 2 is almost proper if each component of -1 (8) is compact for each compact subset 8 of T 2 • THEOREM 4. Let {Lj} be a finite family of closed central subsets of a weakly holomorphically convex partially analytic space K. Let the dimensions of m on the structure manifolds of K be bounded and have the maximum value n. Let y = y (j) be the maximum of the dimensions of 9f on the structure manifolds of K which are subsets of L j • Then the set of elements f = (f,,' . " fn) in n9f such thai for each j the mapping

fi:

p~

(f,(P),' . ',f'Y(p»

of L} into G'Y is almost proper is dense in nm. Proof. It is permissible to assume that L, = K. Let {8d be a sequence of compact sets whose union is K with 8, C interior 8 i +1 • Let Ti be an open set containing 8 i whose closure is compact and whose boundary does not intersect Si. For each point P in bdry T i , there exists h in W with

[ h(p) [ > 1

> maxqEs,

[h(q) [.

By the compactness of bdry T i , there therefore exist a finite number h" , hm of functions in W such that [hk(q) [ < 1 for q in 8 t and 1 1 for at least one k if P E bdry T i . If we let Q be the union of those components of

Qo={P: pEK,[hi(p)[
be given functions in W.

We show that if the complex constants {ail and the

174

PARTIALLY ANAL~TIC SPACES.

positive integers {til are chosen properly then the functions", defined by the series

h=

. a.(gi

~

+ Ak,

k ) tt

i;;:l

h

have the required properties and that To this end, write m

fll'k

~

=

a, (gik) t,

i=l

1 <: k<:n,

is arbitrarily close to Ak in

+ A/,.,

~.

1 <: 7c <: n, 1 <: m

<

00 •

Let mo be an arbitrary positive integer. Choose a,,' . " am. to be zero and t,,' . " tm~ to be 1. We now give the rules for choosing am and t m, assuming that the previous ai and ti have been chosen. Choose

I am I > max{ I

f,"-l k

(p)

I:

1 <: 7c <: y(j), p E bdry Qm;, all j}

+ m + 1.

Then choose t,n so large that

This can be done since for s

<m

we have

so that I gmk(p) I < 1 for p E Q'i and 7c <: y(j). The sequences {am} and {tm} having been chosen in this way, we show first that the series defining the funC'tions h converge uniformly on compact subsets of K, so that fk is welldefinerl and belongs to ~L It is clearly sufficient to consider the compact sets Qn:J S,. Now I am(gml.-(p)) 1m 1 <: 2- m for m > 8 and p in Q81' so that the series for h converge., uniformly on Q81' Therefore h E~. To see that the mapping fi is almost proper, consider a compact set [{ C C1'. sufficient to assume that II

=

{

(Zl'

" , z'Y) : I z, I <: <x, 1 <: i <: y}

for some <x> O. Let r be all)' component of (fi)-l(H). Choose i such that Qii is non-void. Choose a positive integer m larger than <x, i, and mo. Then I g"'k(P) 1= 1 for some le, 1 <: 7c <: Y (j), if P E bdry Qmi' Therefore

r n

1h(p) 1= 1f m -

1

k(P)

+

.

~ a,.(grk(p) tr r=m+l

+ am (gml.-(p» 1 1m

~

>[am l-lf m- 1 k(P)I-

~

la r (gr k (p») tr l>m+l-

.

~ 2- r >m>a. r=m+l

r=m+l

Thus we see that (fi) -1 (H) does not intersect bdry Qmi' We therefore have r C Qmj. Thus r is a closed subset of the compact set Qmi' Therefore r is

175

JHut.!'.:'!"!' Hll:!liU.I:'.

compact. It follows that fi is almost proper, as was to be proved. It is also clear that fk can be made arbitrarily close to Ak by taking mo sufficiently large.

4. Special analytic polyhedra and integration. In order to study the local properties of an analytic ployhedron, it is helpful to impose certain restrictions. Definition 8. A reduced analytic polyhedron P in a partially analytic space K with frame L defined by functions fl' . " fn in ~r will be called special if the level sets of fl, . , fn on P are countable and if the set A of all a such that M", C L, .M", intersects P, and dimM",=n has the property that ( U M",) n P is dense in P and that M", n P is open in L for each a in A, 'lEA

It is clear that each component of a special analytic polyhedron is a special analytic ployhedron.

Definition 9. If X is a topological space we define vY, called the k-fold unordered product of X with itself, to be the set of all unordered Ie-tuples of elements of X. '1'he natural map of the k-fold Cartesian product kX of X with itself into",X is denoted by Ox, or Okx if k is not clear from the context. A set in kX is defined to be open if and only if its inverse image under the map Ox is an open set in kX. It is clear that kX is Hausdorff if X is Hausdorff and compact if X is compact. THEOREM 5. Let P be a special analytic polyhedron with frame L in a partially analytic space K defined by the functions f 1," , f n in W. Let En be the subset of On consisting of points whose coordinates are less than 1 in absolute value, and let f be the map

',f,,(p» of P into En. For each a in A let M"'l be the subset of M", consisting of all points in Ma n P at which (fl, . ,fn) is a coordinate set of functions. Then the set M1 = U MIa is dense in P and H = f(P-1I-f 1 ) is closed and nowhere a

dense in En. There exists an integer A > 1, called the multiplicity of P, such that for each z in En-H there exist exactly A distinct points p in M1 with f (p) ~ z. Th e map w from En - H into "P which takes z into the unordered A-tuple of such points can be uniquely extended to a continuous map w of En into "P. The set P has only a finite number of components. Proof.

If a E A the set M 2 a = (M", n P) -MIa is a locally analytic

176

.I:'Al1TIALL:r

A.N AL:r IlL

!!.I:'AU ..... !!.

subset of Ma , since locally it is given by the vanishing of a certain Jacobian. Also M2a=FM(JI.nP because 1I1 2 (J1.=M a nP would imply that locally one of the functions f" ,fn depends on the others, contradicting the fact that the level sets of f = (f,,' " fn) on MOl n P are countable. Thus M 2 a is the union of a finite family {N"'./l} of analytic submanifolds of M a , with dim NOl./l < dim J[ a = n. Since 111, Of. is clearly dense in M and U 111 is dense (JI.

(JI.

(JI.EA

in P, }'1, is dense in P. Now each NOl./l is a countable union of closed sets, as is each 11101 n P, (X ¢ A. Thus f(NOf../l)' 2 E A, and f(M", n P), (X ¢ A arp countable unions of closed sets of topological dimensions < 2n - 2, since the analytic image of a closed set of dimensions < 2n -2 is of dimension < 2n - 2. I t follows that real dim H < 2n - 2. Therefore En - H is connedcd (see r'l] for these theorems) . Since f (.11.) is of real dimension 2n, f( .~[ 1) - II is nOll-void. For each z in En-II let l(z) be the number of points p in 111, with f(p) =Z If l(z) were not iinitt', the set of such points p would have a limit point po in P. Actually Po is in P, since f(po) E En implies by Lemma 3 that po is not in bdry P. If po E JIa for some (X E A, the fact that ."'fla n P i~ open in P implies that the level set of f on Jla containing po also contains the points p which are sufficiently near to Po. Thus this level set has po as a limit point and is therefore not O-cl.imensional; it is therefore uncountable, contradicting the fact that P is a special analytic polyhedron. Thus po is not in JI, C U JI a. Also po is not in P - Jl, SlIlce z is not in H. Thus po <",A

is not 1Il P at all. This contradiction shows that I (z) is finite. positn'e lIlteger y let TI'={z: zEEn-11,I(z) >y}.

For each

If Zo E T 1', there are distinct points P" " PI' in .Il, with f(Pi) = Zoo Choose 2j in A with Pi E .~I,"", For each i, 1 y. Thus TI' is an open set in En-H. To see that T I' is closed in En - H, consi der Zo in T I' n (En - H) . Let {Zj} be a sequence of points in TI' converging to zoo Let p'j, " Pl'i be distinct points in M, with f(pij) = Zi' We may assume that for each i, 1
In

177

ERRETT BISHOP.

230

Thus f is not one-to-one on any neighborhood of pi, contradicting the fact that pI E MI. Thus the points pi, 1 <: i <: 'I, are distinct. Therefore Z E T 'Y. Thus T'Y is both open and closed in En - H. Since En - H is connected it follows that either T'Y is void or T'Y = En - H. Now T'Y is certainly void for some integer y; take 'I larger than I (z) for some z in En - H. Let'\ be the greatest integer such that T" is not void. It is clear that l(z) >,\ for all z, since Tx=En-H. If l(z) >,\ for some z, then T"+1 is non-void, contradicting the choice of,\. Therefore I (z) =,\ for all z in En - H. This integer ,\ is called the multiplicity of the analytic polyhedron P. To see that H is closed in En, for each r < 1 let

Now j}[1 is open in P since each MIa is open in P. Therefore f(P r -M1 ) is closed in En. But

f (P r -

M 1)

=

f (P -

M I)

n {( ZI'··

,

Thus P r -."'1[1 is compact.

Zn) E En: I Zi I < r, 1 <: i <: n}.

It follows that H = f (P - M d is closed in En. It remains to show that w is continuous on En - H and to extend w continuously to En. The extension of w will perforce be unique because En-H is dense in En. To see that w is continuous on En-H, let {Zi} be a sequence of points in En-H converging to a point Zo in En-H. Let W(Zi) be the unordered A-tuple {p\,. ., pi.,.} in "P and let w(z) be {P1, .. , p,,}. If w is not continuous on En-H, the sequence {zd and the point Zo can be chosen so that W(Zi) converges to a point JL#w(zo) in "P. As above we see that JL consists of A distinct points P in lrf, with f (p) = Z00 Therefore p. = W (zo), a contradiction, proving that W is continuous on En - II. To show that w can be extended continuously to En it is enough, since "P is metnzable, to show that w has a continuous extension to (En-H) U {zo} for each Zo in H. To do this, let P1, .. , Pk be all points in P with f (Pi) = zo, so that actually Pi E P, 1 <: i <: le. Let U 1, • . , U" be dlsjoint neighborhoods of P" ,p" respectively. Assume that there exists a sequence {z;} of points converging to Zo such that for each i there exists a pomt qi in P - U 1 _ .. -Uk with f(gi) =Zi. Then any limit point q of the sequence {gd has the property that f( q) = Zo and q E P - U 1 . Uk. This contradiction shows that the seqeunce {Zi} does not exist. Therefore f-1(V) nP

c

U 1 U·

U Uk

if V is a small enough neighborhood of Z00 Since V is open in En and real dim H <: 2n - 2, the set V - H is connected. For each Z in V - H let Il i (z),

178

PARTIALLY ANALYTIC SPACES.

1 <: i <: 7c, be the number of elements in the set w(z) n U~ = t-l (z) n Ui . Since w is continuous on En-H, 8i is continuous. Since V - H is connected, 8i is therefore constant on V-H. Clearly ~8i=.\. Define w(zo) E ~P to consist of the points Pl, .., Pk with the respective multiplicities 81 , . . . , 8k· This choice is clearly independent of the neighborhood V, and the extension of w so defined to (En-H) U {Zo} is clearly continuous at zoo To see finally that P has a finite number of components, note that each component of P is itself a special analytic polyhedron, and therefore by the analysis just given contains a point p with f(p) = o. Since by the analysis just given P contains only A such points p, it follows that actually P has at most A components. LEMMA 5. Let P be a special analytic polyhedron in a partially analytic space K defined by functions f 1,· . ., f.... Let s be a function on O~ which is a symmetric polynomial in the variables (Zl,· .. , z~) and let s be the unique continuous function on ~O (where 0 is the complex plane) such that so Ox = s. Let g be a continuous function on P. Let ~g be the continuous function on >.P which ta7ces the unordered A-tuple {Pl'· . ., p~..} into the point {g (P1),· .. , g (p~)} in >.0. Let 9 = s 0 ~g 0 w. Then 9 is a continuous function on En which is analytic on En if g is analytic on M", n P for each IX in A.

Proof. Since g is continuous >.g is continuous. Therefore 9 = s 0 >.g 0 w is continuous. Assume that g is analytic on each Ma n P for IX E A. Consider any point z in En-H. There exists a neighborhood U of z such that r(U) n P is the union of disjoint neighborhoods U 1 , · . , U~ of the points of w(p). We may assume that each Ui lies in some M"" IXE A, and that U; is mapped homeomorphically onto U by f. This makes it clear that 9 is analytic at z. Thus 9 is analytic on En-H. To show that 9 is analytic on ~ we need only show that 9 is analytic on all one-dimensional coordinate planes: As typical of such coordinate planes, consider the planes

,cn-d }, where c is a point of En-1. Let G consist of all c in En-1 such that H n Lc is countable. Since 9 is continuous on En and analytic on En - H, 9 is analytic on Lc for each c in G. Clearly, then, 9 is analytic on Lc for all c in G. Thus we need only show that G = En-1. To do this we shall show that En-l _ G is of first category in En-1. Let {Ud be a countable basis for the open sets in E1. Let {J6} be the family consisting of the manifolds M", n P, IX ¢ A, and N"'{J, IX E A, where the N"'p are defined as in Theorem 5. Thus He U f(J6). For each 8 let {Gn'Y} II

179

232

ERRETT BISHOP.

be a countable family of open sets in J o which have compact closures in J,l and which cover J o. Then

For each 8, each '1, and each positive integer le let

FO,),k= {c: cE En-t, {c} X Uk C f(G o')')}. The set FO'Yk is closed because Uk and f (Go'Y) are closed. Also F{hk has no interior points, since otherwise f (Go'Y) would have an interior point in En, contradicting the fact that real dim f (Gli'Y) < 2n - 2. It follows that Y = U FO'Yk is of first category in En-l. To complete the o,')',k proof we show that En-l - G C Y. Consider c E En-l_ G. There exists Go'Y such that f (Go')') n Lc is uncountable. Thus tn assumes an uncountable number of distinct values on the analytic subset

of Go'Y' It follows that f.,(8) contains some open set in El, so that rIle C fn(8) for some le. Thus {c} X Uk C I (Go'Y)' so that c E FO'Yk' Thus En-l - G C Y, as was to be proved. Definition 10. Let K be a partially analytic space and P a speC'ial analytic polyhedron in P defined by the functions fl" . " In. For 0 < r < 1 let P r = P n {p: ! ti(P)! r, 1
<

Let B r =Pr n{p:!fi(p)!=r,l
Let

. , zn) : ! Zi ! < T, 1 < i < n} and V r = f (Br) C bdry Enr- Let s be the function on 0)0. defined by 2)0.. If 9 is a continuous function on P, define s (2 1 ' . • ,2)0.) = Zl !J as in Lemma 5, so that

+ . .+

where w (2) = {PI, . . ,P)o.}. Thus!J is continuous and analytic on each Mcx n P, IX EA. Now the linear functional

9~ IS

r

..JVr

!J (z) dz l

'

.

.

IS

analytic if 9

IS

dZn

a continuous linear functional in the uniform norm on the space of all

180

PARTIALLY ANALYTIC SPACES.

continuous functions 9 on By. It follows by the Riesz representation theorem for such linear functionals that there exists a unique finite complex-valued Borel measure p. on BT such that

J'

g(p)dp.(p)

=

[

aJv r

Br

for all continuous functions 9 on By.

g(z)dz, · .. dz",

The measure dp. is written df,

dfn.

We now give an integral formula for special analytic polyhedra which, while trivial, is fundamental. THEOREM

6.

n

function on Ma

(271"i)-"

r

)B r

Under the conditions of Definition 10, if 9 is an analytic P for each ex E A and if q E P r, then

g(P)[f,(P) -f,(q)]-'

.. [f.,(p) -fn(q)J-1df,(P)·

=g(f(q)). Proof. Write h=gLf,-f,(q)]-l . .. [f,,-fn(q)]-l. The proof depends on computing h, which is defined on Vr because h is continuous on B r. Consider z in Vr and let IlJ (z) = {PI'· .. , p~}. Since Ii (pj) = Zi for all j, where z = (ZI'· . , zn), it follows that

We therefore have

(2.".i)-n

r h(p)df,(q)· .. dfn(P)

)B r

=(271"i)-n

f

.ll(z)[z,-f,(q)]-I .. [zn-f",(q)J-'dz, · . ·dz". Vr

Since (f 1 (q), .. , f n (q)) E En. it follows from the n-times iteration of the Cauchy integral formula that this quantity is just fj (f (q) ), as was to be "\ proved. The integral formula of Theorem 6 is apparently only good for representing functions which have equal values at points of P mapping into the same point of En. That this is not the case is shown by the following theorem. THEOREM 7. With the notation of Lemma 5 and Theorem 6, let w be any analytic function on P (continuous on P and analytic on Man P for each tZ E A). Let w be the function on P X P defined by

w(p, q)

=

[W(P2) -w(q) ] ...

[w(p~)

w

-w(q)],

where w(f(p) ={P=Pl'· . ·,PA}. Then is continuous on PXP and analytic in q for each value of P and analytic in p for each value of q. If w

181

J<;HH.I!:'l"l' BU!HOl'.

is in W, then W(p, g) is in W when considered as a function of g for each fixed p in P. For each function 9 analytic on P and each g in P r (2wi)-n

r

J B•

g(p)w(p, g)[Mp) - fl(g)] - 1 . .

[f,,(p) - f .. (q)J-ldMp)· .. df,,(p)

= w(g)g(g),

where the function 1V(q) =w(q,q) is analytic on P. Proof.

Clearly A-I

w(p, q)

=

~

-t-

1=0

a.(p) [w(q) J',

where ai (p) is the A - i -l-th elementary symmetric function of

{W(P2),' . " w(PA)}' Since P ~ {P2' . . ,p,.} is a continuous map of Pinto A-IP, it follows that the a. are continuous functions on P. Thus ill is continuous on P X P. Clearly ill is analytic in q for each fixed P and (p, q) E W for each fixed p if w E W. To show that w is analytic in p, it is enough to show that the elementary symmetric functions of {w (P2),' . " w (PA)} are analytic as functions of p. It is a consequence of Lemma 5 that the elementary symmetric functions of {w (PI),' . " w (PA)} are analytic as functions of p. Now each elementary symmetric function of {w (p2),· . ',10 (PA)} is a linear combination of an elementary symmetric function of {w (PI),' . " W (PA)} and the product of w (PI) = W (p) by an elementary symmetric function of {w (P2),' .. , w (PA)} of lower order. ThuB it follows by induction that the a., are analytic functions of p. By Theorem 6,

w

(271'i)-n

J

g(p)w(p, q)[fI(p)-fl(q)J-l

. [fn(P)-f.,(q)J-ldfl(P)··

dfn(P)

Br

=

li(f(q»,

where h is the function on P defined by

h(p) =g(p)w(p,q). By Definition 10,

li(z)

=

h(pd

+. .+ h(PA) =g(Pl)W(PI,q)

Thus to compute li(f(q»

we replace {PI"

+ .

+ . ·+g(PA)w(PX,q). ',px} by {ql,

. ',qx}, obtaining

li(f(q» =g(ql)W(ql,q) '+g(qx)w(qX,q) =g(q)iiJ(q,q) =g(q)w(q)

182

l'AN'.l'.lALLX

Al'oIALX .lll'

!:)J:'A"..,.<>.

since q, = q and iii (qi, q) = 0 for 2 <: i <: A. Since h is analytic on P, n is analytic on En, so that h 0 f = gw is analytic on P. Taking g = 1 it follows that iii is analytic on P.

5.

Mappings into higher dimensional spaces.

The following lemma

continues the study of special analytic polyhedra. LEMMA 6. Let P be a special analytic polyhedron in a partially analytic space K defined by functions f,,' . ,fn in ~. Let

Pr

P n {p: I fi (p)

=

I < r, 1 <: i <: n}

< <

for 0 r 1. Let ~r be the uniform closure of ~ on Pr , so that ~r is a Banach algebra in the uniform norm. If a is any bounded mUltiplicative linear functional on ~" then there exists a point po in P r such that .. a(g) =g(po), all g in ~rIf in addition 1 E ~r and ~r separates points of P" there exists a function cp in ~r which has the value 1 on any open and closed set in P r and the value 0 on its complement. Proof.

Since a is bounded, Zo

=

I a(fi) I <: r

for 1 <: i <: n.

Thus

(a(f,),' . " a(fn)) E En r.

We wish to show that there exists Po in Pr with f(po) =Zo and a(g) =g(po) for all g in~. Assume that this is not the case. Then there exists gi in ~ with gi(pOi) =;1= a (gi) for each pOi in ",(zo) = {pO,," ,p\}. There therefore exists a linear combination go of g" ", gA such that go (pOi) =;1= a(go), 1 <: i <: A. :For each z in En and each i with 1 <: i <: A write

\ where {PI, " , PAl = " , (z) and" here j,,' ,ji runs over all combinations ' of the integers 1, ,A taken i at a time. By Lemma 5, 8; is an analytic function on En. If we let Si = 0; 0 f, then clearly A _

(gO)A

+.L Oi(gO)A-i =

0

i=l

identically on P, since A

[go(p)]A

+ .L 8 (p)[go(p) ]A-i i

A

=

i=l

II

(go (p) -

gO(Pi»

=

0

i=l

for all p in P. Since 8i is analytic on En, it can be uniformly approximated on f (P r) = Enr by polynomials F in z,' " , Zn. Therefore 8i can be uniformly

183

236

ERRETT BISHOP.

approxImated on P r by polynomials F in f1" is continuous, we have

. " fn.

Hence

S; E ~r'

Since a

0'(8;) =O'[FU1,' ',fn)] =limF(
=

[O'(go) J"

A

+ ~ [8 (zo)] [a (go) F-i. i

i=l

Since ill (zo), . , il" (Zo) are the elementary s} mmetric functions of go (pol), ,go(p\), this implies that a(go) = go (pOi) for some i. This contradictlOn shows that a point po III Fr with the desired property exi~ts. Assume now that 1 E ~r. According to what we have just proved, P r is the ma.:.imal ideal space of the Banach algebra ~(,. Therefore according to a theorem 01 Silo, -Calderon-Arens, (see L1]), the function 1> exists. TIlEOHE~I

8. Let J{ be a lI;eakly holonlO1 jJh~cally con~ex partially analytic the 1n(uilnum of the dimensions of lI;ho~e ~tructure manifolds i~ n Let each n-dimenstonal structure manifold be open and let their union .l/ o be denle in K. Let ~r contain the unit fundion and separate points of K. For each [,ub:,et r of ~r let each compact component Yo of each level set Y of r onK con~i~t of Just one point, which i~isolated in Y. Let f= (f1' . ',fn), fi E ~r, be an almo~t proper map of K into Let {8d be a sequence of compact :,ubsets of K, with U 81. = K and 8/" C interior 8/"+1, each of wI! ich has a finite number of component~. Let 2;, 1 < i < n, be po~itice functions of a real rariable x, with limHOC2,(r) =00. F01 r > let ~}Jace,

en.

°

D,={z'

zEC",

[z,[
For each positive intege! 7c let HI, be the union of tlwoSe componentl of j-1(Ih) which intersect 8/... lI'rite

for 7c > 1 Write Go = H l ' For each non-negatil e integer 7c let A.k be any function in ~r and Ek < 1 a positire 1 eal number. Then H k is compact and there e.rists fnn in ~r wch that 7c=O,1,2,

The map ( #)

1=

f X fn+1 of ]{ into e,,+1 inf

pE

G.

i~

proper if

[ A.h (p) [ ~ 00 as k ~ 00 .

184

:.\is'l

PARTIALLY ANALYTIC SPACES.

If S is a countable subset of M o at each point of which rank f < nand rank m: = n, f"+1 can in addition be chosen to have the property (*) rank (/,,' " f,H1) = 1 rank f at each point of S. The function f"+1 can also be chosen to have the property (:) of taking distinct values at the points of a given countable subset of K.

+

Proof. Let Q be any component of j-1 (D j ). Since Q is a subset of some component of j-1 (Dj), Q is compact. Thus Q is an analytic polyhedron with frame K defined by multiples of the functions f,,' . " tn. Since each level set of on Q is closed in Q, and therefore compact, it follows from the hypotheses of the theorem that such level sets are finite. Therefore Q is a special analytic polyhedron. For k given, choose j so large that f(SI.) CD;. Thus each component of Sk is contained in some Q, so that a finite number of such Q cover Sk. The union Q of these Q is an analytic polyhedron. Clearly IIk C Q, so that II.,. is compact. "Ve conbtruct by induetion functions go, g" . in il( such that for all k=>l I gdp) I < £k2-(k+l) for p in Ih and for all k => 0

t

For this purpose there is no loss of generality in assuming that £k+1 < £k for all k. Choose go = '\0, and a'>sume that go, . ,gk-l have been chosen with the abovc properties. Let r be a real number with k <:: r < k 1. From the compactness of IIk+1 it follows that

+

pr=II k +1 n j-l(D r ) is an analytic pol}hedron with frame K, defined by multiples of the functionSx ,tn. It follows, as above for the analytic polyhedron Q, that pr is a, special analytic polyhedron. By Theorem 0, pr has only a finite number or component" .•\.1;;0 by Theorem 5, applied to pr, it is eabY to see if k < r < k 1 that Hk+1 n j-1(iJ",) = {p: p E pr, I /i(p) I <:: lXi(k), 1 <:: i <:: n}

t,,··

+

is the closure of the set pk = j-' (D k ) n H 1.+1. number or components, the set

GkU II k =

t-

1

Since pk has only a finite

(Dk) nH/"+1

has only a finite number of components. By the definition of II k , Hk is the union of certain components of Gk U H k; thererore Gk is the union of the remaining components of Gk U H k. Thus Gk and H k are disjoint closed sets.

185

238

ERRETT BISHOP.

+

Since Gk U Hk is the intersection of the nested compact sets pr, k < r < k 1, there therefore exists a value of r in the range such that the finite number of components of G k U Hk all belong to separate components of pro

Then

Gk belongs to an open and closed set W in pr and HI< belongs to p~ - W. It follows from Lemma 6 that there exists a function T in m which is arbitrarily near to 0 on Hk and to 1 on Gk • Write

gk = -T[go

+ ... + gk-l-Ak].

It is then clear that gk has the desired properties if 1 on Hk and Gk respectively. Define

T

is near enough to 0 and

'l'his series converges uniformly on HI. "ince for j >- k

N ow if ~ is any com pact subset of K, there exists j wi th ~ C 8 j. If k > j is chosen so large that f(8 j ) C D k , it follows that ~ C 8 j C H k • Therefore the series for fn+l is well-defined and lies in m. For p in Gk we have

I Adp) -fn+l(P) I < I Ak(P) -

k

~ gj(p) ;=0

< t~k + ;=k+l ~ ,,,2-

j

<

I+

""

~

I g;(p) I

;=k+l

fIe·

Thus the function f1l+1 satisfies the required inequalities. An obvious modification of the proof yields a function f7Hl which in addition has properties (*) and I t remains to show that f X f n+1 = 1 is proper if the condItion (#) is satisfied. Consider p in 1-1 (F,,), where Fk = D" X {z: z E Ct, I z I < k}. Let j be the smallest integer >- k such that p E H j . If j> k, then p is in Gj - 1 • Therefore

un.

It follows from condition (#) that this can happen for only a finite number of values of i. Therefore 1-1 (F,,) is contained in the union of H k and such H j . Since H j and HI.. are compact, this implies that 1 is proper, as was to be proved. The hypothesis in the last theorem, and also in the following theorem, that compact components Vo of V consist of just one point" which is isolated in V, is in fact unnecessary if K is a complex space. This can be seen from

186

PARTIALLY ANALYTIC SPACES.

239

results of Stein [l1J, which states that if f is an analytic mapping of a complex space K having the property that the components of the level sets of f on K are compact then it is possible to introduce an analytic structure into the class k of such components in such a way that the natural map of K into k is analytic. However, we will not pursue this question here. THEOREM

9. Let K be a weakly holomorphically convex partially analytic

space, the maximum of the dimensions of whose structure manifolds is n. Let each n-dimensional structure manifold be open and let their union Mo be dense in K. Let ~ contain the unit function and separate points of K. For each subset r of ~ let each compact component Va of each level set of V on K consist of just one point, which is isolated in V. Let K be the union of a sequence Sk of compact sets, each having a finite number of components, with Sk C interior Skn. Let J.I be a particular n-dimensional structure manifold of K, such that ~ has rank n at each point of M. Let G be the set of functions (fl'· . . , f2Ml) in 2n+l~ such that

(i) (ii)

f

=

(fl'·

., fn) is almost proper,

1= (fl' . . ,ffl+l)

is proper,

(iii)

the rank of (fl,· .. , f2") at each point of M is n,

(iv)

the functions (fl,· .. , f211+l) separate points of M.

Then G is dense in

2n+l~.

Proof. By Theorem 4, the elements f in n~ which are almost proper are dense Fix such an element f. If V is a level set of f, each component Va of V is compact because f is almost proper. It follows from the hypothes~s of the theorem that Va consists of one point, which is isolated in V. Therefore V is countable, so that all level sets of f are countable. For 1 < i < n let A at consist of all points in M at which rank f < i. Each of these sets is analytic in M and A C A °i+l. Clearly

0.

dim A 0"

< dim M =

n,

since f, whose level sets are countable, must have rank n at some point of lrI. We now show inductively, starting with i = n and working backward, that dim A °i < i. Assume therefore that dim A °in < i 1. Let the analytic set A °i+l be written as the union of submanifolds {Ni+l'Y} of JI, and let {Ni+lll} consist of those Ni+l'Y of dimension i. To show dim A < i, it will be sufficient to show that A 0, does not contain any of the Ni+lll . For if this is the case the inclusion A io C A °in implies that A °i is contained in the union of certain

+

0.

187

240

ERRETT BISHOP.

proper analytic subsets of the Ni+lf3 and of the Ni+1'Y of dimension less than i. so that dim A °i < i. To see that Ni+1f3 q: A °i, note that t has rank i at some point of Ni+1f3, since its level sets are countable and dim Ni+1f3 = i. This completes the proof that dim A °i < i. Let S be a countable dense set in M. For each Nif3, 1 < i < n, choose a point p i f3 in Nif3 at which the rank of t is at least i - I . Let II be the f'et of all tn+1 in ~r such that 1 = t X tn+1 is proper, sueh that 1 has rank i on N at the point pif3 for each i and (3, and such that tn+1 takes distinct ,alues at the points of the set j-1(S). By Theorem 8, there exists tn+1 in H such that

I tn+1(p) -A(p) 1<

£

for all p in Go = H1 where II1 is defined as in Theorem 8, where A is any function in ~r, and where £ > O. Since by an appropriate choice of the sequrnce {Sd and the functions IXi of Theorem 8 the set H1 can be made to contain any compact subset of K, it follows that H is dense in W. We now choose the functions tn+2, . ,t2n+1 by induction. Assuming that t n+1, . . ,tn+'" haye been chosen, let A ki' 1 < i < n, consist of all points in 1',1 at which t1' . ,tn+'" have rank less than i. Clearly each A\ is an analytic subset of 1',[. Let B~ consist of all points in J[ -A"'n which are identified with some other point of ]1,[ by the functions t1' ,tn+"" Since 1 is proper, the points in K which are identified with a given point pin Bk by the functions t1)' . " tn+k form a compact set L. By the hypothesis of the theorem it follows that L is finite. Let p = P1," ,Pi be the points of J[ n L. Let £ be a small positive number and write

gi =

£-1

(fi - ti (p) ),

1
Since t is almost proper g = (gl," ,gn) IS almost proper. Therefore each component of g-l (En) is an analytic polyhedron defined by g with frame K. Let Pm be that component of g-l (En) which contains pm, 1 < m < j. For a fixed m the intersection 1m of the Pm, taken for all values of £, is clearly a subset of the countable discrete set j-1 (f (p) ). Since 1m is also the intersection of the Pm, which are nested connected compact sets, it follows that 1m is connected. Hence 1m = {pm}. It follows that £ can be chosen so small that P 1, " Pi are disjoint subsets of M. We can also choose £ so small that (f1," ,tn+k) is one-to-one on P 1, since p,=pEM-Akn. Write 1
n+ k,

h = (h " . " hn+k)' It follows by an argument similar to the one just useel that there exists a neighborhood U C P 1 of p such that

188

PARTIALLY ANALYTIC SPACES.

Now h map" Pm properly into En X OJ.., since P'" is an analytic polyhedron defined by g= (hl' ·,hn). Let W",=h(P m). Since h is proper on Pm, it follows from a standard result ([9J, Satz 23) that Wm is an analytic subset U Wi is analytic. ThU'l Un h-' (ll') of En X OJ.. Thus W = lV, U = U n B .. is an analytic subset of U. Since 1) was any point in B ,,, it follows that B " is locally analyti(·. We choose the functions t n+2," ,t2n+l by induction to be arbitTarily close to given functions in ~r on a given compact subset of K and to have the properties uimA\ < ma"\: {O,i-7c}, uim B ..

< n-

7c

+ 1.

t

'We first check that thc function n+l already cho<;en has these propertics. Since 1 has rank i on !vi at p i f3' it follows that 'P i intert'ects Ni/3 in a proper subset, which therefore has dimension at most i - 2. Since A \ c A °i, it then follows as above that dim A \ < i - 2. Let}J be any point in S - A. In. By the choice of tMl we see that ]J ¢ B l • Thus Bl is a proper analytic subset of the manifold M -Aln; it follows that dimB l < n. "\.ssnme now that t" . ,t/Hk have been chosen so that the above dimension inequalities are true. Write A\=

U Nf3 il" /3

a countable union of submanifolds of M.

Write

B .. =UJ/3, Il

a countable umon of submanifolds of J1. N ow if i -lc > 0, consider Nil." of dimension i - lc - 1, the maximum possible dimension by the induction hypothesis. Since Nf3 i .. q::Aki _ 1, (f,," ,tn+/") have rank at least i - I at some point pf3i l.. of Nf3ih • Thus tn+/"+l can be chosen to be arbitrarily close to a given function in ~ on a given compact subset of K anu to have the property that tl' . ,tll+k+1 has rank i at pf3ik for each pllil'" It follows then that dim A /.,+\

< max {O, i-7c -I}.

The function t n+I..+1 can be chosen to have the additional property that each J /3 contains some point p not identified by n+/"+l with any of the points which f" . ,t,,+l.. identify with p. Thus P is not in B k+1, so that Bk+1 intersects J/3 in a set of dimension at most (dim J/3) - 1 < (dim B .. ) - 1 < n-7c. Since

t

it follows that 3

189

242

ERRETT BISHOP.

dimBk+l < max{dimAk", dimB,,+!

n J p } < n-7c =n- (7c + 1)

+ 1.

Thus the functions fl" . " f"Hl with the desired properties are dense in 2"+1~r. Now (iii) follows from the fact that dim An" so that A"" is void. void and that dimB,,+!

< max {O,n-n} =0,

Finally, (iv) follows from the fact that A "+1.. cAn..

< n - (n + 1) + 1 =

IS

0, so that Bn+l is void.

This completes the proof. UNIVERSITY OF CALIFORNIA, BERKELEY.

REFERENCES.

[1] R. Arens and A. P. Calderon, "Analytic functions of several Banach algebra

elements," Annals of Mathematics, vol. 62 (1955), pp. 204-216. [2] H. Behnke and K. Stein, "Modification komplexer Mannigfaltigkeiten und Riemannscher Gebiete," Mathematische Annalen, vol. 124 (1955), pp. 1-16. [3] S. Bergmann, "tiber eine in gewissen Bereichen mit Maximumflache giIltige Integral darstellung der Funktionen zweier komplexer Variabler," Mathematische Zeitschrift, voL 3'9 (1934), pp. 76-94. [4] H. Cartan, Seminaire Ecole Normale Superieure, 1951 1952. [5] - - - , Seminaire Ecole Normale Superieure, 1953-1954. [6] H. Grauert, "Characterzierung der holomorphvolstiindigen komplexen Rii.ume,"

Mathematische Annalen, voL 129 (1955), pp. 233-259. [7] W. Hurewicz and H. Wallman, Dimension theory, Princeton, 1948. [8] R. Remmert, "Sur les espaces analytiques holomorphiquement separables et holo-

[9]

[10] [11] [12] [13]

morphiquement convexes," Comptes Rendus Hebdomadaires des Seances de l'Academie des Sciences, Paris, vol. 243 (1956), pp. 118-121. - - - , "Holomorphe und meromorphe Abbildungen komplexer Raume," Mathematische Annalen, vol. 133 (1957), pp. 328370. H. Rossi, Analytic spaces, U. S. Air Force technical report, Princeton University, 1960_ K. Stein, "Analytische Zerlegungen komplexer Raume," Mathematische Annalen, vol. 132 (1956), pp_ 63-93. - - - and R. Remmert, .. tiber die wesentlichen Singularitaten analytischer Mengen," ibid., vol. 126 (1953), pp. 263 306. A. Weil, .. L'integrale de Cauchy et les fonctions de plusieurs variables," ibid, vol. III (1935), pp_ 178-182.

190

SOME GLOBAL PROBLEMS IN THE THEORY OF FUNCTIONS OF SEVERAL COMPLEX VARIABLES. * By ERRETT BISHOP.'

1. Introduction. In a previous paper [2] the global structure of a complex analytic variety was examined ill some detail The present paper is a sequel containing some application of those results. In order that thi~ paper be ~elf-colltained. the definitions and results of [2] are summarized in SectIOn 2 Consider HOW an analytic !>ubset L of a Stem manifold K. It is a consequence of Theorems A and B of [3] that

I The set L is the set of common zeros of Rome family r of analytic functions on [( II. If L IS a sllbmanIfold of K then every analytic function on Lean be extended to an analytic function on K PrevIOus work on Problems I and II was done by Oka [5], r6], and others. In Section 3 we give new proofs of I and II which are independent of sheaf theory. The proof of I contains an explicit formula for the functions in the class r, and in thiR respeC't gives information which is not available from the sheaf-theoretic proof. Slmilarly the proof of II contains a specific formula for the extension of a function analytic on L to a function analytic on K. This formula makes it clear that II continues to hold tor analytic functions whose values lie in a Banach space, thus answering a question posed by H. Rossi in a conversation with the author. In Section 4 we prove that the first cohomology group of the shea:! of germs of holomorphic functions on a Stein manifold is trivial. Again tb~s result follows from sheaf theory [3]. Previous work was done by Cousin [4], Oka [5], [6], [7J, and others. Since it is the purpose of this paper to proceed independently of sheaf theory, this result is formulated and proved without mention of sheaves or cohomology. As is well-known, the EOlutions to Cousin's first and second problems are consquences of this result. The techniques which we use for solving these problems are all essentially similar. Since all of the problems considered have previously been solved, • Received December 27, 1960. 'This work was supported by the Sloan Foundation and the National Science Foundation

479

191

480

ERRETT BISHOP.

we have not attemptt'd to push the theory to the limit of its generality. It should however be clear to someone familiar with the subject that the

methods employed here are capable of handling the same problems in greater generality. These methods, which seem not to afford so powerful a general approach to these questions as does the theory of sheaves, nevertheless have the advantages of explicitness and simplicity. The gain in explicitness comes from the use of explicit formulas to construct the various analytic functions which arise. These formulas make it clear, for instance, that all of the above mentioned results hold for Banach space valued analytic functions. 2. Preliminaries. In order not to burden the exposition with too much generality (as was done in [2]), we confine our attention to Stein manifolds, merely remarking that there are obvious possibilities for generalizing the results to one type or another of analytic space.

Definition 1. A Stein manifold K of dimension n is an n-dimensional separable complex analytic manifold K having the following properties where A (K) denotes the set of analytic functions on K (a) (b)

For each pair (p, q) of distinct points in K there exists f in ..L (K) with f(p) =l=1(q)· For each p in K there exist 11, ,I.. in A (K) which map some neighborhood of p homeomorphically onto an open set in n-dimensional complex affine space K is holomorphically convex. This means that for each compact subc;et S of K the set

a...

(c)

s=

{p: p E K, I f(p) I <sup

{I I(q) I:

q E S} for all

f

in A (K)}

is compact. We now state, for the special case of a Stein manifold, some of the definitions and results of [2].

Definition 2. Let K be a Stein manifQld and L an analytic subset of K Letf" . ,/k be functions in A(K). Write

Po = {p E L: I f,(p)

I < 1,1 <

i < 7c}.

Let P be an open and closed subset of Po such that P is a compact subset ,fk' of L. Then P is called an analytic polyhedron in L defined by f1" The analytic polyhedron P is called special if L has the dimension 7c at each of its points.

FUNCTIONS OF SEVERAL COMPLEX VARIABLES.

1. If P is an analytic polyhedron in a homogeneous dimen~iol1al analytic subset L of a Stein manifold K, if S is a compact subset of P, and if T is a compact subset of K at each point of which each of the defining functions for P has absolute value les~ than 1, then there e r;ist~ a special analytic polyhedron Q in L with THEOREM

SCQCP ~1lch that each of the f1lnctio11' defining Q has abloltIle value le9s than 1 at each point of T. The function~ defining Q can be cho~en to be polynomials in the f1tnction~ defining P. Since it is eaRY that a homogeneous-dimemlional analytic subset L of it Stein manifold K can be exhausted by an increasing sequence of analytic polyhedra, it follows from Theorem 1 that L can be exhausted by an increasing sequence of special analytic polyhedra. Thus the study of L can be reduced for most purposes to the study of special analytic polyhedra in L. It turn~ out that a special analytic ployhedron P can be studied in detail by mean.;; of the map f of Pinto d-dimensional complex affine "pace Cd, where d = dim Land f is defined by

f(p) = (fl(P),· . ',fd(P» with fl, . , fd the functions which occur in the definition of the &pecial analytic polyhedron P. Before we state the main results about special analytic polyhedra another definition is necessary. Definition 3. Let X be a topological space. Let I,X be the k-fold Cartesian product of X with itself. Define an equivalence relation on kX by taking (Xl, , Xk) ~ (Yl, , y",) to mean that (Yl, , YIc) is a reordering of (Xl, , Xk)' Then "X, the k-fold unordered product of X with itself, is the set of equivalence classes of kX under this relation. A set in k.Y'iz; called open if and only if its inverse image in kX with respect to the natural map 7r: kX -----') leX is open. The image under 7r of the element (Xl' ,J'd in kX will be denoted by {Xl'" , Xk}.

The following three theorems give the fundamental facts about analytic polyhedra. 2. Let P be a special analytic polyhedron in ad-dimensional wbset L of a Stein manifold K, defined by functions fl" ,fd in A (K). Let f be the map THEOREM

, !d(P)) 6

193

482

ERRETT BISHOP.

uf K intu Cd,

SO

that f(P) CEil, where Ed={(Zl,'

,zd):[zi[
Then there exists an integer >.., a continuous map closed nowhere dense subset H of Ed such that (a) (b) (c)

w

from Ed into AP' and a

For all z in Ed the points Pl' ,lh in w (z) = {Ill' mapped b~' f into z. No other points in P are mapped by f into z. For all z in E r l - H the points Pt.

THEOREM 3. n'ith the condition;; uf The07pm 2 let .l be the sub~et of P consisting of (Ill 1 egular point~ }J of ]' wch that the funrtion, fl' ,fd lire a coordinate let of function~ at earh of the }Joint~ Pl' ,px of lJ)(f(p») = {PI, ,PA}' Let 9 be a continuou,1 function on l' which i~ analytir at each point of.l, Then 9 is analytic at each regular point of ]'. If also g(pd =g(P2) whenever PI and P2 are points in P for which f(Pt) =f(P2) and if go is a function on an open subset U of {-I (Ed) such th lLi go (g) = 9 (p) whenever 9 E U, pEP, and f(p) = f(g), then go is analytic on U and thp function h on Ed for 11,hich 9 = h 0 f i,1 analytic on Ed. THEOREM 4 Let T be a compact subset of a 8tein manifold K and [,T (( neighborhood of T such that for each p in U - T there exists f in A (K) with [f(p)1 >sup{lf(g)l: gET}.

Let C(T) be the 1wifol'mly-normerl Banach 'pare 1'((lued function~ on T. Let ~ be the rlo~w'e of Then the spectrum of the Banach algebra ~I is T. homomorphism rj> of ~r into the compleX' number" T with rj> (f) = f(po) for aU f in ~L

of all continuou., rompleJ'A(K) in the spare C(T). That is, to any continuous corresponds a point po in

The proof of Theorem 4 (Lemma 6 of [2]) was for certain special sets T, but by means of Theorem lone can easily approximate T by such sperial sets and thereby obtain Theorem 4 for T. In order to study a d-dimensional analytic· subset L of a Stein manifold K globally it is sometimes necessary to ronsider a map of K into Cd whose restrirtion to L is almost proper. This ronrept is explained in the following n.efinition.

Definition 4 A continuous map f of a topological space X into a topological spare Y is raIled almo~t proper if every component of {-I (8) is compact for every compact subset 8 of Y.

194

FUNCTIONS OF SEVERAL COMPLEX VARIABLES.

4!M

The following theorem is easily proved from the fact that an analytic subset of a Stein manifold can be exhausted by special analytic polyhedra. THEOREM 5. If L is a d-dimensional analytic subset of a Stein manifold K then K admit., an analytic map into Cd whose restriction to L is almost IJroper. The mapping can be chosen to be non-singular on L at each point of any O-dimensional set of regular points of L.

The final result we shall need guarantees the existence of a function whjch will be used as a convergence factor for constructions involving an almost proper map f of an analytic subset D of a Rtein manifold.

f d+l

THEOREM 6 Let f = (fl, ,fa) be an analytic map of a Stein manifold K into Cd. uho,e 1"estriction to the analytic subset L of K is alm05t proper. Let L have dimension d at each point. Let {Sd be an increasing sequence of compact subsei.s of L, each of which ha.s finitely many components, the union of whose interiors i5 L. Let a be a function from the positi'L'e integers to the po~itive real numben such that

SA C F" = {p E L: [ fi(p) [
for all k. Then H h

..'I"

H

com pact. For each Gk

=

po~itiL"e

(H h + 1

in teger 7c write

n F , ) -Hh •

Let GO=Hl Let {g,,} be a sequence of functio71~ w A(K) and {Ed a Mquence of posit1ce Teal numbel~. Then there e.ri~ts fd+l 111 A (K) such that [ fd+l (]I) -

gh (p) [ <

Ek

In addition fd+l Crln be cho~en to l/ace distinct ralup, modulo 27ri at the of any countable ,ubset of K

point~

Since it is not clear that Theorem 6 ab stated follows from the results of [2J, we indicate the modifications necessary in the proof of Theorem 8 of [2J to give Theorem 6. The only point in question is the convergence on K (ann not just on L) of the series

used to define f a+l on p. 238 of [2]. To see that this series can be made to converge on K, let D be any compact subset of K, with D = D. We shall show that for k sufficiently large the g" can be chosen so that [g" (p) [ < 2- k for all p in D. It will be clear that this can actually be achieved for any "equence of 'Ouch compact sets D which exhausts K, from which the uniform

195

484

ERRETT BISHOP.

convergence of the series for fd+l on all compact subsets of K can be obtaineu. To this end it is sufficiently to show that the function T constructed in the proof of Theorem 8 of [2] and used in the construction of gk can be made arbitrarily small on D if 7c is sufficiently large. That is, we wish to find T in W which is arbitrarily close to 0 on Hie U D and arbitrarily close to 1 on Gk. Choose 7c so large that D n L e H k. Thus there exists a neighborhood U in K of T=H k U G k U D such that Hk U G k =Fk n U. Clearly G k is open in the set T. Thus to show that T exists with the desired properties it is sufficient by Theorem 4 above to show that for each p in U - T there exists h in W with I h(p) I> sup{1 h(q) I: q E T}. If P E U n L then pEL - F" so that h can be taken to be one of the functions fl" .. ,fd' Thus we need only consider the case p E U -L. It is well known, and a new proof will be given below (Theorem 7), that there e~ists an analytic function hI on an arbitrary open relatively compart set VeK such that h, (p) = 1 and hI vanishes on V n L. N ow since D = jj there exit\ts h2 in W with h2(P) = 1 and I h,(q)h2(q) 1< 1 for all q in D. Now if V is sufficiently large then by Runge's theorem for several complex variable" (a new proof of which will be given below which does not depend on this theorem), the function h,h2 can be uniformly approximated on T U {p} by functions h in W, so that h will have the desired property. ,{'hi" completes the proof that the desired functIOn T can be found.

3. Analytic sets in Stein manifolds. We wish to proye Theorelllb I and II of the introduction. We first prove somewhat weaker results. I, amI Il, below, which are sufficient for most applications. THEOREM I. Let L be an analytic lJubset of a Stein nUlIlifold K. U be an open subset of K whose closure is compact. Let L hare the dimension d at each of its point~. Then

II' III'

[~pt ,{(!IIC

For each Po in U - L there e:riiit~ an analytic fUIl( lion h on (~, defined by (*) beloUJ, which vani~hes on U n L, with h (Po) =1= 0 If in addition L is a submanifold of K then every analyt'ic function u on L can be extended to an analytic function ii on U, gil'en by (a), (d), and (e) below.

Proof· Since K is a Stein manifold and (j is compact there exists an analytic polyhedron Q in K with (j e Q. Thus Q n L is an analytic polyhedron in L. By Theorem 1 there exist functions fl' ,fa in A (K) defining

196

FUNCTIONS OF SEVERAL COMPLEX VARIABLES.

un L CP

an analytic polyhedron P in L with and 1 <: i <: d. Let f be the map P~

(f,(P),

and

I /i(p) I < 1

485

for P in U

" fa(p»

of U into Ed and let fo be the restriction of f to P. Let A and ~J respectively be the multiplicity and multiplicity function, as defined in 'rheorem 2, of the map fo of P into Ea. Consider Po in U - L and write Zo = f(po). Since K is a Stein manifold there exists w in A(K) such that w(Po) #w(p) for all p in OJ ( Zo ) • , q~} Define the For an arbitrary point q in U let OJ (f (q) ) = {q" fUllction h on U by ~

h(q)

(*)

=ll (w(q)

-W(qi))'

,:::1

'1'hus h is a polynomial in w whose coefficients aj, 0 <: j <: A, are symmetric functions of the q,. Clearly aj is a continuous function on P which is analytic on L at each point of the set J defined in Theorem 3. By Theorem 3 the functions aj are analytic on U. It follows that h is analytic on U. Clearly h(po) #0 and h vanishes on L n UcP. '1'his proves 1" To prove Il" define f" . , "" and P as above. Choose 1"0 with 0 < To < 1 such that I f, (p) I < To, 1 <: i <: d, p E U. Choose a real number T with 1"1l < r < 1. Write T = {p E P : I fi (p) I <: T, 1 <: i <: d} whenever 0 < r < 1. Let po be any point of T. Rince K i8 a Stein manifold there exist h" . , ha in A (K) whose restrictions to L are a coordinate set of functions on L at the point Po Thus for all sufficiently small f # 0 the functions 1 <:i<:d are a coordinate set of function" on L at the point po. It follows from the choice of ro that I gi(P) I < rD. 1 <: i<: d. for all pin tT if f is small enough. )\" ow fix r, with r < 1", < 1. ronsider the i'cb. and

R,

=

{p E P: I gi(P) I <

,-,.1 <: i <: d}

If f is small enough then Ro and R, are analytic polyhedra in L defined by multiples of the functions g,,' ,gd with RoCTCR,cP. Let g be the map (g,,' . ,gd) of K into Cd and let {j be the restriction of g to R,. From Theorem 2 it follows that the map {j of R, into r,Ed has a definite multiplicity

197

486

ERRETT BISHOP.

Since g" ,ga form a coordinate system at po, the point po occurs with multiplicity 1 in the set w(g(po». Let w be any function in A. (K) whose value at po is different from it» values at the other points of w(g(po». For an arbitrary q in g-' (I",Ed) let q" ,qx be the points of the "et w(g(q» = {q,,' " qx}' counted according to multiplicity. Let u be anv function analytic on RD' Define the function ii on g-' (roEd) ::J U by

A and multiplicity function w.

x

(a)

fi(q) =~ [U(qi) i~'

II (w(q)

-w(qj»]'

joFi

Now 'U (q) is a polynomial in w (q) whose coefficients U o (q), ,ax (q) analytic functions of q at each point q in g-' (ToEd) for which q" ,qx dismct points of RD. By Theorem 3, the functions ai are analytic on Thereforp 17 is analytic on U. For q in un LeR o the point q is one of points q" ,qx in w(g (q) ), Ray q = q,. [n this ca~e it i'l dear that

(b) where

are arp U. tlll'

17 ( q) = u ( q ) 1V ( q) x

(c)

w(q)

=II

(w(q) -W(qi»

i=2

is an analytic function on R" it being under'ltood that q, IS chosen to equal (I for q in R,. Remembering that po oc('ur~ with multiplicity 1 in the "et w(g(po» and that 10 takes on values other than w(po) at the other points aT w(g(po», we see that w(po) #0. Thus for each po in T there e:..ists a function 1V analytic on R, with (Po) # V such that for evpry analytic function u on Bo thprp exists an analytic function i1 on U which agrees with uw on Un L. Let B he the uniform closure on T of all functions analytic on R,. From Theorem -1 we see that the spectrum of B is T. Since the function W, which is analytic on R" can be cho»pn to be non-zero at any giyen pomt of T, there exist a finite number iL" , ii \ of such functions which have no common zero in T. Thus from the genpral theory of Banach algebras there pxist elements c" • ,C... of B with (d) ii)

For pach j, 1 <j< N, let ii j be the function defined by (a) with replaced by w; and u replaced by CjU. Thus N

(e)

ii,= ~Uj j~1

198

1/'

FUNCTIONS OJ<' SEVER.iL COMPLEX VARIABLES.

487

is analytic on U, and for all q in Un L we have, by (b), (d), and (e) a(q) =LUj(q) =LCj(q)U(q)Wj(q) = [LCj(q)Wj(q)]u(q) =u(q). Thi~

completes the proof of III'

Remalk. Theorem III holds III the following ~!Teater generality. Let V be a complex vector space and 1V some set of linear fundionals on V.

Define a function u with values in V to be analytic if cf> a U is analytic for all in W. Then Theorem III holds for analytic functions with valuei' in 17 We now proye I and II.

cf>

TIIEORKM 8 Let L be an analytic ~ubset of (/ 8 teill manifold K. L hal e dimen8ion d at each of its point~. Then

II.

Let

Fo!" each Po in K - L there exist!> an allalytic function h on K, defined by (*) and (~) below, lilhich zani,he, on L but doe!:. not vanish at po. If L is a wbmanifold of K then corre8ponding to every analytic function u on L there exi~ts an analytic function II on K with ii (q) = u (q) for 'all q in L. TlI e function fl is given by a conglomeration of formulas which are summarized by * below.

By Theorem 5, there exists an analytic map f = (fl' ,fd) of K into Cd whose restriction 1 to L is almost proper If p E K then J-l(f(p» is countable and consists of isolated points. Let PI, . , pn, be the points of 1-1 (f (p) ), each being counted a number of times equal to ib! multiplicity for the mapping 1- 'rhe multiplicity of a point q for the mapping 1 is defined by considering a special analytic polyhedron Q in L eontaining q defined by multiples of fl' " fd and taking the multiplicity of q for the analytic polyhedron (.); thib multiplicity cloe,> not depend on the ehoiee of Q. Proof.

Let {8d be a bequence of compact subsets of L, each of whil'h has finitely many components, with 8", C 8~+1 for all k and U ~ int 81. = L. For each positive integer lc let HI.. be the union of those components of J-l(rx(lc)£rl) which intersect 8~, where rx(lc) > lc 1 is chosen so that f(81.) C rx(lc)Ed. Thus 8~ C H~. Write

+

for all positive integers k. Now int Hie = HI. n 1-1 (rx (k) Ed) is an analytic polyhedron defined by the functions rx(k)-lfl" ,rx(lc)-lfd' Let Ak be the

199

ERRETT BISHOP.

multiplicity of int H k • Thus for each P in K the number of points in the sequence Pl, P2,· belonging to intHk is either A" or 0, depending on whether f(p) is in a(Je)Ea or not. By Theorem 6, there exists fa+1 in A (K) such that

I frl+l(P) for all p in Gk and all Je.

for all p in G/c. audition

-

(Al'+1

Therefore

If po is any point in K -

1- ' (f (Po».

L, we may choose fd+l so that in

=F exp (frl+1 (g»

exp (fd+l (Po» for all g in

+ Je2 + 1) 1< 1

Write

w = exp(- fd+l) ;"0

that wE A (K) and

lor all p in Gk • Choose JeD::> 2 with f(T) C koErl, where T is a compact sub~et of K to be fixed later. Since {lId is an increasing sequence of sets whose union is K, there exists for each g in '-1(f(T» a smallest le with g E HI... If k> ko then g E Gk - 1 ; in fact, g E int Glv-1 because f(g) E koErlC kEd so that g E 1-' (leEd) n Gk - l C int G1v-1. If k = ko then similarly g E intHko. Thus 00

1-1(f(T» C intHko U

U int Gk • k=k.o

Now for each p in K and k::> 1 define

where the product is taken over all Pi in F k , with F 1 = H 1 and F" = H k - H 10-1 for k> 1. Since at mm,t Ak of the Pi belong to Ih and since w does not ,anish on K, we see that tllf' functions 11 'e are well-defined. We shall f'how that the product h (p)

=

"" II hk (p) k=l

lOnverges uniformly on all compact subsets of K and defines a function h in A (K). Assuming for the moment that this is so, it follows from the fact that P = Pi for some i whenever pEL that h vanishes on L. It follows from the fact that w has values none of which equal w (Po) at the points of 1- 1 (f (Po) ) that h (Po) =F o. Thm; h has all of the desired properties. rt remains to

200

FUNCTIONS OF SEVERAL COMPLEX VARIABLES.

4S!J

prove the <.;onvergence of the product U) to a function h in A (K). For this llllrpose it is sufficient to consider an open subset U of K whose closure T is compact. Fix T to be the compact set considered above. Write ho(p)

=

IT(1-w(p;) [w(p) ]-1),

where the product is taken over all p, in int H k o ' that for all p in T

Thus by the above we see

leo

ho (p)

IT hie (p) .

=

k=1

Similarly

lor all p in T and lc > lco, where the product is taken over all Pi in int Gk • X ow for lc > lco, hk is a polynomial in [w (p) ] -1 whose coefficients we denote by aj (p) . Clearly aj is an analytic function of p for all p for which the points Pi in GJ.. are distinct. Since int Gk is an analytic polyhedron defined by a multiple of the functions f" . ,fd, it follows from Theorem 3 that hie is analytic on t'(f(int Gk»)::J U. Similarly ho is analytic on U. Thus we need only check the uniform convergence of the product

.

IThk k=1eo OJ]

Z'.

To no this we may equally well consider the series

.

}: (}:{Pi in Gk

:

I w (Pi) I}),

k=ko

'Illce I u, I is bounded below on the compact set T. The sum over Pi in Gk (ontain.s at most A'+l terms, each bounded by (lc 2"\'/..+I)-" and so is at most lc- 2. Thus the series converges absolutely and umformly on T. This completes the proof of 1. We turn to the proof of II Define {SIc}, ,AI,., fd+l, and w as above. For P in L dIOo~e f" ,fd to be a coordinate system on O-{hmensional subset of L. Also choose w to points of 1- ' (f (p) ). For each positive integer K by J1 2.

f,,"

,fd, /,

Gk, H k, T, P"

choose PI = p. In addition L at p for all p in a given have distinct values at the lc define the function hie on

"here the product and sum are taken over points Pi and Pi in H k. Here c is a ,!!ivE'n function in A (L) and rZ iR a function in A (K) which will be deter-

201

490

ERRETT mSHOP.

mined later. Let U CK have compact closure T. Consider lc so larg<' that f(T) Ca(lc)Ed. Now if f(p) E a(le)Ed. then f(Pi) = f(p) E a(le)Ed for all i. Thus we see that piE intHIe whenever f(p) E a(le)Ed and piE HI.. From this it is clear that each of the coefficients aj (p) in the expansion of hdp) in powers of [W(p)]-1 defines a continuous function Uj on [-1(a(k)Ed) which is analytie at every point p of [-1 (a (le) Ed) which has the property that the pointR jJi which lie in Hie are distinct. By Theorem 3, aj is analytic on uct1«(l(le)1~'(I). 'rhus hI. is analytic on U for all le sufficiently large. 1£ j > le then

I hj(p) -

hl.(p) I
+ CD(R -1),

where A is the sum ~

taken over

[Ii

in L -

~ I U(Pi)

=

HI., B

IS

II C(Pi) II d(Pi) I

the product

II = II(l + I W(Pi) II w(p) 1-

1)

taken over all 'Pi, C is the sum ~ taken over 1); in Ih, D is the product II taken over Pi in II. and E is the product II taken over Pi in L -II !.. Thus to show uniform convergence of the sequenC'e {hd on thp set IT we mmt show that the terms AB and CD (E - 1) converge to 0 as le ~ 00 \V e consider only the term AB, Slllce the treatment of the other term IS entIrely similar. Now the product B is finite and bounded independently of 11 if P E U, ab may be seen by considenng the sum 00

I

~ w(pj)

I,

j=1

where the factors I W (p) 1- 1 have been left out because I w I itS boundeu independently of P in U. Thus to show that the sequence {hd converges uniformly to 0 on U as 7c ~ 00 we need only show that A ~ 0 uniformly on U as k ~ 00. This will be done by an appropriate chOIce of the function d. Let

Ih=sup{IU(p)C(p)l: pE Gd. By Theorem 5 there exists

f'd+l

in A (K) such that

I f'd+1 (p) -

le 2f3IeAIe+1 - 1

I< 1

for all pin G k, so that ~(f'd+l(p))>7c2f3IeAIe+l. Let d=exp(-f'd+l), so that dE A (K) and d vanishes nowhere on K. Also

for all p in Gle •

Now if P E U and leo is chosen so large that f(T)

202

C

leoEd.

FUNCTIONS OF SEVERAL COMPLEX VARIABLEI:l.

each Pi in L - H ko is in some G/,. for lc > lco. Therefore the sum .l1 splits into partial sums Ak taken over the sets G/,. for 7c:> lco. Now at most '\/.+1 of the points Pi lie in Gk • It follows from the choice of d that

Thus A

.

<: ~

k- 2 •

Hence {hd converges uniformly on U.

It follows that

k=ko

{hd converges uniformly on compact subsets of K to a function u in A (K). For p in L we have, remembering that in this case Pl is chosen to be p, hk(P) =u(p)c(p)d(p)

II (l-w(p;)[w(p)]-l),

;7'1

where the product is taken over all Pi in H k.

Thus, passing to the limit,

u(p) =u(p)c(p)d(p)1V(p),

where 00

w(p)

=

II (l-w(p;)[w(p)J-l). ;=2

By the choice of fl' ,fa, p; =7= p for j:> 2 it P = p. it follows that W(p) =7= O.

By the ehoice of

10

We have thus found a funrtion iv = 17\ on L such that for any C1 in A (L) there exists d 1 in A (K) vanishing nowhere such that there is a funrtion u1 in A (K) which equals uc 1d11V 1 on L. Wp may choose iT-l not to vanish at a O-dimensional subset of L which rontains points in each component of L. Thus the set Ll of zeros of 1 is of dimension at mm,t rl-l. N ow choose iV 2 to have the same propertie;., as 11\ and to be non zero at a O-uimensional subset of Ll containing points ill each analytic component of L 1 • Continuing in this fashion we find 11\, . iCd haying the aboye propertil's and having no common zero on L. By a theorem of ~\.rens r 1] l it follows that there exi",t c1, . ,Crl in A (L) with "

w

By the choire of Wi, there exists d; in A (K) Yallishing nowhere on A (K) such that uCidiWi has an extension from L to an analytic function Ui on K. Write rJ

U=~(di)-1ii,b i=l

, The possibility of using Arens' theorem here was pointed out by Kenneth Hoffman

203

4l:Ji:l

ERRETT BISHOP.

so that u E A (K).

For P in L we have d

U (p)

=

L

(di(p) )-lU(p) c.(p) t4(p )w.(p)

i=1 d

=u(p) LCi(P)W.(p) =u(p). ,=1

This completes the proof of II. The final formula reads •


'"

u(p) =~ (d .. (p»-l~u(pi)c .. (pi)d,.(pi)[II (l-w,,(pJ)[w .. (p)]-l)] . ..=1 '=1 J-F'

Remark. As is well-known, II (in fact Ill) can be used to prove Runge's theorem for Stein manifolds. The proof goes as follows. Let T be a compact subset of a Stein manifold K and h a function analytic in some neighborhood U of T. Runge's theorem says that h can be uniformly approximated on T by functions in A (K) if for each p in U - T there exists in A (K) with I g (p) I > max {I g (q) I: q E T}. To show this notice that the assumption implies that therr exiRt~ an analytic polyhedron P in K with T

c

Pc U.

Let fl' . h be the functions defining r. Choose h+l' . . ,fn in A (K) to beparate points of U and to have rank equal to the dimension of K at each point of U. We may assume that I fi(P) I < 1 for all p in T, k 1
+

4. The resolution of Cousin distributions. _\s is well-known, the solution on a Stein manifold of Cousin's first problem, which consists of the global construction of a meromorphic function whose principal parts are locally given, can be obtained from the fact that on a Stein manifold the sheaf of germs of analytic functions has a first cohomology group which i~ trivial. Cousin's ~econd problem, which states that a globally defined meromorphic function exists whose quotients with certain locally given mcromorphic functions are analytic functions without zero'l, can al~o be sol"er1 uqing this statl'ment about cohomology. Thus we shall re;,trict ourselves in this section to showing that the cohomology group in questIOn is trivial. ThIS will lJP dom without thr usr of cohomological terminology.

204

FUNCTIONS OF SEVERAL COMPLEX VARIABLES.

4!J::I

Definition 5. Let U be an open subset of a Stein manifold K. A Cousin distribution on U will consist of a covering {U a} of U by open sets and an assignment for each a and f3 of an analytic function tap to the intersection U a n UfJ such that (a) and on Ua n Up n U'Y'

(b)

for all a, (3, and y. By a Tesolution of the given distribution {Ua}, {fap} on an open set S C K we mean a family {fa}, where fa is analytic on S n Ua, such that on sn Uan U for all a and (3. By a Tefinement of the distribution {Ua}, {tap} we mean a Cousin dibtribution {V'Y}' {g')'5} and a mapping a of the index set of {V')'} into the index set of {U a} such that (c) and (d)

for all y and 8. If T is a compact subset of K then a Cousin distribution on T means a Cousin distribution on some neighborhood U of T, and a resolution on T means a resolution on some neighborhood of T. The same definitions will be taken to hold if the functions f ap etcetera are replaced by who<;e coefficients arf' polynomials of given degree in an indeterminate analytic functions on the appropriate bets. ~ We make use below of the trivial fact that a Cousin (listribution whirh has a refinement with a resolution it~elf ha~ a resolution. The cohomology theorem mentioned above is just thp statement that every set of Cousin data on a Stein manifold f( has a rf'solution on K, ana it is this fact which will be proyed below. As a tool '\E' ..,hall use the fact that every Cousin distribution on the subset En oi Gil hu'i a lP.;olution. 'l'hi-; is well-known to be a rompletely elementary fact anrl incleptl wa~ p,,"f'ntiallv proved by Cousin [4] For completeness howeyC'l' WE sketrh thp ploof at thp end of this section. The following lemma contain<; the p'<'lence of the theorem to be proved.

.-r

205

ERRETT BISHOP. LEMMA 1. Let T be a compact set of a Stein manifold K and A a neighborhood of T such that f01' all p in A - T there exibts f in A (K) with

f(p)

> sup {J f(q) I:

q E T}.

Then any Cousin distribution on T has a resolution on the interim' M of T. Proof. Let the given Cousin data {Ua}, {tap} be defined on an open set U::J T. Let P be a special analytic polyhedron defined by functions fl, . ',fa in A(K) with T c PcP cU. Choose a covering {V')'} of Ed so fine that each component of W')' = P n [-" (V')') lies in one of the sets U a for each y. Since P is a special analytic polyhedron, each W')' has only a finite number of components, which we denote by WI')',' , Wt')', where t depends on y ]'or each of these components Wi')' choose Uu-(i')') in {Ua} with Wi')' C UaWrl' For each y and 8 definl' thl' analytic function h')'6 on W')' n W6 by

for all p in Wi')' n TV}. It tribution on P. For each Let w(z) = {PI' . ,PAl indeterminak FOl eaeh y

is easy to see that {W')'}, {h')'B} is a Cousin dispoint z in Ea let PI' . , px be the points of w be any function in A(K). Let X be an and B and each z in V')' n VB define

By Theorem 3 w(> sel' that the coefficient of each power of X in the polynomial g')'6 in X iR analytic on V')' n VB. Since {W')'}, {h')'6} is a Cousin distribution on P we sre that and

for all y, 8, and TllU& {V')'}, {g')'6} is a Cousin distribution for Ed. By the statement madl' above (whose proof will be outlined below), there exists a resolution {g')'} of this Cousin distribution. Now {Wi'}'} is a covering of P. For l'aeh WI'} dl'fine thl' function F i ,), an Wi')' by (T.

These functiom are the resolution of a Cousin distribution {W;Y}, {Fi')'j~} on P, where F i')'16 is the function

206

4lJ5

FUNCTIONS OF SEVERAL COMPLEX VARIABLES.

on W.'Y n Wl. If for each p in P we write w(f(p)) = p, it follows that

=

{P1'

,p~},

with

P,

~

Fi'YJB(p) =~h-y<5(Pi) .=1

~

=h-y<5(p)

II

j""i

(w(p) -W(pj))

II (w(p)

-W(pj))

i=2

=

W (p) h'Y~ (p) = w (p) fa (i'Y)a(j6) (p),

whE'l"E' ii; i'l the anah·tic function on P defined by ~

w(p)

=II

(w(p) -w(pj)).

i=2

'l'hus {Fi'Yid is a resolution on P, and therefore on M, of the distribution {lV,'Y}, {Wfa(i'Y)a(j~d. Since this distribution is a refinemE'nt on III of the dH,tribution {r',,}, {wfap}, it follows that thE' latter distribution has a resolution on M. If po ib any point in T we may assumE' that f" ,frt were chosen so that f has multiplicity 1 at Po and that 1IJ was chosen to distinguish Po from thE' other points of w(f(po)), so that iii (Po) #0. Thus we can find a finitp number of funetions W1' ,W n, each analytic in some neighborhood of T and having no common zero on T. sueh that for 1 < i < n the distribution {U,,}, {wifap} has a resolution on J1. Thull for E'ach Ci analytic on M thE' distribution {U a}, {wicd"p} has a resolution on N. By Theorem 4, the Ci may be chosen ith ~ CiWi = 1. It follows that distribution {U a}, {tap}, which on At is therefore the sum of distributions which have rE'solutions on M, has a resolution on "11. This completes the proof. TUEOTIElI1 9 1'p,ulutiUrI on K.

"1 ny Cousin

di~tl'ibulio1! 011 1/

Stein manifold K ha> a

Pruof. Sim'e K is Stein, then' E''(i~tl-\ all Jlll'l'eaBing H'quell('E' {Mn} of open sets whose union i" K and whose elo-.ures T .. all ha\'e thp pro~ of the bet T of Lemma 1. Thu" for each /I In Lemma 1 tllP gi\'en Cousin distribution {Fa}, {tap} ha;-, a re~olutioll {f",,} all ]I n , We may assume that thp closure of each Tla is compaet. Lpt {Sn} bp all inerea;:ing ,-,E'quenc'p of C'ompact sets whosp union is K. whieh haye thp propE'rty of thp ~et T in Lemma 1, and which satisfy the ineluRion S" C J1", We show now by indul tion on n that for each n there exists a resolution {hall} of {U,,}, {tap} on 111" such that I h"n-1 (p) - han (p) I < 2- 11

for all p

1Il

8 n-1

Assume that thE' han have been found for all integer"

207

ERRETT BISHOP.

smaller than a given integer n. and write

For each p in MTlr-l choose

It is easily seen that go is an analytic function on MTlr-l. there exists g in A (K) with

1%

with p

In

U '"

By Runge's Theorem

for all p in Sn ,. It follows that the functions {han} have the requlrf'll properties, where ha"=fo/'+g on UanM". Since the closure of each C" is compact, it follows that for each fixed 1% the sequence {han} converge" uniformly on U a to a function {fa}. These functions clearly form a re"olution on K of the given Cousin distribution. This completes the proof. Remark. If a meromorphic function with values in a Banach spacp i" defined to be one which is locally a quotient of an analytic Banach-space valued function by an analytic scalar-valued function, then Cousin's fir"t and second problems can be solved for such functions by applying Thf'orf'l1l £) for Banach-space yalued Cousin distributions. To finish we indicate the technique for obtaning a resolution of a Cousin distribution on E. Let D be the set

{(Z,,· .. , za) E Cd: z = x"

+ iYn, I x" I < 1, I Y" I < 1,1 <

n < d}.

Since Da is conformally equivalent to Ed, we may equally well consider the resolution of a Cousin distribution given on Dd. In fact we may assume that the distribution is given on the closure Ha of Dd, as the proof of Theorem 9 shows. Let the given distribution be {Ua}, {fap} , given on Hd. We fir"t introduce the notion of a soft resolution for a Cousin distribution. This it-. the same as a resolution except that the functions fa are permitted to be the arbitrary (not necessarily analytic) function". It is easy t.o see t.hat every Cousin distribution has a soft resolution Let {F}", be a soft resolution of the given distribution. If U is an open set then it. is easy to check that the given distribution has a resolution on U if and only if there exist a function F on U such that F - Fa is analytic on U a n U for all 1%. Let z" = x" iy", 1 < n < d, be the coordinates in ('d. Let N be a positive integer and write

+

8,,(k)

=

{zE Hd: - 1

for 1 < k < 2N and 1 < n < 2d. 1 < le j <: 2N, write

7c-1 7c + --.v<x" <-1 + N} For each choice of k ,,·

208

. ,le2d, with

FUNCTIONS OF SEVERAL COMPLEX VARIABLES.

Choose N tiO large that the covering {8 (lei' of the covering {U a} of Hd. For each le 1 , o
41H

, le 2d )} of Hd is a refinement ,let, with 1 < le j < 2N and

that 8(0) =Hd. We shall show that the given Cousin di~,tributIOn haa resolution on each 8 (le 1 , , k t ). This is true if t = 2d becamc the covering {8(le 1 , ,le 2rt ) } is a refinement of {rTa}. Assume therefore that it i~ true for all t greater than some given t, so that we must Rhow that there if' a r('~olutioIi on each 8 (le 1 , , let). To this enrt write for each 1 < 7c < 21'-..'

1'0

I.

8 (lei'

k

,kt;le)=8(lei, . ,le t )nU8 t + i (j)=U8(le i , · J=i

·,led)

;=1

Thus

,7c t

,k+ 1) =8(k

1,

and

8(le 1 ,

,let, 1).

By the indudion hypothesif> on t, It follows that the given distribution ha'> a resolution on 8 (le 1 , , k t ; 1). We now show by induction on k (t remaining fixed) that the given distribution has a resolution on 8(le 1 , · ,kt;k). L\ssume therefore that this is true for all 7c smaller than the given k. Thu'by the above we see that there exists a neighborhood V 1 of 8(kl' ,k t ; k -1) and a function G1 on V 1 "uch that G1 - Fa IS analytIc on V 1 n U r>. for all IZ. Similarly, there exi;;ts a function Go on some neighborhood V 2 of 8 (le 1 , . , le" le) such that G2 - F r>. is analvtlc on V 2 n U a for all IZ. rfhus G1 - G2 is analytic OIl V 1 n V 2. It is well known and easy to see that therefore there exist open sets 11'1 and 11'2 with

8 (kl'

, k t ; 7c -

1) C Wi C V 1

and 8 (kl'

, kt, k)

C W2 C

V2

/

-I-

and analytic functIOns .11 and A, on W1 and W 2 respectively such that .12 Ai and G1 - G2 are equal on W1 n W 2. It follows that the function B on W1 U W 2 defined as Al G 1 on W1 and A2 G 2 on lV, has the property that B - Fa is analytic on (W 1 U W 2) n (J a for all IZ. Thus the given distribution has a resolution on

+

8(kl'

. ,k t ; 1c)

=

8(kl'

+

, let; k - 1) U 8 (kl'

Thir, ('ompletes thc induction on k

. ,kt, k) C Wi U 11!

It follows that there cxists a resolution on ,k t ) .

8(kl' 'I

209

ERRETT BISHOP.

This in turn completes the induction on f. It follows that there exists a resolution on S (0) = Htl, as was to be proved. Added in proof Let L be an analytic submanifod of dimension d in a Stein manifold K. Let U be an open relatively com part subset of K. Then the contruction used in the proof of Theorem Il, shows that there exists a multiple-valued projection of U into L in the following sense There exists an integer A and an analytic map P-+ {(P"Z,),

ot U into the A-fold unordered produl't An of the "pal'e n = L X C' with itself having the following propertie'l. For earh II in C

}i~or each p in L n (. alld eal'h i with 1 < i < A tor whil'h Iii =1= Ji we have Zi = O. This fact in tum rlearly implies Theorem IT 1, since u ran be extended from L to U by the formula A

u(p)

~

=

ZiU(P.).

i=l

Similarly the proot of Theorem TI ean be L1.'-ed to show the e'l:istenre of certain multiple-valued analytir projl'dions of K into L. UNIVERSITY OF

C \LII'OR'HA,

BERKEL'" Y

REFERE~CES

[11 Richard Alens. "Dense inwI8e limit ring-~.' Mirhiqall Mathematiral Jou1'nal, vol 5 (1938). pp IBn 182 [2] Errett Bishop. ")Iappin,!!"'; of partiall~ analytic ~pa('e~," A~mericaJ1 ·Journal of Mathemati('s. vol R3 (19tH). rp 209 242 [3] Henri Cartan, "Seminaile E. N S ," H)51·32 [4] P. Cousin, "Rur Ie'! fonctions de n vaJiable" complexe~." A.cta Jlathematica, vol. 19 (1895 ) [;)1 Kiyosi Oka, "Sur les fonctions Itnalytique~ de plu~ieur" variable~ I Domaines convexe" par rapport aux fonction~ rationelles," Journal of Science of Hiroshima f"J1iversity, vol 6 (11)36), I'P 245255 - - - , "SUI les fonction" analjtiques de plllsieurs variable~ II Domaines d'holo morphie," .Journal of Science of Hi"o~hima f'niversity, vol 7 (1937), pp 115 130. [i I "flUl les fonction. analytique" de plm,ieUl s variables III Deuxieme prob Ierne de Cousin," .Journal of ScienGe of Hiroshima Universit1/. vol 9 (1939), pp 719

210

PARTIALLY ANALYTIC SPACES.* By ERRETT BISHOP.

1. Introduction. The concept of a partially analytic space was introduced in a previous paper [11. The present paper will refine and extend the theory. To this end it will be necessary to modify the definition of a partially analytic space whIch was given in [1]. Although formally quite different, the modified definition contains only one essentially new feature-local connectedness is no longer required.

DEFINITION 1. A pa1"iially analytic space consists of a separable locally compact Hausdorff space K and a subalgebra ~ of C (K), where C (K) is the algebra of all continuous complex-valued functions on K. The algebra ~r contains the constant functions and is closed in the topology of uniform convergence on compact subsets of K. ~ will be called the set of analytic functions on K. In the previous paper certain structure manifolds were also postulated in the definition of a partially analytic space. The stmcture manifolds are now permitted to occur willy-nilly. DEFINITION 2. A structure manifold J[ of a l'artially analytic space K is a subset of K which has the structure of a complex analytic manifold such that (a) the manifold topology and the subset-of-K topology for J[ are the same, and (b) all functions in ~ are analytic on }1. By manifold we mean connected separable complex analytic manifold. Our first task is to show that the results of [1] remain valid for the more generally defined partially analytic spaces considered in this I)Rper. This can be done by simple modifications of the proofs given in [1]. ,Proofs are only sketched. After this has been done we study a special kind of partially analytic space K, called an embryonic space, which is shown to have the property that the image of K under a proper analytic map of K into a manifold M is an analytic subset of M. This property of embryonic spaces is then applied to obtain simple proofs of two known results, (i) and (ii) below. (i). Let A be an analytic set of dimension k in a manifold M. Let B be an analytic set in the manifold M - A whose dimension at each point of * Received May 11, 1961.

669 7

211

670

ERRETT BISHOP.

A is at least k

+ 1.

We show that jj is an embryonic space. Since the identity map of jj into M is proper, we obtain as a corollary of the proper mapping theorem for embryonic spaces the Remmert-Stein theorem that jj is an analytic subset of M. For this theorem see [9J. JI.[ -

(ii). Let K be a holomorphically convex analytic space and m: a closed algebra of analytic functions on K with respect to which K is holomorphically convex. It is easy to see that the image Ko of K under the natural map of K into Cw. (C denotes the complex numbers) is a separable locally compact Hausdorff space. The algebra m: can be considered as an algebra m:o of functions on K o, and together ~o and Ko constitute a partially analytic space. We show that this space is embryoniC'. From this fact we obtain Remmert';; proper mapping theorem [8J, which states that the image of an analytie bpace under a proper analytic map is an analytic subset of the image space. If p is a point in a partially analytic space K, an n-tuple tl' . ,tn of functions in ~r will be said to be coordinates at the point p if there exists a neighborhood U of p which t = (tl' ., tn) maps properly into an open set V C Cn suC'h that for each h in ~ there exists 9 analytic on V with hjU = 9 0 f. Then K will be said to have rank at most n at p. We show that an embryonic space admits a set of coordinate functions at each point. It follows that an embryonic space can be canonically given the structure of an analytic space in the sense of Serre. Using this fact and the map described above of a holomorphically convex analytic space K onto a canonically associated embryonic space K we show that if ~r is any closed algebra of analytic functions on K with respect to which K is holomorphically com ex then there exists a canonical map cp of K onto an analytIc space Ko such that if ~o denotes the set of functions on Ko such that cp E ~ and if ~Ko denote8 the set of analytic functions on Ko then ~o = ~Ko. This theorem extends work of Stein [10], and has also been proved by Cartan [4], using other methods.

t

to

The theory of the imbedding of analytic spaees has been studied by Remmert, Narasimhan [7], and the author [1]. In this paper we study (Theorem 11) the problem of finding a proper analytic map of a holomorphically convex embryonic space K into complex Euclidean space CN of sufficiently high dimension N which separates certain set of points of K and which locally generates all analytic functions at certain points of K. The imbedding theorems of [1] and [7] then follow as corollaries. The methods used for these constructions are basically those of [lJ but also have something In common with the methods of [7]. The author is indebted to Hugo Rossi for many discussions about the

212

PARTIALLY ANALYTIC SPACES.

671

topics of this paper. Rossi has proved by different methods some results which are closely related to some of the results given below, in particular to Theorem 6 and to Corollary 2 of Theorem 8. 2. Partially analytic spaces. Since a new definition has been given of partially analytic spaces it becomes necessary to redefine analytic polyhedra. DEFINITION 3. A subset L of a partially analytic space K is called a frame if it is contl!ined in the union of some countable family of structure manifolds of K. The dimension of L is the least integer d (possibly d = 00 ) such that L is contained in the union of a countable set of structure manifolds on each of which 2f has dimension at most d. Let fl." ,fn functions in 2f and let L be a frame. Write

Po={p: pEL,

IMp)1


Let P be a subset of Po which is both relatively open and relatively closed in Po. If the closure P of P is a compact subset of L then P is called an analytic polyhedron with frame L defined by the functions fl,' ,fn. The dimension of P is the dimension of P when P itself is considered as a frame of K. P is called reduced if n < dim P, otherwise unreduced. An unreduced analytIc polyhedron P is called prepared if it can be co\ered by a C'ountable family PI,,} of structure manifolds such that for each IX (a) The dimension of & on J1" does not exceed dim P, (b) For each (f 2f 1-1, • •

IX •

the algebra 9£ has dimension 0 on 111" over the set

f fi-1) ,n

a

Bdry P denotes the boundary of P when C'onsidered as a subset of its frame L. The proof of the following theorem is the same as the proof of Theorem 3 of [lJ, except for one modification which will be discussed below. THEOREM 1. Let {Lj} be a finite family of frames in the partially akzYlic space K. Let the functions fl," ,f.. in & define for each j an analytic polyhedron P j with frame L j. Let y = y (j) < n be the dimensions of Pj' Let Sj be a compact subset of Pj. Then there exist functions It', . , f n' in &, which can be taken to be polynomial functions of the functions fl'" ,fn, such that for each j the functions ft', " f ../ define an analytic polyhedron P/ with frame L j such that Sj C P/ C Pj' If I fi I < 1 for 1
213

.M!.H.t<.:'l"l' HH!HU.P.

Remarks on the Proof of Theorem 1. We need only consider the one place at which the proof of Theorem 3 of [1J needs to be modified. The modification is necessary because we have dropped the local-connectedness condition when defining the frame L of an analytic polyhedron P. Becau&e of the absence of local connectedness for L, a union of components of Po Heed no longer be an analytic polyhedron. This means that the proof of Theorem 2 of [1J must be modified. Theorem 2 of [1J states, in part, that if a }nepared analytic polyhedron P is defined by functions fl' ,fll' if S C l' is compact, and if r > 1 has the property that scpn{p·lfi(p)l
then for .Y supfliciently large the functions (rfi(p))-V-(rf,(p))oV

define an analytic polyhedron Q with SeQ

2
P.

To see this write

Qo={P: pEL,I(rfi(p))lv-(rfl(p))l"1 <1,

2
In [1J it was sufficient to take Q to be the union of those components of Qo which intersect S, but this need not give an analytic polyhedron if L is not locally connected. Thus a different comtruction of Q is needed. Write Ql = Qo n P. As in [lJ it follows that if ~V is sufficiently large then Sand (bdry P) n Q, are disjoint closed subsets of the compact Hausdorff space Q, which are simultaneously intersected by no component of Q" It follows (see [6J, p. 158) that there exists an open and closed subset Q2 of Ql which contains S and is disjoint from bdry P. Thus Q = Q2 n Qo is an open and closed subset of Qo n P which is disjoint from bdry of P. It follows that SeQ C P and that Q is open and closed in Qo. Thus Q is the desired analytic polyhedron. Thus Theorem 2 of [1J can be established even WlWll L is not locally connected. The rest of the proof of Theorem 3 of [IJ ib the same. Theorem 3 of [IJ is just Theorem 1 above, except for the remark that the f/ can be taken to be polynomials in the fi' To see this note that the theory holds if ~ is replaced by the closed subalgebra of W generated by the functions fl' " , fn. This implies that the functions f/ can be taken to be uniform limits on compact sets of polynomials in the functions fi, from which it easily follows that they can be taken to be polynomials in the f;. Finally, it is clear from the proof of Theorem 3 of [IJ that the f( can be chosen to have absolute values less than 1 at all points of T. The following definitions and theorem correspond to Definitions 6 and 7 and Theorem 4 of [1]. The proof is the same.

214

PARTIALLY ANALYTIC SPACES. DEFINITION 4. A partially analytic space K is called weakly holomorphically convex if K itsel£ is a frame and if for each compact set 8 C Keach com ponen t of the set 8={pEK: If(p)l<maxq(slf(q)l, all

f

in~}

is compact. DEFINITION 3. ~\. continuous mapping cp of a topological space T1 into a topological space T2 is almost proper if each component of cp-1 (8) is compact for each compact set 8 C T 2' THEOREM 2. Let {Lj} be a finite family of closed frames of a weakly holomorphically convex partially analytic space K, with L1 = K. Let the dimension y=y(j) of L; be finite for each j. Let n=y(l) be the dimension of K. Then the set of elements f = (f1," ,fn) of the n-fold Cartesian product ~rn of ~ such that for each j the mapping

of L j into C'Y is almost proper is dense in

3. Special frames. of [1].

~n.

Definition 6 which follows replaces Definition 8

DEJ<'IXITION 6. A continuous proper map f from a subset 8 of a partially analytic space K to an n-dimensional complex analytic manifold M is called special of dimension n if all of the level sets of f on 8 are countable and if there exists a countable family {Ma} of n-dimensional structure manifolds of K which are subsets of 8 such that (2)

f is analytic on each Jfa and non-bingular on ]I a

(f3) Each AIa is open and U (y)

},fa

is dense in 8,

at eyery point of M a , /

The set H = f (8 - U .Ma) is a countable union of locally analytic subsets of .'If, each of dimension at most n - 1.

It is worth noting that condition (2) may be replaced by the weaker condition (2') f is analytic on each .'I/o. and non-singular on

]fa

at some point of Jfa.

To see this equivalence it is sufficient to note that if {Ma} satisfies (2'), ({3),

215

J~.IlUt~~

1.

.Hl~.l1..ur.

and (y) and if A", denotes the singular set of 1 on M", then the :family {JI,,'}, where M",'=M"-A",, satisfies (ex), ((3), and (y). The only condition that is not obvious is (y), and ('Y) follows from the inclusion f(8 -

U .1[,,') C U f(Aa) U 1{8 -

U Ma)

and the fad that f(Aa) is a countable union of locally analytic sets of dimensions at most n-l. Similarly we see that if the mapping 1 admits an extension to K f.ueh that 8 - U .lIa is contained in the union of a eountable family o:f structure mani:folds of K of dimensions at most n - 1 on eaeh of which f is analytic then f satisfies condition (y). Before <>tating the next theorem we recall from [11 that If X is a topological space then IX denotes the k-fold unordered product of X, consisting of all unordered k-tuples of elements of X with the natural topology. THEORE~f 3. Let f be a special map from a subset 8 of a partially analytic space K into a complex analytic manifold M, and let {.ilIa} be (( c01wtable family of n-dimensional structure manifolds satisfiyng (:z), «(3), al1(l ("I). Let G = U Ma. Then H = f(S - G) is closed and nowhere dense in .11, and .11 -H is connected. There exists an integer A >1, called the multiplicity of the map f, such that for each point z in 1f[ - H there exist exactly A distinct points p in G with f (p) = z. The map w from J[ - H into AS which takes z into the unordered A-tuple of such points can be uniquely extended to a continuous map w of Minto h8.

Proof. The proof is the same as the proof o:f Theorem ,j of [1], except where the proof o:f said theorem is incomplete. In the penultimate paragraph of the proof of Theorem 5 of [lJ it is shown that w can be continuously extended to (En - H) U {zo} for each Zo in H. To do this it is assumed that there are only a finite number of points p in P with f (p) = Z". This fact is not evident without further consideration. Assume therefore that the set r of points in P mapping into ZD is infinite. Since r is a countable compact set in P, we see by the Baire category theorem that r has an infinite number o:f isolated points. Choose points pi' . 'Ph+l whieh are i"olated in r, and for 1 < i < A 1 let Ui be a neighborhood of Pi with (Ji n r = 1); such that the sets lJi are disjoint. As in the proof of Theorem ,j of [lJ we see thllt there exists a neighborhood V of Zo such that j-l (V) n bdry is \ oid for alli . .For each z in V -H let 8,(z), 1
+

+

216

"-i

PARTIALLY ANALYTIC SPACES.

0'"

Also Oi#O because if p is any point in P_[-l(H) sufficiently near to Pi then f(p) = z E Ui so that Oi(Z) # o. Thus

V-H.

A> ~8i> A+ 1,

a contradiction.

Thus r is finite, as was to be proved.

COROLLARY. If f is a special map from a subset S of a partially analytic space K into a complex analytic manifold J[ then the family {Ma} of Definition 6 can be taken to be a finite family of at most A manifolds, such that f(111a) c M - H for all <X.

Proof. Let A be the set of points p in JI ~uch that w(p) consi~ts of A distinct points, so that A is open and dense in JI. Because the complement of A has topological dimension at most 2n - 2, A is connected. Now [-1 (A) is a A-sheeted covering space of A. Since A is connected, the set pIa'} of components of [-l(A) is finite. It is then clear that each 1.1,/ has the structure of a complex analytic manifold and that pfc/} satisfies conditions (a), (/3), and (y) of Definition 6.

Corresponding to Lemma.) of [lJ we have the following theorem and corollary THEOREM 4. Under the assumptions of Theorem 3, let h be a continuous function on M which is analytic on 1.1 -H. Then h is analytic on M.

Proof. The proof is essentially contained in the proof of Lemma 5 of [1]. H. Rossi suggests the following simple proof of Theorem 4 by induction on the dimension n of M. If n = 1 then H is a countable closed subset of M, so that the set IIo of points in M where f is not analytic is countable and closed. Thus H 0 is void since otherwise II 0 would contain isolated points, contradicting the fact that a continuous function does not have isolated points of non-analyticity. This proves the theorem for n = 1. Consider n> 1 and assume the theorem true for n - 1. We may aRsume thaVM is En = {(Zl' .. , zn) : I Zi I < 1,1 < i < n} Thus the intersection of H with all but a countable set of the varieties

is a countable union of locally analytic subsets of L e, each of dimension at most n - 2. By the induction hypothesis, f is analvtic on all such Le. Thus f is analytic on 31. COROLLARY.

Under the

a~';Umption~

217

of Theorem 3, let 9 be a continuous

ERRETT BISHOP.

UIU

function on S which is analytic on each };fa and let s be a symmetric poly nomial in variables Z,,' . ., ZII.. For each Z in };f let P" .., PII. denote the points of w(z), counted according to multiplicity. Then the functio'f1 h on Jl defined by h(z) =s(g(pd,' ',g(pll.)) is analytic on };f. Proof. By Theorem 4, it is sufficient to show that h is analytic at each point Zo of };f-H. Let P,o,' ·,PII.° be the points of w(zo). Thus Zo has a neighborhood U such that there exist analytic homeomorphisms i(Z) E w(z) for all Z in U. Thus for all z in U we have

so that h is analytic at Zo, as was to be proved. We now study conditions under which the image of a partially analytie spat'e under proper maps will be an analytic subset of the image space. LEMMA 1. Let f = (ft. . ,fn) be a special map of a subset S of a partially analytic space J( into an n-dimensional manifold .11. Let g be a continuous function on S with values in C" which i~ analytic on each Ma, with {.'lf a } as in Definition 6. Let h be the map (f, g) of S into the manifold ji=MXCk. Then h(S) is an analytic subset of .iI.

Proof. Consider a point r in ]1- h (S), so that r = (q, t " . , tk ) where q E M and t,,' . " tk are complex numbers Let q" , qA be the points in S which map into q under f, i e., the points of w (q) counted according to multiplicity. Thus for each i, 1 < i < A, there exists j = j (i), 1 < j < lc, with gj (qi) -¥= tj • There thus exists a polynomial ~i in one variable such that ~i(tj) = 1 and ~.(gj(qd) = O. Define the analytic function ~ on Ck by II.

~(zJ'

,zl.J=II~i(zj(i). i:::l

Thus ~ (t " .. , t k ) = 1 and ~ (g (qi» = 0 for 1 < i < A. It follows that 6. = ~ a g is a continuous function on S which is analytic on each JI a , with 6. (qi) = 0 for 1 < i < A. If for each P in J[ we let P" . . , PA denote the points of w (p) then by the corollary of Theorem 4 we seee that each of the elementary symmetric functions of the quantities 6. (Pl)" . ',6. (pA) is an anlJ,lytic function of the point P in Jf. Thu'> the function F on if defined by A

F(P,Z"

. ·,ZI")

=II

(~(Zl"

i==l

218

P ARTIALL Y ANALYTIC SPACES.

is analytic on lII.

67'7

Clearly A

F(r) =F(q,t,,·

·,tT<)

=TI .=1

(l-Ll(q.» =1.

Also if (p, Zl,· . , ZIe) is in h (8) then there exists i, 1
c

L for all a,

(a)

M"

(b)

Each M" is an open set in L,

(c)

The dimension of the frame L - U Ma is at most n-l,

(d)

U lila is dense in L,

( e)

There exists a set N ::J L such that the level sets of m: on N are all ('ountable and such that N is a countable union of structure manifolds of K.

The family {111 a} is called a structuml family for the frame L. It is clear that if L is a special frame of dimension n in a partially analytic space K then a proper map i = (fl,·· ,in) with fi E m:, 1 < i < n, from L into an open subset of On whose level sets on L are all countable is special whenever L has a structural family {1I1a} such that f is non-singular at some point of each ]I,]a.

·,gle), with giE~I, l
219

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ERRETT BISHOP.

is a compact analytic subset of En X U C Cn+k. Therefore T is finite. Thus f (g-l (Po» is finite. Since f is at most A to one, g-l (Po) is finite. Thi,; proves the lemma. THEOREM 5. If g = (g" ,gk) is a proper map of an n-dilllelllional special frame L in a partially analytic space K into an open set [T C Ck, where gi E ~ for all i, then g (L) is an analytic wbset of U.

Proof. Since g is proper, g (L) is closC'd in G 'fhus we need olll~ check that g (L) is analytic at each point po in g (L ). Let ~o be the closure in the topology of uniform convergence on compact subsets of K of the set of all , g1.., so that (K, ~o) is a partiall! analytic polynomials in the functions gl,' space. Let V be a small neighborhood of po with V C U. Since the map g is proper there exists an analytic polyhedron Q in (K, ~o) with frame L with L n g-l(V) C Q. By Theorem 1 there exists an analytic polyhedron P definC'd by functions f,," , fn in Wo with Lng-I (V) C P C Q. Thus f can be changed slightly such that each fi is a polynomial function of gl' ,g1..' Thus f is a proper map from an analytic polyhedron P ::J Lng-I (F) into En, . Il n on ('I< with such that there exist polynomial functions Ill' T

•

for all p in K. Since P is an open subset of the special frame L it i~ Itself a special frame, so that by Lemma ~ f is a special map of Pinto EIII. By Lemma 1 we see that (f, g) (P) = A is an analytic set in En X C, which clearly consists of all (Il(p),p) for pEg(]», where 1l=(IlI' ·,Iln). Let Yl," ,'/j be analytic functions defined on some neighborhood n in En X U of the point (Il (Po), Po) whose set of common zer05 is Ann. Choose a neighborhood W of po with W C V such that (Il (p) , p) E n for all p in W. Let B be the set of zeros on lV of the analytic functions YI(Il(p),p),' ·,y;(Il(p),p). If pE B then (Il(p),p) E A so that pE g(P) cg(L). If pEg(L)nW=g(P)nW then (ll(p),p)EAnn so that YI(Il(p),p) =' '=y;(Il(p),p) =0 and therefore pE B. Thus geL) n W = B so g (L) is analytic at the point po, as was to be proved. 8. A frame L in a partially analytic space K which is the union of a countable family of closed special frames only finitely many of which intersect any compact subset of L is called an embryonic frame. If K itself is an embryonic frame then K is called an embryonic space. DEFINITION

For embryonic frames we have the following immediate corollary of Theorem 5. COROLLARY.

If L is an embryonic frame and f"

220

. , fk in

~I

define a

PARTIALLY ANALYTIC SPACES.

U 1;,

proper map of L into an open set U in Ok then f(L) tS an analytic subset of U. As a corollary of this corollary, we obtain the following theorem.

THEOREM OF REMMERT-STEIN. Let A be a k-dimensional analytic subset of an open set U C On. Let B be an analytic set in U - A of dimension at least k 1 at each of its points. Then ii n U is an analytic subset of u.

+

Proof. Let K = A U B and let ~ be the closure on K in the topology of uniform convergence on compact sets of all functions analytic in U. Thus (K,~) is a partially analytic space. Clearly L = ii n U is an embryonic frame in K. Let f be the inclusion map of L in ll. By thE:' aboye corollary, ii = f(L) is an analytic subset of U, as was to be proYE:'d.

4. Contractions of analytic spaces. The term analytic used in the sense of Serre (see [5J, p. 280).

~pare

will be

DEFINITION 9. Let K be an analytic space and ~ Lr a closed algebra of functions everywhere analytic on K. Assume that K is holomorphically com'ex in the sense that the set S of Definition 4 is compact for all compact S c K. Let Ko be the image of K under the natural map ep of K into O~. Since K is holomorphically convex, the map ep is proper and Ko is closed in C~, so that Ko is locally compact. Let ~o be the subalgebra of C(Ko) conSIsting of all f in O(Ko) such that foepE ~L It is clear that (Ko,IlIo) is a holomorphically convex partially analytic space which we call the contrarlion of (K,Ilr).

Of course Definition 9 could be made for an arbitrary partially analytic space K, but we shall only apply the construction to analytic spaces. The object of this section will be to show that (Ko, IlIo) is actually an embryonic space. LEMMA 3. With the notation of Definition 9, let JI be a ~tructuTe manifold of K on which Ilr has dimen ,ion n. Then ep (JI) i~ a frame in K 0 of dimension at most n. / Proof. The proof is by induction on the dimension d of l.I. If d = 0 the lemma is clearly true, so assume that the lemma is proved for all d smaller than the actual dimension d of M. Choose f,,' . ,fn in Ilr to have rank n at some point of M, and let A be the analytic subset of M at which these functions have rank less than n. Thus A is a countable union of structure manifolds of K each of dimension less than d on each of which III has rank at most n. By the induction hypothesis ep(A) is a frame of dimension at most n. On the other hand, M -A can be covered by a countable family

221

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ERRETT BISHOP.

{.lla} manifolds each of which is mapped by fl'" ,fn onto a connected open set La in an and on each of which h," ,fn and I)( have the same level sets. Thus La is naturally homeomorphic to cp(Ma) ; the cp(Ma) are thus a countable family of structure manifolds of Ko on each of which 1)(0 has dimension n. Thus cp(M -A) is also a frame of dimension at most n. This completes the proof. LEMMA 4. Let S be an analytic space whose set R of regular pain ts is connected and let I)( be an algebra of analytic functions on S of dimension at most n at each point of R. Let M be any structure manifold of S. Then ~f has dimension at most n on M.

Proof. If d is the dimension of S then every level set of 2f on R has dimension at least d-n at each point. Therefore, by a known result ([3], Exp. XIV), every level set of I)( on S has dimension at least d-n at each point. Assume that the dimension of I)( on M is larger than n. Thus there ,fn+1) in I)(n+l having rank n 1 at some point of M, exists f = (f 1," so that f(M) contains an open set U of en +1 • ~ow Scan hp coyered by a countable family pIa} of structure manifolds such that for each O! either f has rank n 1 at every point of Ma or at no point of Ma. Let {.vIa'} be the 1. subfamily of {M a} consisting of those ]I a on which f has rank less than n If P is any point in U then tl (p) contains some level spt of 2f on Sand therefore has dimension at least d - n. By the theorem on the dimension of a countable union of closed sets, tl (p) n Ma has dimension at least d - n for some a. Since the dimension of Ma is at most d it follows that f does not have rank n 1 at all points of JIa, so that JIa is some Ma'. Thus U is covered by the sets f(M a'}. Since f has rank less than n+ 1 at all point" of M a', f(M a') is a countable union of closed sets of topological dimensions at most 2n. Thus U is a countable union of closed sets of topological dimensions at most 2n. This contradiction proves Lemma 4.

+

+

+

+

THEOREM

6.

With the notation of Definition 9, (Ko, Wo) i~ an em-

bryonic space. Proof· The analytic space K is the union of a countable family pIa} of irreducible sets, only finitely many of which intersect any compact subset of K. Thus {cp(Ma}} is a covering of Ko by closed frames, only finitely many of which intersect any compact subset of Ko. Thus it is sufficient to show that each 4> (Ma) is an embryonic frame in Ko. It is therefore enough to show that 4>(T) is an embryonic frame in Ko for each irreducible analytie Imbset T of K. We will do this by an induction on the dimension d of T. The case d = 0 is clear. Thus we assume that the statement has been verified for

222

PARTIALLY AN~\LYTIC SPACES.

uo~

all T of dimension smaller than the dimension d of the given T. Let 8 be the set of singular points of T, so that dim 8 < d. Since 8 is the union of of a countable family of irreducible analytic sets of dimensions less than rl, only finitely many of which intersect any compact set, we see by the induction hypothesis that ep(8) is an embryonic frame in Ko. Thus if ep(8) = .p(T) then ep (T) is an embryonic frame, as was to be provecl Therefore we may assume that ep(8) # ep(T). We shall prove that the set 8 1 = T n ep-1(ep(8)) is an analytic subset of K. To this end, let p be any point in K, and let TJ be a neighborhood of p such that r: is compact. Since K is holomorphically convex there exists an analytic polyhedron P with frame KC) defined by funr'tions fl' . f n in ~ro such that ep (C) c P. ~ ow let 9 br am' element of ~[V, where N is any po<;itive integer and let rJ be the element of 9[0 v t'orrespon(lin~ to g. Sincr ep(S) is an embryonic frame in K o, ep(S) n P is an embryonic frame in the partiaJly analytic spat'P (1', ~[') obtained by taking ~r' to be all uniform limits on compact subsets of P of functions in ~ro. By the corollary of Theorem 5 it follows that (fl' ,f", g) = ~ maps ep (8) n P onto an analytir set By in En X ON. Hence the 8ubEPt

of U

j"

analytie.

~ow

elrarly

un ep-l (ep (8) ) = n AI}' where the intersection is taken oypr all finite familieH 9 of functions in m:. Thus Un ep-l (ep (8) ) is an analytic subset of U. It follows that ep-1(ep(8)) is analytic subset of K. Thus 8 1 = T n ep-l(ep(8)) is an analytic set. N ow let 7c be the smallellt of the local dimensions of all of the level sets of m: on T. By a known theorem ([3J, Exp. XIV), the set J of p in T such that the level set of m: on T which passses through p has dimension 7c at p is an open set in T. Thus J intersects T - 8 1 so that ~r has rank d -7c at some point of T-8 ,. Now 8 1 is the union of a countable family pIa} of structure manifolds of K. Since the level sets of m: on 8 1 have dimension at least 7c at each jl9lnt, by Lemma 4 we see that m: is of dimension at most d - 1 - 7c on each J.fa. By Lemma 3 it follows that ep (8 1 ) is a frame in K ° of dimension at most

d-7c-l. N GW let 8 2 be the set of all points in T - 8 2 at which m: has rank d -7c - l o r less. Clearly 8 2 is an analytic subset of T - 8 1 , Since T - 8 1 is a (connected) manifold, 8 2 is a proper analytic subset of T - 8 1 , N ow if m: had rank d - 7c or more on 8 2 at some regular point of 8 2 then m: would have rank at least as great on T - 8 1 at that point. Therefore

223

ERRETT BISHOP.

uo'"

~

has rank at most d - k - 1 at all regular points of 8 2 , Thus the dimensioll~ of ~ at every regular point of 8 2 is at most d -lc - 1. By Lemma 4 it follow!! that 8 2 can be covered by a countable family of structure manifolds on each of which ~ has dimension at most d - lc - 1. Thus (8 2 ) is a frame in K 0 of dimension at most d - k -1. If 8 a = 8 2 U 8 1 it follows that the frame (8.) in Ko has dimension at most d-lc-1. If 8 4 =Tn-I((8s » it follows that H = T - 8 4 is open and dense in T and that