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0
< oo.
and
IIP ll
p for m  O ,
1,.
Then lim s u p ( a m ) ~

lim sup(llPmll

Lr.
m~(N~
m~oo
By the root test of the series of nonnegative numbers, ~ a~ converges if L r < 1. Therefore oo
!
rnO
for
s
converges normally and < ~. This shows t h a t y ] . ~ = o P , ~ ( x  a ) 1 < hence uniformly on the ball B   ( a , r) for each r < ~. Thus 7  P "
7.2 P O W E R
SERIES
IN FSPACES
On the other hand, if r > ~, then F~ of natural numbers, we have
141
L>
!
IIPmilq >
1 Then for some infinite set r.
1 ~FTM
whenever m E Ft. As in case (1), we can find Ym e E, I l y m l [  1, for each m C F~ such t h a t 1 1
IIr.~(ym)ll~ >  r. m Then for Xm  a + rym, we have 1
IIPm(xm a)ll
 r m l l P ~ ( Y m ) l l ~ >
r~(1) m
1
which shows t h a t Em~__0P.~(xa) does not converge uniformly on B  ( a , r ) . This shows t h a t p <_ 1/L. Hence p  1/L. (3) For L  0 " Let r > 0 ande
1 Then there exists N > 0 such t h a t 2rq/v" IIP.~II _< e "~
If x E B  ( a , r ) ,
,for m >_ N.
we have
IIP,~(x a)ll <__IIPmll.llx all ~q/p ~ ( erq/P) m
m
Vm>N.
Notice t h a t IlP,~(x)ll _ IIP~ll.llxll ~q/p for phomogenous polynomial (see section 2.2.4.). This shows t h a t the power series converges uniformly on B  ( a , r). Since r > 0 is arbitrary, we conclude t h a t p  oe. / C o r o l l a r y 10 I f p is the radius of uniform convergence for a) and O < r < p, then
E~%0Pro(x
oo
IIP~ll ~ / ~
< o~
(7.14)
rn0
P r o o f . As in the case (2) of the preceding proof, apply the root test with am I[Pmll r mq/p t o obtain the result, m
CHAPTER
142
7 QUASIHOLOMORPHIC
MAPS
F is a qBanach space, the following statements are equivalent 9 (a) E m ~ = 0 P m ( x  a) converges uniformly on B  ( a , r ) , r > O.
C o r o l l a r y 11 If
1
(b) lira sup IIPmll~ < oo. (c) The sequence ( IlPmll ~ ) is bounded, (d) There exists co > O and c > O such that lIPmll ~ coc mq/p
liP011 ~
(In (d), to avoid
V m  o, 1, 2, ..
(7.15)
1, w~ u ~ the constant co)
P r o o f . We note that the implications (a)=~ (b) =~ (c) =~ (d) are clear. For ( d ) ~ ( a ) ' l e t r > 0 be such that r c o < l . Then OO
oo
I I P m ( x  a)ll _< ~
co(cr) mq/p < c~
(7.16)
mO
mO
for x e B(a, r), which shows that the series converges normally on B  ( a , r). We used here the fact, IIPm(x)ll <_ IIPmlI.IlXllmq/p. Hence the radius of convergence is strictly positive, m In the definition of the radius of uniform convergence and the generalized C a t c h y  H a d a m a r d formula, we used the power series ~~.m~__oPm(x a) with polynomial coefficients. Let us now use the power series OO
A m ( x  a) m,
with
Am E L~(mE, F).
rn0
We define the radius of uniform convergence as before. Let p  be the radius of uniform convergence for this series. Then we can prove that m p1 = limsup II Atoll ~
if F
(7.17)
is qBanach space. In what follows we have the following relations between p and
T h e o r e m 69 [36]
Let F
be a q  B a n a c h space and let
p.
7.2 P O W E R
SERIES IN FSPACES
(x)
143
oo
Pm (x  a) = ~ rn0
Am (x  a) m
m=0
where Pm  f~m, Am E Ls(mE; F), then
_P _
(7.18)
e
in particular, p  p  if F  K. P r o o f . Since by Theorem 28, (subsection 2.2.4 p.41)
mm(q/P)
It Pml] _< II Atoll _<
m!
Pmli,
Vm E~g
we have
p > p > p lim (m!)l/m
Vm E ~
(7.19)
m
Hence by Stirling's formula" m~ lira = 1 m,c~ mmem(27rm)l/2
(7.20)
we have m
lira ~~ (m!)i/m
~ze
(7.21)
and hence
P<_p<_p. e
(7.22)
m
The strict inequality in example.
p  <_ p
can be achieved
as in the following
CHAPTER
144
7 QUASIHOLOMORPHIC
E x a m p l e 26 Let E  l P , ( O < p < 1 ) , F  K . 0, 1, 2, ...there exists A,~ e L , ( m E , F), II A,~ II  1
and
MAPS
Then for each
m

m!
[[ Atoll 
?Ttm. 1 lp
see example 8, p.42. Now by the Cauchy Hadamard formula, we obtain 1
p = limsup II A~II 1/~p,
p
1,
7.2.4
pe
since lim(m!)l/m = e ,
Radius
m
of Normal
D e f i n i t i o n 45 Let F E~=0Pm(xa) from 0 <_ r < oo, such that We call Pn the r a d i u s
i.e.
p
Convergence

(7.23)
< p. A
p~
be a q  n o r m e d space. Consider the power series E to F, and let p,~ be the supermum of all r, the power series converges normally on B  ( a , r ) . o f the n o r m a l c o n v e r g e n c e of the power series.
In the proof of the generalized CauchyHadamard formula for the radius of the uniform convergence, we in fact proved the following theorem "
T h e o r e m 70 . If F is a q  B a n a c h space, then the radius of the normal convergence is equal to the radius of uniform convergence. 7.2.5
Radius of Absolute
Convergence
D e f i n i t i o n 46 A number p~, 0 <_ Pa < oo a b s o l u t e c o n v e r g e n c e for a power series
fla is said to be the r a d i u s o f
oo
P m ( x  b), rn0
if Pa is the largest number such that the series converges absolutely for II x  b I1< Pa" The radius of absolute convergence is equal to radius of uniform convergence if E  K. It is clear that p <_ fla, but p can be strictly less than #a for some infinite dimensional metric space, as the following example shows.
7.2 P O W E R
Example
SERIES
IN FSPACES
27 Let E  1
p, (O < p < 1 ) .
Pro(x)
(Xm) m,
145 For
(Xl, X2, ...) E l B.
X
(7.24)
we have
II Frail = 1,
Vm
and hence p = 1. On the other hand, since x m + 0 as m + oo, the series ~ ( x , ~ ) "~ converges absolutely f o r all x C lp. So this shows that Pa = +oo, and hence P < Pa. A
F o r m u l a for R a d i u s o f A b s o l u t e
Convergence
Pa
D e f i n i t i o n 47 Let F be a q  B a n a c h space.For a p o w e r series ~m~176 P m ( x a) and u E E , Ilull  1, we define the f u n c t i o n p(u) to satisfy
1
P(~)
(7.2s)
= lim sup IIP~(~)II m
T h e o r e m 71 [36] Let E be a p  n o r m e d space and Then Pa  inf
F
a q  B a n a c h space, and
{p(u) ; liul[
 1}.
uCE.
(7.26)
P r o o f . Let R
inf {p(u);
I1~11 1}.
We may take a  O. Suppose R > O. T h e n if Ilxll < n, u ~ E, Ilull 1 and k >_ 0 such t h a t x ku. We now have l i m s u D l , P ~ ( k u ) l l  ~ . q 
kp
kp <   < 1.
p(~) R
It follows immediately by the root test t h a t the series converges absolutely. Hence I~ ~_ Pa" On the other hand, if kp > R, we can find 5 > 0 such t h a t Since
we can find
(7.27) E~%0 P~ (x) (7.28) k p > R +5.
146
CHAPTER
R  inf there exists u e E ,
I1~11 1,
7 QUASIHOLOMORPHIC
{p(u);llull
 1},
such t h a t
kP > R + ~
p(u).
>
Then limsup [[P~(ku)ll&~ and hence the series that
E [IPm(ku)ll

kp
p(~)
>1
diverges by the root test.
Pa < R Hence we have proved Remark
7.2.6
17 If E  K ,
Uniqueness
MAPS
p~  R ,
This shows
(7.29)
by(7.28) and(7.29), m
the formula for Pa reduces to p. of Power
Series
If two series ~~=0 P m ( x  a), ~~m~=0Q m ( x  a) converge absolutely and have the same sum on a n e i g h b o r h o o d of a, then we show t h a t P,~ = Qm for all m  0, 1, 2, ...This is an i m m e d i a t e consequence of the following theorem.
T h e o r e m 72 [36] If the power series ~~m~=oP m ( x  a) converges absolutely and its sum is equal to zero on some neighborhood of a, then
ProO,
mO,
1,...
(7.30)
P r o o f . We first prove the t h e o r e m for E  J R . For simplicity assume a  0, and let }~m=0UrnXm be a power series from K to F , where um E F. Suppose t h a t ~~,r umxm converges absolutely and its s u m is equal to zero for I x l < ~ for some ~ > 0 . If we put x 0, then u0  0. Inductively assume t h a t u 0  Ul ... = Un1 0 for some n > 1. Since }~~=0ur162  0, Ilu.~[15 "~q + 0 as m + oo; hence {llu~ll~q
9~

o,
1,2,
...}
(7.31)
7.2 POWER
SERIES
IN FSPACES
147
is bounded above, say by c. Now for x ~ 0, I x l < 5 , (N:)
I1~11 ~ ~
Ilumll Ix I(r~~)q<
re=n+1
c Ix Iq
(7.32)
 (~nq(~_ I X Iq )"
Thus we obtain u,~ = 0 by letting x ~ 0. We are now ready to prove the theorem for the general case. Assume the power series converges absolutely to 0 for Ilxll _< ~, r r 0. Let t e E be fixed but arbitrary. Then for ]A J_< ~/lltll, oo
oo
Pm()~t)  ~ mO
Pm(t))~ TM  O .
Hence from the case where F  K above, we have Pro(t)  0 m  O, 1, ...Since t is arbitrary, we can conclude that Pm  0 m  O, 1, ..m 7.2.7 Theorem
QuasiDifferentials 73
(7.33)
mO
of Power
for all for all
Series
[34]
Let F be a qBanach space and let ~~m~=oAm(xa) m be a power series whose radius of uniform convergence p is strictly positive, and let (X)
f(x)  ~
A m ( x  a) m
(7.34)
m0
be its sum on B ( a , p ) . If b E B ( a , p ) , }~.k~__oB k ( x  b) k such that
then there exists a power series
0(3
f(x)  ~
B k ( x  b) k
(7.35)
k=0
fo~ IIx bll < p  lib all, and its radius of uniform convergence is > p l i b  all; the coefficients Bk are given by
m
k)Am(b
a)m_ k
.
(7.36)
mk,
Proof. Let 0 < r < p. T h e n Em~__011A.~llrm~ < oe by Corollary 10, (subsection 7.2.3, p.141) and E m ~ = o A , ~ ( x  a) m converges uniformly on
CHAPTER 7 QUASIHOLOMORPHIC MAPS
148
B(a, p). For b e B(a, p), apply the multinomial formula (2.2) ( sbsection 2.2.2, p.36) to Am [(b  a) + (x  b)] m to have oo
m
A m ( x  a) TM  ~
( k ) A m ( b  a)mk(x  b ) k
mO
Then Oo
k ) IIAml "lIb  all
~ llx  bll ~ < IIAmllrm~
rn~O
for
IIx  bll < r 
lib  all. Thus
oo
oO
][Am(x  b)mll < ~ m=O
Oo
~
IIAmllq lib all
k
~.llx bll
m = O k=O oo
~
IIA~llqr~
< +c~.
mO
Since the iterated series in the preceding inequalities converges, we can interchange the order of summation to get ~
m=
m
k
IlAmllq'llb  all
(~k)
~ "llx 
bk k
II N < + o o
Therefore, if we set oo
Bk  ~
m
( k ) Am(ba)(mk)~,
rnk
then it follows that
IIBkll <
+oo.
Hence, Bk E Ls(kE; F). Thus we have
Am(x  a) m  ~ m0
Bk(x  b) k
k=0
and the convergence in the last series being uniform for IIxbll < r  liball. This shows that the radius of uniform convergence for ~k~176B k ( x  b) k is at least p  lib all since r was arbitrary. This completes the proof, m We are ready now to show, as a consequence of the preceding theorem, that a function represented by a power series is a C q function within the circle of convergence.
7.2 P O W E R
SERIES IN FSPACES
149
T h e o r e m 74 [36] Let F be a qBanach space. If ~m~=O A m ( x  a) m converges to f ( x ) on B(a, p), where p is the radius of uniform convergence of the series, then f is C~q function on B(a,p) and
ld~ Am = m!
f(a)
(7.37)
for m  O, 1, 2, ...
d
sup{llAmllr

Since f ( x )  ~~,~oA m ( x 
rrttl
Em=0 I I A m l l r
P r o o f . Let 0 < r < p. Then (subsection 7.2.3.p.141). Let
p ;
"
mO,
1,2,
< oo by Corollary 10
}
....
(7.38)
a) m, we have f(a)  Ao and oo
f(x)

f(a)


Al(x
a)
 Z
Am(x

a) m.
m=2
Hence c~
llf (x)  f (a)  A ~ (x  a) ll <_
d
~
m~
m~ I]x  all
m   2 /*
p
2 q
dllxall
"
r~(r~IIxall
P
~ )
for x C B(a, r). This proves that f is quasidifferential at b and D f ( a ) A1. Repeating this argument for b E B(a, p) and oo
f (x)  ~
Bk(x  b) k
k=O
for I I x  bll < p  l i b  all, see the preceding theorem, we conclude that is quasidifferentiable at b, and (X)
Df(b)  B1  ~
mAm(x
a) m1
m1
This proves that
f is quasi differentiable on B(a, p) and Oo
D f(x)  ~ m1
Oo
m A m ( x  a) m1  ~ m1
D A m ( x  a) m
f
150
CHAPTER
7 QUASIHOLOMORPHIC
MAPS
for x C B(a, p). Since lira sup]]mA,~[] 1/m  l i m sup[[Aml[ 1/'~, the radius uniform convergence of the power series ~~,~0r m A m ( x  a) "~1 is equal to p. By repeating this argument, we conclude that D f is quasidifferentiable on B ( a , p ) and (x)
(x)
d2 f ( x )  ~
m ( m  1)Am(x  a) m2 = ~
m=2
d2Am(x  a) m
m=2
where the radius of uniform convergence is equal to p, see the definition of d "~f in subsection 6.2.1, p.107. Notice that
A2  ~.1 d2 f(a). Inductively, B(a, p) and
we conclude that
f
1 Am = mi. dmf(a),
(7.39)
is infinitely quasi differentiable on for m  O, 1, 2, ...
(7.40)
m T h e o r e m 75 [3~] Let f be as in the above Theorem oo
oo
dk f ( x )  ~
dkAm( x  a) m  ~
m=k
and
oo
dAkPm(x  a)  ~
d^k f (x) mk
where P ~  f l
C o r o l l a r y 12 Let f
dkAm+k( x  a)m+ k
m=0
(:x)
for x C B ( a , p )
7~. Then
dAkPm+k(x  a)
m=0
m. II
be as in Theorem 7~ with
ldAk~ k~ (.dAmf)(a)
Profire,
ldAm( 1  ~. (k + m)' dAk+mf(a)) .
P r o o f . In the notation of the preceding Theorem 74 with have 1 d/~k(d/~ m 1 1 kU f)(a)  dAmPk+,~ = m! dAm((k+m)! m
we have
Pm
dAk+mf(a))"
Am, we
(7.41)
7.3 Q U A S I  A N A L Y T I C M A P S
151
C o r o l l a r y 13 Let f be as in the Theorem 7~ and a C U. Then for any k, m, we have the following relations between Taylor's expansions Tk,d^mLa  d Am [Tk+m,L~] 9
(7.42)
m
7.3
QUASIANALYTIC MAPS
The theory of holomorphic functions based on complex quasidifferentiability is called here Q u a s i  H o l o m o r p h y . It refers to the extension of Cauchy viewpoint. The theory based on power series is considered here as extension of Wierstrass viewpoint and is called Q u a s i  A n a l y t i c i t y . The view points are equivalent for functions for one complex variable. This part is devoted to the theory of quasianalytic functions based on the wierstrass approach and to study the p r i n c i p l e of quasianalytic continuation. As we noticed in our study of one variable, that the Weirstrass view point of analytic functions is more general than the Cauchy view point. It unifies both real and complex cases in development. As usual E and F will denote pnormed and qnormed spaces over the field K respectively (0 _< p, q __ 1), and U will denote a nonempty open subset of E. We will assume that F is always complete, i.e. a qBanach space.
7.3.1
Q u a s i  A n a l y t i c and Q u a s i  H o l o m o r p h i c M a p s
D e f i n i t i o n 48 ( T a y l o r s e r i e s ) W e say that a function f:U~F has a Taylor Series expansion at a point a C U if there exists a power series O0
cO
( x  a) = 119,'0
( x  a) rrt o
from E to F about a such that the power series converges to f uniformly on the ball B (a, r) C U for some r > O. The series, if it exists, is unique by Theorem 72 (subsection 7.2.~, p.1~6).
152
CHAPTER 7 QUASIHOLOMORPHIC MAPS
It deserves to point out that if f has a Taylor Series expansion (or representation) at a, then the function f is infinitely quasidifferentiable in a neighborhood of a as shown in Theorem 74(subsection7.2.7, p.149). Moreover we have the following relations:
Pm = m!l~ f (a)
,
Am  ~dm
(a)
for m = 0, 1, 2 ..... Consequently, if f has a Taylor Series expansion at a, then 1
f (x)  ~
~
1
~..dmf (a)(x  a)  ~
lrt O
m
~.dmf (a)(x  a) m O
in some neighborhood of a in U. Since the open ball of uniform convergence of the Taylor Series at a and the domain of this function f could be different, we usually write OO
f (x) _ ~
~1. dAmf (a) (x  a )  ~
1 ~..dmf (a)(x  a) m
?Tt" O
to indicate formally, the relationship between f and its Taylor Series expansion of a. D e f i n i t i o n 49 A function f : U ~ F
is said to be quasianalytic(or Bayoumi analytic) on U if f has a Taylor Series expansion at each point a E U. If K = (~, a quasianalytic mapping will be specially called quasiholomorphic(or Bayoumi holomorphic). It is clear that any linear combination of quasianalytic mappings is quasianalytic, and hence we have a v e c t o r s p a c e of all q u a s i  a n a l y t i c m a p p i n g s f r o m U t o F . This space will be denoted by QA (U; F ) . In caseK =(Y, we write QH (U; F) instead of QA (U, F ) . For simplicity, we denote QA (U) = QA (U;K) and QH(U) = QH(U,(T). Since QA ( U ; F ) is a vector subspace of Qc$, ( U , F ) , the space of infinitely q u a s i  d i f f e r e n t i a b l e m a p p i n g s from U to F, we have for f C QA (U; F) the following quasidifferentiable mappings.
din f : x e U ~ d ~ f (x) e L~ (mE; F)
(7.43)
d m f " x E U ~ dmf (x) e P (mE, F)
(7.44)
We also have from Ch.6(sec.6.2 ), the following quasidifferentiable operators
7.3 Q U A S I  A N A L Y T I C M A P S
153
rim: f E QA (U; F) ~ d ~ f E QA (U; L~ (mE; F))
~"
f E QA (U; F ) ~ ~ f
E QA (U; P (mE; F))
R e m a r k 18 (Local Property)QuasiAnalyticity is a local property in the following sense: If f E QA ( U ; F ) and V is a nonempty open subset of U; then f i y E QA (V, F) conversely if U   U V~, where V is an open nonempty subset of aEI
oc
U and f : U , F is such that for each a E I, fiv~ E QA ( V ~ , F ) , then f E QA(U; F). The following theorem gives a condition on a power series to be a Taylor series to its sum.
T h e o r e m 76 Let ~ Pm ( x  a) be a power series between locally bounded Fspaces whose radius of uniform convergence r is positive and f (x) be its sum for II x I1< r. Then f is quasianalytic in B (a,r). Proof. This is a restatement of Theorem 74 (subsection 7.2.7, p.149). I 7.3.2
Principle of QuasiAnalytic Continuation
To solve the problem of q u a s i  a n a l y t i c c o n t i n u a t i o n means: " G i v e n a q u a s i  a n a l y t i c m a p f in a connected open set U a n d a connected open set V c o n t a i n i n g U, find a n a n a l y t i c m a p g o n V s u c h t h a t giu = f ". The following theorem guarantees that such a mapping g is u n i q u e if it exists. This is the principle of analytic continuation in its s t r o n g form.
T h e o r e m 77 ( s t r o n g form)[34] (2002) Let E and F be a pnormed and a qnormed spaces respectively, (0 p, q <_ 1). Let U be a connected open subset of E, f E QA (U; F) and a E U; The following are equivalent : (i) d m f (a)  0 Vm = 0, 1, 2 ..... (ii) f is identically zero in a neighborhood of a. (iii) f is identically zero in U.
154
CHAPTER 7 QUASIHOLOMORPHIC MAPS
P r o o f . It is obvious that (iii) =~(i). It remains to show that (i) =v(ii)=~(iii) (i)=~(ii). Assume (i) is satisfied. Since f E QA (V; F ) , the Taylor series at a converges to f uniformly on some neighborhood of a. Thus f is zero identically in this neighborhood. (ii) =v (iii): Let W = {x E U; f (x) = 0} Then by the assumption (ii), W is nonempty and open since the Taylor series of f at each x C W represents f in a neighborhood of x. Furthermore W is closed since f is continuous. Thus W  U, since U is connected and W is a closed and open subset of U. This shows that f is identically zero on U. I The following theorem gives the weak form of the principle of quasianalytic continuation. T h e o r e m 78 (weak form)[34](2002) Let U be a connected open set and f,g ~ QA (U) . If f and g are identical on a nonempty open subset of U, then they are identical on U. P r o o f . Let V be a nonempty subset of U and assume that f and g are identical on V. It follows then from the preceding Theorem 77 that f  g, onU. I C o r o l l a r y 14 Let U and V be two connected open subsets of E, and f C QA (U; F ) . Then there is at most one g E QA (V; F) satisfying f  g on UNV. I R e m a r k 19 If f, g E QA (U), where U is a connected open set i n k and if f  g on an infinite set A with an accumulation point, where A C U, then fgonU. The analogous statement for dim E >_ 2 does not hold. For example, let g  ~ 2 , f (Xl, x2)  Xl and g (Xl, X2)  0 Then f , g E QA (E) and f  g on { ( x l , x 2 ) ' X l 0}, but f # g. In general, if M is a vector subspace of E which is not dense in E, then we can find a bounded linear functional f satisfying I] f ]I 1 and f  0 on M , (see Ch.1, p.12).
7.3.3
Integral D o m a i n QA (U)
If f, g ~ QA (U) then the multiplication
f.g ~ QA (U) as it can be shown from multiplication of power series. Therefore, QA (U) is a ring.
7.3 Q U A S I  A N A L Y T I C
MAPS
155
T h e o r e m 79
QA (U) is an integral domain if and only if U is connected. P r o o f . It is clear that, if QA (U) is an integral domain then U is connected. Assume that U is connected and f , g C QA (U) are such t h a t f.g = O. If f ~ O. Then there exists a nonempty open subset V of U such t h a t f 5r 0 on V. Thus g must be zero on V. Since g is analytic, g must be zero identically on U by Theorem 77 on the strong form of analytic continuation. In what follows we give an organogram (schedule) which shows the generalized theorems in the book obtained by the author between 1979 (the year he has obtained his Ph.D.) and 2003 (the year of writing the book).
GenralizedTheoremsinourbook (1979.2003) ,,
CompleAnal x ysisi ,I
I
,
Schwatz Levip~oblem 'e~= i Symmetric Steinhaust Theorem Theorem I _L_
!
~vex ~es [/I  Separable T.V.S
,,
I
Boundingi Sets t ! ! I
t
!
I
I
,
I
,
,
.os i1..msI .anva.ue,,
Lagrange IFund, an oe~,llF,~o,,s a, I ,n,eg~,i' i on~ MeanValue nterrTled~I Theorems ofCaculus Theorems[ V~ Thee
ied I We dy'
_
hn' lach
.
.
.
.
1
Realani l Cmplex ! =t~!
bdedSpaces~
....
....
............ 1.
.
.
I .
!
....
Krien . Milman Milman 1 Theorem
J
.
!
l e,s II; , s, ' B~
,
FunctionalAnalysis
!
,,
]
.
1 ........
FSpaces i S:~:s~ ! .actor ! Spaces I
Chapter 8
NEW VERSIONS THEOREMS
OF MAIN
This chapter is devoted to v e c t o r  v a l u e d continuous or quasi h o l o m o r phic m a p p i n g s . The reader should be familiar with the two equivalent ways of finding a holomorphic mapping, the one based on complex q u a s i d i f f e r e n t i a b i l i t y , and the other based on convergent p o w e r series. We point out here that the concept of vectorvalued holomorphic mappings arises naturally, as in the study of Banach algebras as well as in some other contexts. We have already enlarged the classical definition of holomorphic mappings from complex to vectorvalued ones in locally bounded Fspaces. We use some of the new concepts introduced here, as the concept of quasidifferentiability, to present more generalized fundamental theorems in complex and functional analysis such as : F u n d a m e n t a l Theorem of Calculus, Integral M e a n  V a l u e Theorem, and B o l z a n o ' s I n t e r m e d i a t e  V a l u e Theorem.
8.1
FUNDAMENTAL CALCULUS
THEOREM
OF
In this part we extend one of the most celebrated theorems of mathematical analysis, the Fundamental Theorem of Calculus~ to vectorvalued functions of pBanach spaces using the concept of quasidifferentiability. Hence, we have established a new general form of the fundamental theorem 157
158
CHAPTER
8 NEW
VERSIONS
OF MAIN
THEOREMS
of calculus suitable for applying to the integral forms of continuous vector valued maps of locally bounded Fspaces, not necessarily locally convex. In fact, using the curvilinear integrals, the theory of vectorvalued holomorphic mappings, like that of complexvalued holomorphic mappings can be developed most efficiently. We first need to generalize the Riemann Integral to vector valued functions. 8.1.1
Riemann
Integration
on [a,~]
D e f i n i t i o n 50 ( S t e p M a p s ) . Let I = [a, ~] be a closed interval in ~, and E be a pBanach space (0 < p < 1) over K . A s t e p m a p
f.
E
is a map for which there exists a partition P. aand elements Y l , . . . , Y n E E
ao <
0~1 < . . .
~
~n
 ~,
such that
f
n ~ yiX(OLi_l,OG)
(8.1)
1
where >CA is the characteristic function on the set [a,/3]. Let S(I, E) be the v e c t o r s p a c e of all s t e p m a p s from I to E. Then S ( I , E ) is a v e c t o r s u b s p a c e of the vector space B ( I , E ) of all bounded maps from I to E. Having the psup norm, [[fl[  sup
{[f(x)[[
xEI
on the space B ( I , E ) , B ( I , E) becomes a p  B a n a c h s p a c e . D e f i n i t i o n 51 ( R e g u l a r F u n c t i o n s ) L e t R(I, E)  S(I, E) be the closure of S ( I , E ) in B ( I , E ) . A map in R ( I , E ) is said to be regular. Since a continuous map f " I ~ E is uniformly continuous, we can conclude that
C(I, E) C R(I, E)
(s.2)
8.1 F U N D A M E N T A L
THEOREM
OF CALCULUS
159
Definition 52 (Integra 0 We define the integral of a step map f " [~, ~] , E,
n
f  ~
yi)~.(C~i__l,(~i ) 1
by
(8.3)
f  ~ ( o ~ i  oLi1)Yi. 1
It is easy to see that the integral
J Z.S(I,E)~E is linear and
II
E
fll 1/p <_ IIfll(~
~).
(8.4)
Note that IIT(x)lll~q < IITll.llxll~p for any T C L ( E , F ) being E , F pnormed and qnormed spaces respectively, see Chapter 1,( section 1.1, p.5) Hence f ~ f is continuous. In fact, 1/p
 II
n
n
1
1
~~(~ ~l)y~ll 1/" <__~~ I~
_< c m a x
l
lly~llI/p < cllfIl~/p
= (9  ~). So ,,,~ can ~zt~nd f [
~11 .lly~ll1/~
to ~ f~,~t~on on th~ clo~,~ R ( I , E ) , of S(• E), by the linear extension,, Theorem 2, and Theorem 3 (see section 1.1, p.5, 6). H ~ ~ if f ~ R(I, E), ~ c ~ d ~ ~
f
~f
and call it the integral o f f. D e f i n i t i o n 53 If a < c < d <_ 19, we define
idf
 /d~ f
160
CHAPTER
8 NEW VERSIONS
OF M A I N T H E O R E M S
Then for any three points c, d, e in any order, f C R ( I , E), we can prove easily that
8.1.2
lying in I = [c~,/3], and
Curvilinear Integrals
D e f i n i t i o n 54 ( p a t h ) . A p a t h in the complex plane~T is a continuous mapping =
(8.5)
of closed interval I of ~ (not reduced to a point) into (~, which is piecewise continuously differentiable. A path ~ : I ~ (~ is said to be contained in an open set U C (~ if ~/(I) C U, where U may not be convex. The point ~(c~) is called the initial point of the path ~/ and 7(fl) the terminal point. If ~(tl) ~ ~(t2) unless tl = t2 or tl and t2 are the same as c~ and ~, ~ i s said to be s i m p l e . If ~ : I ~ (~ is a path and ~/(~) = ~/(~), then ~/ is said to be closed. If a point a lies in the image of'~, we often say that a is on the path ~.
Let 7 : I ~ ~T be a path and let f :~(I) ~ E be continuous, where E is a complex pnormed space. Then the complex composed function t e I ~ f ( 7 ( t ) ) 7 ' ( t ) e E
is piecewisecontinuous and hence its integral exists on I by the definition of integral. E x a m p l e 28 As a particular case is the Arc segment A b  { A1/Pa + (1 joining a and b in(~.
/~)I/pb);
1 > A > 0}
(8.6)
Let us call it here a p a t h o f t y p e Ab.A
D e f i n i t i o n 55 (Curvilinear Integral )The curvilinear integral of f over the path ~/ is defined by ]~ f (z)dz  / ~ ~ f ( 7 ( t ) ) ~ / ( t ) d t . Note that the curvilinear integral above is an element of E.
(8.7)
8.1 F U N D A M E N T A L
THEOREM
OF CALCULUS
161
The following theorem will show t h a t T(f~ f ( z ) d z )  f~ T ( f ( z ) ) d z for every T C L(E, F), t h a t is the c o m m u t a t i v i t y between the curvilinear integral and continuous linear maps.
T h e o r e m 80 [22](1999) Let E and F be pBanach and qBanach spaces respectively (0 < p, q <_ 1) and T E L ( E , F ) . If ~ : I = [a,/3]+(T is a path and f : "y(I) + E is continuous, then
Proof. P u t
gn = fn O')' for some f~ C S ( I , E ) , where f = l i m f~. Let g = lim gn be the uniform limit of a sequence (gn) of step maps. T h e n each T o gn is a step m a p from I into F. So the sequence (T o gn) converges to T o g and hence Tog C R(I, F). Using the definition of the integral for a step m a p g~ we have
Tgn
T(]a gn).
i.e. f~ T f n  T(f~ fn). Since the integral is continuous, taking the limit we will get the required identity. This completes the proof, m
8.1.3
Fundamental
Theorem
of Calculus
T h e o r e m 81 ( F u n d a m e n t a l T h e o r e m o f Calculus)[g2] (1999) Let 9/" I  [0, 1] , ~ be a path of type A b, i.e. "y(I)=A b, a, bc4T and let f: 7(Z)+F br ~onti~o~,~ at a p o ~ t "r(t) of'r(1), wnr162 F i~ a pSa~a~h ~p~r the map
H(t) 
f(z)dz J a
Then
f(y(A))y'(A)dA 0
is quasidifferentiable at t and H'(t)  f(tl/pb + (1  t)l/Pa). [ (tl/plb  (1  t)l/pla)] P i.e.
H'(t)
f(9/(t)) "y'(t).
(8.9)
V
V
i_~
m
9
~
~"
I~
9
~
~ ~
o
m
~
r~
..
.
I
~
o
c::~
~.
.,"~
~"~"
9~
~
~
.
I
I
I
I
I
~
~'~

.
9
1~o
i,~ o
~"
~
~ T"
o .
Oo ~
~
~
Oo
I
~
~ ~ ' 1 ~ ~
"J
~
o~
II
~
~
~ ~
~
t::;I ~
~ ~
~
~
I
I
~
+
"<..
~
IA
I ~.,
~
+
+
.2
II +
q.
R
I
2
II
I
4
4
O
rh
IA
IA
o
..,o
I
k~
II
~<
d~
O O
t~
= t~ 9
9
9
t~
L'~
8.2 B O L Z A N O ' S
INTERMEDIATE
THEOREM
163
and V(I)" [a,/3] ~ (T be a path of type A b, a, b e (T, I  [0, 1] C ~, i.e.
V(I)  {x C (T; x  tl/pb + (1  t)l/Pa,
t E [0, 1]}.
Set H(t)   / ~ f (z)dz  /: t
f(7(A))v'(A)dA,
?(t) =
e ?(z) e r
OL
Then using the preceding theorem, we have H'  f ( v ( t ) ) v ' ( t ) ,
v(t)  x e r
= f(tl/Pb + (1  t ) l / p a ) . ( l t l / p  l b 1(1  t ) l / p  l a ) P P  (tl/pb + (1  t)l/pa, 0 , . . . ) . p1( t l / p  l b (1  t ) l / p  l a ) which is an element of 1p. Now if p  1, f ( x )  (x, 0 , . . . ) e 11/2, then for x  x(t) e 7 ( I ) ,
H'(x) 
(t2b + ( 1  t) 2a, 0 , . . . ) . 2 ( ( t b  (1  t)a))
i.e. H ' ( x ) = ( x x ' , 0 , . . . ) .
8.2
BOLZANO'S
INTERMEDIATE
THEOREM
In this part we extend the intermediatevalue theorem (or Bolzano's theorem) to locally bounded Fspaces which are not necessarily locally convex. In its simplest form it says t h a t : A r e a l  v a l u e d c o n t i n u o u s m a p f o n a c l o s e d i n t e r v a l [a, b] s u c h t h a t f(a) a n d f(b) h a v e d i f f e r e n t s i g n s , h a s a z e r o in ( a , b ) .
8.2.1
Finite Dimensional
Spaces
Shih [192] extends the intermediate value theorem to the complex plane ~T. He first remarked that Bolzano's theorem can reformulated as follows : Assume a < 0 < b and f(a) < 0 < f(b) and consequently the condition f (a). f (b) < 0 becomes
x. f (x) > O,
for x E OU
(8.12)
164
CHAPTER
8 NEW
VERSIONS
OF MAIN
THEOREMS
where OU denotes the b o u n d a r y of U  (a, b). T h e n f has at least one zero in U. He then proved t h a t " I f U is a b o u n d e d d o m a i n o f (T c o n t a i n i n g t h e o r i g i n , a n d
(s.13) is a n a l y t i c in U a n d c o n t i n u o u s o n Re z  f ( z ) then result. If closed
> 0,
U and such that for z e / ) U
(8.14)
f h a s o n e z e r o in U. He has used Rouche's theorem to prove that It says that : f and g are each functions which are analytic inside and on a simple contour C and if the strict inequality
If(z)
g(z)]<1 f ( z ) l
for each z e C
then f and g must have the same total number of zeros (counting multiplicity ) inside C.
8.2.2
Degree
Theory
Shih [193] extends then his above result to analytic functions in d~ n. His approach is an application of degree theory. Therefore it deserves and fruitful to discuss the degree theory first. We point out here t h a t all the results written here in this section have been obtained from the book by Smart [200] of fixed point theorems. Consider a simple closed curve C in ~2 and a m a p p i n g
g . C~ S 1
Figure 3" A m a p p i n g
of d e g r e e
1 of C o n t o
S1
8.2 B O L Z A N O ' S
INTERMEDIATE
THEOREM
165
In simple cases the following facts are clear from the above diagram. L e m m a 15 Let g : C ~ S 1 have the above figure. Then (1)Most points of S1 are covered a finite number of times nl in the pointwise sense, and a finite number of times n2 in the negative sense. (2) The number n   n l n2
is the same at all points where nl
and n2 exist.
g(x)

f(x)
II f(x) II
for a nonzero vector field f on C, then the rotation of g about C exactly n.
is
D e f i n i t i o n 56 The number n given in the above lemma is called the degree o f the m a p p i n g f.
In what follows we mention the main properties of the degree( for mapping of sn). L e m m a 16 ( P r o p e r t i e s o f Degree o f M a p s ) (1)The degree is an integer.
(2)The degree is unaltered by continuous deformation(homotopy) of the mapping. deg ( S T ) degI
1 ,
deg S . deg T
deg(I)
(I)n+l
The following lemma discusses the degree of a mapping f without fixed point. Lemma (1)If
17 ( F u n c t i o n s w i t h o u t f i x e d p o i n t s ) f .S~,
S~
and has no fixed point, then deg f  (  1)n+l. (2)If (  f )
has no fixed point, then deg f 
1.
166
CHAPTER
8 NEW VERSIONS
OF MAIN THEOREMS
T h e D e g r e e for m a p p i n g s of o p e n sets : Consider the open subset M of a pBanach space F and a continuous m a p p i n g f of M into F i.e.
f:
M*F
First ( b o u n d e d o p e n sets) :For the m o m e n t assume that M bounded and t h a t F = ~n. We illustrate the particular case in ~2.
is
i/~ /,., .//
f,' ,',,'/
If"tl 11
t
/
~ ~ \
Figure 4" A m a p p i n g of M b y f . region is s h o w n .
T h e v a l u e of d e g r e e in e a c h
We observe in this particular case the following : The images of OM i.e. f ( O M ) divides F into (connected) regions u~; in each region the algebraic number of times that a point is covered is constant (except of relatively few exceptional points where the image is folded ). This is true for all simplicial mappings, or differentiable mappings. In b o t h these cases the exceptional points in a region ui form a set of lower dimension, hence of measure zero.
D e f i n i t i o n 57 Let f of M into ~n i.e.
be a simplicial mapping (or a differentiable mapping) f :M~
~n.
The degree of f (with respect to M ) at a point x E ~  f ( OM ) is the algebraic number of times (almost all) points are covered, in the region ui containing x. This integer is written
8.2 B O L Z A N O ' S
INTERMEDIATE
THEOREM
167
deg(f , M , x ) or deg(f, M, ui). The degree of an arbitrary, continuous mapping is defined by approximating by simplicial mapping (or differentiable mapping). S e c o n d ( u n b o u n d e d o p e n sets): If M is unbounded open set we must assume that f ( x )  x is bounded on M; in other words, ( f  I) is compact. If F is infinite dimensional, we must approximate f by finite dimensional mappings; this requires some compactness assumptions. The usual theory defines the degree for mappings f such that ( f  I) is compact. The main properties of the degree are given in what follows 9 L e m m a 18 ( M a i n properties o f the degree) (1)deg ( f , M, x) is an integer, defined if x E f (OM). then deg ( f , M , x )  i if x E M, and deg ( f , M, x)  O (2)If f  I if x ~ M .
(s) deg (f, UMi, x) 
deg ( f , M~, x) i
if Mi are disjoint regions and both sides of the equation are defined. (4)If deg (f, M , x ) r O, then x e f ( M ) . (5) ( H o m o t o p y ) : If f (x, t) is a continuous function compact mapping of [0, 1] x M into F, that is f ( x , t ) ' [ O , 1] • M ~ F and if x
f (x,t) ~ y for all x C OM
then deg(I  f(O, .), M, y)  deg(I  f(1, .), M, y) [We say here f(O, .) is fphomotopic to f ( 1 , . ) ] . (6) (Multiplication)" If f (OM) divides F into regions u~ and if g( f (OM) ) divides F into regions wi then deg(gf , M, wi) 
deg(f, M, ui). deg(9, ui,, wi). i
168
CHAPTER
R e m a r k 20 If f
8 NEW
VERSIONS
has no zeros in M
deg(f, M, 0)  ~
OF MAIN
except
Xl , .., Xn,
THEOREMS then
deg(f, Mi, 0) Also if f is a homeomorphism
where the Mi is a small neighborhood of xi. then it has degree •
In fact, Shih[193] has obtained the following extension of Bolzano's theorem to (T n. T h e o r e m 82 [193] ( 1 9 8 2 ) Let ~
be a bounded domain in (T n containing the origin. Let f : f2 . 6~ n
be analytic in ~
and continuous
in ~  , and assume that
Re z  f ( z ) > 0 , Then
f
for z E 0 f t
has exactly one zero in ~.
The proof depends on the following three lemmas. For the first lemma see the book of Bochner & Martin[48] L e m m a 19 Let
f = (fl,,.., be analytic in a domain ~ of (T n are related by the formula
fn) : ~
~
(T n
Then the real and complex Jacobians
O(uj,_vj) ) _[ det(0fJ 2 det(o(Xk,Yk) ' ~zk) [ , where zk = xk + i yk
and
(8.15)
(8.16)
f j = uj + i vj.
R e m a r k 21 Notice that f :(T n __,(T n can be considered as a map f :~2n
~
j~2n.
Using degree theory, we see that the local degree of a complex analytic function at any preimage of a regular value is always +1. Thus the number of points in the preimage of a regular value of f is always exactly deg f.
8.2 B O L Z A N O ' S
INTERMEDIATE
THEOREM
169
Lemma20 Let A be an ( n x n) complex matrix and d e t A = for sufficiently small e > 0, det(A + eI) r 0
0. Then
(8.17)
where I is the identity matrix.
Proof. By Schur's theorem, there is a non singular matrix P such that p  l A p is triangular and the leading diagonal elements of p  l A p are eigenvalues of A. Since det A = 0, for small enough e > 0 we have
det(A + eI)  d e t ( p  l ( A + e I ) P )  d e t ( p  1 A p
+ eI) 7s O.
(8.18)
m
Lemma
21 Let U be an open bounded set in(F ~ and let f, g : U  , (~ n
(8.19)
be two continuous maps. Let w C ~T n and
(8.20)
w ~ f ( O U ) u g(OU). A s s u m e further that e satisfies
0 < e < min
{llf(z) ~11;
c ou}.
(8.21)
all z C OU,
(8.22)
z
If I l f ( z )  g(z)ll < e for then
deg(f, U, w) = deg(g, U, w).
(8.23)
P r o o f . Define the homotopy H : U  x [0, 1] , Cn
(8.24)
by H(z,t) :=(1t)f(z)+tg(z),
forzEU
andtE
[0,1].
(8.25)
170
CHAPTER
8 NEW
VERSIONS
OF MAIN
THEOREMS
By assumption, it is easy to see that H ( z , t ) ~ w for z C OU and t E [0, 1]. By homotopy invariance theorem the result follows, m Proof
of T h e o r e m 82, Define the homotopy H " ~  • [0, 1] ~ 07 n
by H(z,t)'(1t)z+tf(z),
forze~
andte[0,1].
By assumption, H ( z , 0) # 0 and H ( z , 1) ~= 0, for z e / ) ~ and Re z  . H ( z , t ) > Re z.(1  t)z > O,
(s.26)
for z E Ogt and t E (0, 1). By the homotopy invariance theorem, deg(f, ~, 0)  deg(H(z, t), fl, 0)  deg(I, ~, 0)  1.
(8.27)
Notice that f~(0) is a compact subvariety of ~t. We claim that f  l ( 0 ) is finite. For, let M be a component subset of f  l ( 0 ) . Then M is a compact subvariety of ~. Since each projection ~ j ( z l , .., .Zn)  zj is analytic on M, ~j is constant by applying the result of Gunning&Rossi[[ll7],pl06]. So M is a singleton; thus f  l ( 0 ) is discrete and hence finite. Now, let ~1,, ~k denote the zeros of f. Let Aj be a neighborhood of ~j such that the closed sets A~ are pairwise disjoint and Aj C ~. Let
So K is a closed subset of ~t which does not contain a zero of f. By the excision and additivity properties of the degree, Schwartz [[188],p.86] we have deg(f, ~, 0)  deg(f, ~  K, 0)  ~
deg(f, Aj,0).
(s.29)
J We claim that deg(f, Aj, 0) _> 1,
for each j.
(8.30)
If deg(J G (f)) =fi 0, where J~j (f) denotes the complex Jacobian matrix of f at ~j, then deg(f, Aj,0)  1 by Lemma 19. Suppose now that d e t ( J ~ (f))  0.
(8.31)
8.2 B O L Z A N O ' S I N T E R M E D I A T E
THEOREM
171
Then by Lemma 20, det(JCj (g)) # 0 when
g(z)e(z~j)+f(z),
and e > 0
is small enough.
Consequently it follows from Lemmas 19 and 21 that deg(f, Aj, 0) = deg(g, Aj, 0) _> 1.
(8.32)
By (8.27) and(8.29), the theorem is established, m 8.2.3
Infinite Dimensional
Spaces
In infinite dimensional spaces Wlodraczyk[208], obtained the following extension of Bolzano's intermediatevalue theorem for h o l o m o r p h i c m a p s in certain c o m p l e x B a n a c h spaces. B a n a c h Spaces of C o n t i n u o u s Linear M a p s
Let E , F be complex Hilbert spaces; let L(E; F), as before, denotes the Banach space of all continuous linear maps A : E ~ F with the mapping 12orm
IIAII sup IIA(x)ll.
(8.33)
L1 (E, F) C L(E; F)
(8.34)
Ilxll_
Let
be a closed linear subspace of L(E; F). T h e o r e m 83 [208](1991)
Assume ft C LI (E, F) is a bounded domain such that 0 C f~ and let f : A e ~  ~ f (A) e LI (E, F)
(8.35)
Re {A* f ( A ) } > O, for A EOFt
(8.36)
be continuous in ~  . (1) /f
172
CHAPTER
8 NEW
VERSIONS
OF MAIN
THEOREMS
and ( I  f)(f~) is contained in a compact subset of L(E; F), where A'is the conjugate of A, then f has at least one zero in ft, i.e. f (A)  0 for at least one A E ft.
(2) If additionally f is holomorphic in f~ and D r ( A ) inverse for A E ft, then f has exactly one zero in f~. Proof.
has a bounded
To prove (1), let
ht(A)  A 
f(t,A),
where f (t, A)  t [A  f(A)] ;
i.e. ht(A)  ( 1  t)A + t f ( A ) ,
AEft
and0_
(s.37)
Then
ho(A)  A ~ O, hi (A)  f (A) r O,
for A E 0ft.
(8.38)
Moreover,
ht(A)r
for A E 0 f t
and 0 < t < l ,
(8.39)
because
Re {A*ht(A)} 
Re {(1  t)A*A + tA* f ( A ) }
> (1  t) Re {A'A} > 0. Since 0 E ft, we apply the homotopy property to ht(A), see L e m m a 18 (5) or Smart[[200], 10.3.7], to obtain d e g ( I  f(0, .), ft, 0)  deg(I  f(1, .), ft, 0)  1 or equivalently deg(f, ft, 0)  deg(I, ft, 0)  1. Hence it follows that
fl(0)
(8.40)
is nonempty.
(2) Since
A , D f ( A )
(8.41)
8.2 B O L Z A N O ' S I N T E R M E D I A T E
THEOREM
173
is a holomorphic map of ft into L(L1; L1), see Nachbin[[?l,props.3,p.29], by the inverse map theorem for complex Banach spaces, see Rudin [187], the map f is locally biholomorphic in ft, i.e. for each A C f~, there exists a neighborhood UA of A in ft such that f ( U A ) = VA,
is open in L1,
(8.42)
f  1 exists and is holomorphic in VA. We now prove that f  l ( 0 ) contains one m a p p i n g : Let us first show that f  1 (0) is finite. Towards a contradiction, let Ak E f  l ( 0 ) , k = 1,2,... We have h(Ak) = Ak, k = 1,2,...,where h = I  f . Since h is compact and continuous in f~, there exists a subsequence of (Ak), say (Ak), and A C a  such that
IIAAkII~0
as k ~ c ~
and
h(A)=
A.
(8.43)
This yields f ( A ) = O and
Re{A*f(A)}=0.
9
(8.44)
Consequently, A C ft. But f is biholomorphic in UA and At: c UA, for k sufficiently large. This yields a contradiction. Thus f  l ( 0 ) is finite. If (8.45)
f  ~ (0)  {A1, ..., An} n
deg(f, ft, 0)  deg(I, f t / U , 0)  ~
deg(f, Uk, 0),
(8.46)
k=l
where Uk are all small neighbourhoods of Ak such that the sets Uk are pairwise disjoint, U~ C ft and U  ft/U'~= 1Uk. Further we have f  I  h , h has isolated fixed points in Ft; h is compact and L1 (E, F ) is complex. Thus the multiplicity of each eigenvalue of Dh(Ak) = D(I  f)(Ak),
k = 1,2,...,n
is even and by Kransnoseliski [lemma 4.1 and theorem 4.7], this yields deg(f, Uk, 0) = deg(I  h, Uk, 0) = 1, From (8.40) and proof, m
( 8 . 4 7 ) w e deduce that
k = 1, 2, ..., n.
(8.47)
n = 1, which completes the
174
CHAPTER
8 NEW
VERSIONS
OF MAIN
THEOREMS
C o n t i n u o u s L i n e a r M a p s o n N o n locally c o n v e x S p a c e s Regarding the non locally convex spaces we have obtained the following result for the space lp, ( 0 < p < 1). The proof is similar to that given in the preceding theorem, but making use of our recent results here. Let
Ll(lp;K) C L(lp;K) ~ l ~ be a closed linear subspace of lo~. T h e o r e m 84 [30](2002)
Assume ft c L1 is a bounded domain such that 0 E f~ and let f " Ft ~ Ll(lp;K)
(8.48)
Re {A* f ( A ) } > 0, for A E 0gt,
(8.49)
be continuous in f t  . (1) If
and ( I  f ) ( ~ ) is contained in a compact subset of L(lp;K), then f has at least one zero in ~, that is f ( A )  0 for at least one A E ~. (2) If additionally f is quasiholomorphic in ~ and D r ( A ) bounded invese for A E ~, then f has exactly one zero in ~. Proof.
(1) Let
ht(A)  A 
f (t, A),
i.e. h t ( A )  ( 1  t ) A + t f ( A ) ,
where f (t, A)  t [A  f(A)];
AE~t
and 0 _ < t < l .
Then
no(A )  n # 0, h i ( A )  f ( A ) # 0,
for A E Oft.
M or eover,
ht(A) 7~0, since
has a
for A E 0 f t
and 0 < t <
1,
(8.50)
8.2 B O L Z A N O ' S
INTERMEDIATE
THEOREM
175
Re {A*ht(A)} = Re {(1  t)A*A + tA*f(A)} > (1  t) Re {A'A} > 0. Since 0 E gt, we apply the homotopy property to [[200],p.811, to have
ht (A), see Smart
deg(I  f(0, .), ~, 0) = d e g ( I  f(1, .), a , 0) = 1
(8.51)
or equivalently deg(f, a , 0) = deg(I, a , 0) = 1. Hence it follows that
fl(0)
(8.52)
is nonempty.
(2) Since
A ~ Df(A) is a quasiholomorphic map of ~ into L(lp;K), by the inverse map theorem for complex pBanach spaces, the map f is locally quasibiholomorphic in ~t, that is, f and f  1 are quasiholomorphic, see Theorem 46(Ch.4, subsection 4.1.5, p.85). i.e. for each A E ~, there exists a neighborhood UA of A in ~ such that
f(Un) = VA is open in Ll(lp;K), f  1 exists and is quasiholomorphic in
(8.53)
VA.
We now prove that f  l ( 0 ) contains one mapping : Let us first show that f  l ( 0 ) is finite. Towards a contradiction, let Ak E f  ~ ( 0 ) , for all k  1,2,...We have h(Ak)  Ak, k  1,2, ...,where
h=If. Since h is compact and continuous in ~  , there exists a subsequence of (Ak), say (Ak), and A e ~t such t h a t
ilAAkrl 0
as k ~ ~
and
h(A)A
(8.54)
This yields
f(A) = 0
and
R e { A * f ( A ) } = 0.
(8.55)
176
CHAPTER
8 NEW
VERSIONS
OF MAIN
THEOREMS
Consequently, A E ft. But f is biholomorphic in UA and Ak E UA for sufficiently large k. This yields a contradiction. Thus f  l ( 0 ) is finite. If f  1 (0)  {A1, ..., A n } ,
(8.56)
then by Smart[J200], propertiesl0.3.h&10.3.6, p.80)], we have n
deg(f, t2, 0)  deg(I, t2/U, 0)  ~
deg(f, Uk, 0)
(8.57)
k=l
where Uk is a small neighborhood of Ak such that the sets Uk are pairwise disjoint, U~ C f~ and U  t2/(U~=lUk ). Further we have f  I  h , h has isolated fixed points in t2; h is compact and L I ( E , F ) is complex. Thus the multiplicity of each eigenvalue of
D(h)(Ak)  D ( I  f ) ( A k ) ,
k  1, 2, ..., n
(8.58)
is even by Kransnoseliski[[164]Lemma 4.1 and Theorem 4.7], this yields
deg(f, Uk, 0)  deg(I  h, Uk, 0)  1 By (8.51), (8.57) and the proof. I
(8.59)
k  1, 2, ..., n.
we deduce that
(s.59)
n  1, which completes
R e m a r k 22 Every separable locally bounded space Ep is isomorphic to a quotient space of lp, (0 < p <_ 1), i.e.
Ep ~_ l p / M for some closed subspace M of lp. Therefore the above theorem can be applied to Ep whenever its dual separates the points of Ep. We consider a bounded domain f~ of a closed subspace LI(Ep;K) C L(Ep;K).
8.3
INTEGRAL
MEANVALUE
THEOREM
One of the most important theorems of mathematical analysis is the MeanValue Theorem for definite integrals. In the simple case it stats that, if f is a continuous function on an interval [a, b], a, b Eli,, then its integral satisfies
ffa b f dx  f (c).(b  a)
8.3 I N T E G R A L
MEANVALUE
THEOREM
177
for at least one c C (a, b). In this part we establish new general forms of the I n t e g r a l M e a n V a l u e T h e o r e m o f v e c t o r  v a l u e d f u n c t i o n s , by using the concept of curvilinear integral over the arc segment Aba in different spaces. R e a l  V a l u e d Functions"
The following theorem extends the classical MeanValue Theorem for integrals to curvilinear integrals of a realvalued functions on a path 7(I) of type A~ C (F ~_~2 i.e.on (1  t)l/pa + tl/pb; 0 < t <_ 1} Theorem
85 (Generalized MeanValue Theorem for Integrals in
~2) [31] (2000) Let f " 7(I) , 1~ be a continuous realvalued function on a path 7 in (F of the type Abe, where I  [0, 1]. Then, there is zo C 7 such that
/•
f ( z ) i dz I 
j['oI
f ( 7 ( ) ~ ) ) l T ' ( ~ ) i d ) ~  f(zo). I
to/Plb
(1 
p
to)l/pla
I.
(8.60) In particular, if p  1, and 7(I)  [a, b], we obtain the classical meanvalue theorem, thatis,
i
b f (z)dz
f (zo) (b  a).
P r o o f . Since f 0 7 is continuous, f o 7 C R ( I , ~ )
and the curvilinear integral
~ f (z) l dz [ exists. Let M
max {f(7(t)). I 7 ' ( t ) l }
tel
rn  min { f(7(t)). I 7'(t) I}
tCI
Then
rn <_ f(7(t)). I 7 ' ( t ) I _< M
(8.61)
Hence if J(t)  f(7(t)). I 7'(t) I, then by the intermediate value theorem for the function f ( 7 ( t ) ) l T ' ( t ) l o n [0, 1], there exists to C [0, 1] such that
J(t) f(7(t0)). 17'(t0) I
178
CHAPTER
8 NEW
VERSIONS
OF MAIN
THEOREMS
Consequently,
j/0
1 f(~'(t)).
"/(t) l d t 
f('y(to)). I 3/(to) I .(1  O )
= f[tlo/Pb + (1  to)l/Pa].
I t~
(1  to)l/Pla I P
for some zo  to/Pb + (1  to)a E ~/(I)  A~. Now if p  1 we get 7(I)  [ a , b], and we have b
fa
f ( z ) d z  f(tob + (1  to)a). [b  a]
=
f(zo)(b

(8.62)
a)
for some zo  tob + ( 1  to)a E I to obtain on the line segment [a, b] the classical meanvalue theorem for definite integrals. II ComplexValued Theorem
Functions
9
86 ( M e a n  Value T h e o r e m f o r I n t e g r a l s in r
(2000)
Let : " 7(I) r be a continuous vectorvalued function on a path 7(I) of the type Ab~ in a bounded domain U C (T, where I  [0, 1]. Assume II f fl is bounded on y(I). Then, there is zo E "~(I) such that
f llf(z)dzllfo
1
11
11
= IIf(zo)ll.II tg
b(1
p to)~
all .
P r o o f . Similar to that given for the preceding theorem. II
Remark
23 The above theorem can be extended to a vectorvalued function
f " 7(I) ~ E, from a path of arc type ?(I)
to a pBanach space E.
Chapter 9
BOUNDING AND WEAKLY B O UND IN G SETS A subset A of a topological vector space E is called b o u n d i n g if it is bounded for all entire function f on A. That is, bounded for all continuous and Gdteauxanalytic function.
If E (~n every entire function f on (~n i8 bounded on every bounded subset A
of E. T h a t is e v e r y b o u n d e d set i n (T~ is bounding.
If E is an infinitedimensional topological vector spaces, Dineen [86] showed that this is not the case by proving that : the closed bounding subsets of separable or reflexive Banach spaces are compact, and of course a bounded set in E need not be compact. T h a t is b o u n d e d s e t s are n o t necessarily bounding. An example of a bounding subset which is not compact is the set A= {ej;j E N} of the unit vectors in the Banach space l~, see Dineen [87]. Josefson [122] extended this result by showing that a subset of co is bounding as a subset of l ~ if it is bounded. He also proved in [122] that: the bounding subsets of any infinitedimensional locally convex space are nowhere dense. Moreover he proved that the bounding subsets and weaklybounding subsets in lcr are the same. Here a set A in a topological vector space E, with rich dual space E', is w e a k l y  b o u n d i n g if:
179
180CHAPTER
9 BOUNDING
AND WEAKLYBOUNDING
SETS
Cn(X) ~ O, x C E, implies suPn I Cn I ~ O, for every sequence (r E'. The weakly bounding sets are sometimes called l i m i t e d sets. By this he gives a negative answer to the question 9 D o e s there exist an i n f i n i t e  d i m e n s i o n a l B a n a c h space s u c h t h a t its u n i t ball is b o u n d i n g or e v e n w e a k l y  b o u n d i n g ? In this chapter we study and discuss in details the above problems in different classes of topological vector spaces E without having any type of convexity condition.
in
However, the i m p o r t a n t and fundamental results in the theory of holomorphy so far achieved by us ( in brief ) is as follows:
IThe concepts of bounding and weaklybounding subsets are n o t the s a m e , a n d a n e w class o f bounding sets has already appeared. IIThere are three d i f f e r e n t classes o f h o l o m o r p h i c f u n c t i o n s , contained in each other properly, which are found and which are same in locally convex spaces. I I I  M a n y properties of bounding and weaklybounding subsets have been studied. We have given the relations between t h e m , a n d the radius o f c o n v e r g e n c e o f h o l o m o r p h i c f u n c t i o n s a n d the L e v i problem. This study in non locally convex spaces shows a big d i f f e r e n c e and f a r f r o m that for locally convex cases. Consequently one of the points which has been highlighted is the following" "The t h e o r y o f h o l o m o r p h i c f u n c t i o n s in locally c o n v e x and n o n locally c o n v e x spaces are n o t the s a m e . "
9.1
BOUNDING
SETS
This part is devoted to the study of the bounding sets in different topological vector spaces which are not necessarily locally convex E. They are bounded by all holomophic functions on E. By a holomophic function we mean here c o n t i n u o u s a n d G ~ t e a u x a n a l y t i c . We also study the weaklybounding subsets of E. T h e y are bounded by certain type of holomophic functions. It is shown t h a t they are different from the bounding sets, cont r a r y to the locally convex cases. The study shows that: h o l o m o r p h i c p r o p e r t i e s and non locally convex spaces are different.
of l o c a l l y c o n v e x
In fact, the study of bounding sets is of a great interest. It makes a progress wherever we study it. T h e y arise in several problems in intl
9.1 B O U N D I N G
SETS
181
nite dimensional complex and functional analysis. For example in analytic continuation and in the construction of the envelope of holomorphy. It is known that the extension of the notion of the holomorplically convex domain to the infinite dimensional case leads to the notion of the bounding sets which was introduced by Alexander [1]. Indeed the bounding and weaklybounding sets have been studied for locally convex spaces by Dineen [86], [88], Josefson [123], Schottenloher [196].
9.1.1
B o u n d i n g Sets in L o c a l l y B o u n d e d F  S p a c e s
Definition 58 A subset A of a locally bounded space E is said to be bounding if l] f [IA sup If (X) I < OO xcA
(9.1)
for all holomorphic functions f on E. By a holomorphic function we mean a continuous and Gdteauxanalytic function.
Definition 59 A set U in E is said to be p o l y n o m i a l l y convex, if there exists a continuous polynomial P on E which separates the points of U and E/U. That is, for every Xo E E / U there exists P E L ( E ) such that
P (Xo) r P (u) T h e o r e m 87 [15] (1979) The closed unit ball Bzv of lp, (0 < p < 1), is polynomially convex. More precisely, for every point b ~ B~p one can find a contiuous monomial P E p ( m lp) such that sup [ P (x) I< 1 <__P (b).
(9.2)
xEBlp
P r o o f . The construction of P is a finite dimension problem. Obviously for every b ~ B~p there exists an n C N such that 7rn(b) ~ 7rn(BLp). We can assume that all bj > 0, j  1, ..., n by multiplying bj by complex numbers of modulus one and indeed that bj > 0 by ignoring the coordinates for which bi  0. Consider the mapping r __,/~ defined by ...,
where t 
(tl,...,tn) E R n. Let

...,
to)
(9.3)
182CHAPTER
9 BOUNDING
AND WEAKLYBOUNDING
/
B *  d/)l(Trn(Blp)) 
t e ~n; ~
n
eptj < 1
SETS
/
(9.4)
j=l
and b;  05l(b)
The set
 (log bl, ..., log
bn).
(9.5)
B* is closed and convex in/~n. This is because n
t + ~
e ptj
(9.6)
j=l
is a continuous convex function in JRn. Since b* ~ B*, it follows by the classical Hahn Banach Theorem that there exists a continuous linear functional T o n ~]~n such that sup T ( t ) < T(b*) tCB*
where T takes the form n
T(t)  ~ mjtj, j=l
t E ~n
where necessarily m j >_ O,j  1,...n, since point s with 8j < tj. But the hyperplane
/
n
t c ~n; ~ . ~ j t j
B*
 T(b*)
(9.7)
contains with
t
every
/
j1
can be moved so that all the m j become rational numbers, j  1, ...n. Thus there exists a positive integer m such that m m j  kj are all nonnegative integers. The monomial P(x)
kl xk2 .. kn  x 1 .x n
(9.8)
satisfies the following" n
P ( b )  rIjn=l.bknj  exp(~~, kjb;)  e x p ( m T ( b * ) ) ,
j=l
(9.9)
9.1 B O U N D I N G
SETS
183
and sup [ P ( x ) 1 
sup e x p ( m T ( t ) ) <
xEB
tEB*
P(b).
(9.10)
has the desired property. This completes the proof. I
So P I P ( b )
C o r o l l a r y 15 Let such that the sets D
E
be a metric vector space with S c h a u d e r basis (ej)
z E ~T'~ ; d( ~ e x p ( z j ) e j
, O) < r

j=l
= {z e
d(
...,
O) <
are convex. T h e n E has a polynomially convex unit ball. In particular, this valid if, d ( t x , O)  d(x, O) f o r t E 6~, I t I 1.
P r o o f . Notice the formula( 9.3 ) of the convex function r and (8.4) of the sets B*. I E x a m p l e 30 The spaces
l(p~) 
x   ( x j ) ; x j E (~, d(x,O) 
I xj Ipj< c~
,
l > p,~ > O
(9.12) have the property of the above Corollary. Notice that these spaces are locally bounded spaces if p~ /+0 otherwise they are not locally p s e u d o c o n v e x spaces, See Rolewicz[186] . If E is a separable locally bounded space, it is known that E is the image of lp for some (0 < p ~< 1) under a continuous linear mapping T. The following theorem asserts that the kernel of the mapping T is a subspace of lp which has some interesting properties. In fact E is isomorphic to a quotient space of lp for some (0 < p ~< 1), see Rolewicz [186] and Stile [202]. A.
184CHAPTER
Theorem
9 BOUNDING
AND
WEAKLYBOUNDING
SETS
88 [186]
Every separable locally bounded space E with a phomogenous norm Ilxll,(0 < p ~< 1), is an image of lp by a continuous linear mapping T. More precisely, there exists a continuous linear mapping T.l,
, E
(9.13)
i.e. T c L(lp; E), from lp onto E, with E ~ _ l p / M ,
where M 
kerT.
P r o o f . Let {Xn} be a sequence dense in the unit ball BE  {x; Ilxll < 1}. Let T be defined by O0
T(t)  ~
t~x~,
t(tn)
C lp.
(9.14)
n=l
Since the sequence {Xn} is bounded, the mapping T is continuous. To show that T maps lp onto E, let x C BE. Since {Xn} is dense in 1 BE, we can choose a subsequence {Xnk} and a sequence {tk} with Itk I< ~, and m
1
II ~ tkXnk, Xll < 2~. k=l
Let
~
!
tn 0
nk/
elsewhere.
The sequence {t'k} C lp and T({t'k} )  x. This completes the proof of the theorem, m It is not yet known which of the locally bounded spaces E with phomogeneous norms (0 < p < 1) have polynomially convex balls BE (x.r). However, the fact that E is isomorphic to a quotient space of lp (0 < p < 1) will help to overcome the difficulty arising from the absence of the polynomial convexity of the balls of E. The following theorem characterizes the bounding sets of a big class of separable locally bounded Fspaces.
9.1 B O U N D I N G
Theorem
89
SETS
185
[16](1990)
Let E be a complete separable complex locally bounded space which is isomorphic to a quotient space lp/M (0 < p < 1) with Schauder basis, for some subspace M of lp. Then, the bounding subsets of E are relatively compact. Proof. Since E is isomorphic to a quotient space lp/M with Schauder basis (ej), where M is a subspace of lp (0 < p < 1), it suffices to prove that, the bounding subsets of lp/M are relatively compact whenever lp/M has a basis. Notice t h a t a subspace F which is a Fr6chet space in the topology inherited from a topological vector space on E is closed, see Rudin [[187],p.20]. In fact we need the following l e m m a : Lemma
22 Let M be a closed subspace of lp. Then the balls of the quotient
space lp/M
(B are polynomially convex provided that lp/M has a Schauder basis. P r o o f . Consider the quotient m a p 7r 9lp , lp/M and assume lp/M has Schauder basis (ej) and 7r is a continuous linear mapping. It m a p s the Fr~chet space lp onto the Fr~chet space lp/M which is metrized by the metric (Trx, Try) 
inf d ( x ' , y ' ) ~'~(~) yleTr(y)
=
Since all the sets
inf ~ l x }  y ~ l x'~Tr(x) j = l
z e (~n; d ((e zl, ..., eZ"), O) < r
p.
are convex in C n r > 0,
n C N, it follows by Corollary 15, p.181, t h a t the balls/~ (0, r)  7r (B (0, r)) are polynomially convex in lp/M. m P r o o f . of T h e o r e m . We now start proving the theorem. Since the sequence spaces lp/M (0 < p < 1) have polynomially convex balls t h e n the bounding subsets of lp/M are relatively compact. Consequently the bounding subsets of E are also relatively compact. Notice t h a t E is assumed to be isomorphic to lp/M. This completes the proof of Theorem. m C o r o l l a r y 16 [16](1990)The bounding subsets of each complex complete lo
cally bounded space E with Schauder basis (ej) are relatively compact.
186CHAPTER
9 BOUNDING
AND WEAKLYBOUNDING
SETS
P r o o f . E is isomorphic to a quotient space lp/F for some subspace F of lp(, 0 < p < 1). The isomorphic T is isometric since E and lp/F are metrizable. Since x in lp/F is uniquely determined by T ( ( e j ) ) , lp/F has a basis. Now apply the preceding Theorem 89 to obtain the desired result. m R e m a r k 24 The assumption that E has a Schauder basis cannot be dropped out of the above corollary 19, Take for example, the space Lp [0, 1] of measurable functions on [0,1],(0 < p < 1). It is a locally bounded separable metrizable space (without Schauder basis). Therefore it is mapped onto a quotient space l p / M with the property that each continuous linear functional which vanishes on M should vanish on the whole space lp, Stile [202]. Hence subsets of L p [0, 1], which may be unbounded, are bounding. 9.1.2
Bounding
Sets in Separable
Metric
Spaces
D e f i n i t i o n 60 A metric vector space is said to have the b o u n d e d approxi m a t i o n property(b.a.p), if there is an equicontinuous family (r of linear mappings Ct : E ~ Ct(E), of finite rank such that for every ~ > 0 and every compact set K in E, there is a t E T with
d(r
x) < e
(9.15)
for all x C K. T h e o r e m 90 [15](1979) Let E be a complete separable complex metric vector with a translation invariant metric d which has the bounded approximation property and is such that the closed balls in E are polynomially convex. Then a subset of E is bounding if and only if it is relatively compact.
P r o o f . A relatively compact set is obviously bounding since any continuous function on a compact set is bounded. Assume that A C E is not relatively compact. Then A is not precompact, for its completion is equal to its closure in E. So there exists a sequence (xj) of elements of A such that
d(xj, xk) > 5/~ when j ~ k
(9.16)
for some suitable number ~. The proof will depend on whether or not (xj) stays close to a finitedimensional subspace. In fact we have two eases to study.
9.1 B O U N D I N G
SETS
187
Let fit  I (~t, where (d/gt)te T is the given family of linear mappings, and let B ( x , r) denotes here the closed ball of centre x and radius r. C a s e 1 . F o r s o m e t C T w e h a v e pt(xj) e B(O, 2A),for e v e r y j : In this case the sequence of points (yj), (yj) = r satisfies
d(xj, xk) <_ d(xj, yj) + d(yj, yk) + d(y , xk), j T/= k; i.e. d(yj, yk) > d(xj, xk) 
d(xj, yj) 
d(yk, xk) > 5)~  2)~  2)~  ,~, j 7s k.
(9.17) Hence (yj) is a subset of Et  r which is not precompact. Since Et is of finite dimension, we know t h a t there exists a linear form u on Et such t h a t (u(yj)) is unbounded. The linear form q = uoCt is a holomorphic function on E with (q(xj)) is unbounded. Hence { x j } is not bounding in this case. Note t h a t here the family of linear mappings need not be equicontinuous. C a s e 2 . F o r e v e r y t E T t h e r e is a j
such that
the orbit
pt(xj) r B(O, 2A) " By the triangle inequality for the translation invariant metric d we have B(0, A) + B(0, A) C B(0, 2)~).
(9.18)
Choose 8 > 0 such t h a t pt(B(0,8) )C B(0,2A) for every t C T; this is possible in view of the equicontinuity of the family (Pt). Define
Vn  I.jjn=lt~(Zj, (~) where {zj} is a countable dense set in T h e n for every t E T,
(9.19)
E; hence (Vn) is a covering of E.
flt(Vn) C ujn=lB(flt(zj), (~). Let now n be fixed. T h e n for some t e T we have p t ( z j ) C B(O,/~), j  1, ..., n. This is because Ct(zj)  z j  pt(zj) can be brought arbitrarly close to zj,j  1, ..., n. Hence for this t we have
188CHAPTER
9 BOUNDING
AND WEAKLYBOUNDING
SETS
p,(y,~) c B(0, ~) + B(0, ~) c B(0, 2~). But on the other hand we have by the assumption in this c a s e Pt(Xj) B(0, 2A) for some j. By the assumption B(0, 2A) is polynomially convex, so there is polynomial q w i t h l q ( p t ( x j ) l > 1, I q i< 1 on B(0,2A). The polynomial Q = qopt satisfies I Q ( z j ) I > 1 and l Q I< 1 on Vn.
(9.20)
We can now define inductively a covering (Wn) of E and a sequence (a,~) with am E Wrn+l/Wm, ?7"t 1, ...,n as follows" Wm  V~(m), am  Xj(m)
(9.21)
where n ( m ) and j ( m ) are found by the procedure just described. If n n ( m ) has already been found we let j ( m )  j be the index j found above and then we put n ( m + 1) as the smallest integer such that a m = xj E Vn(m+l)  Wm+l 9This is possible since Vn is a covering of E. We now have am E Wm+ l / Wm and polynomials Qm , m  1, 2, ..., with
I Qm l< One can find C~rn c,~Qm, satisfies
Ct m E
1 on
Wn
and
IQm(~m)I> 1.
N and constants am such that the polynomial fm =
m1
I f m l < 2 m on Wm a n d l f ~ ( a m )
l>_m+l+
~ I A(~)I. (9.22) ki
Since (W ~ is an increasing open covering of E, is holomorphic on E and
the function
~~m%1 f r n
(x)
m1
I f(a~)[I
fm(am)+ ~ m1
>_ I fro(am) I ~
k=l
A(a.~) I
fk(am)+
k=l
fk(gm)
k=m+ 1
oo
IIAIIwk > "~k=m+l
f =
9.1 B O U N D I N G SETS
189
Hence { a m } {xj(m)} C A completes the proof. I
is not
bounding in this case.
This
R e m a r k 25 The proof in locally convex cases depends on a similar construction using a cover (Vn) consisting of convex sets. In our notation
g n  {ZI,...,Zn}~B(0,6) is the vector sum of a finite set and a polynomially convex sets. I f B(0, 5) is convex we can use v: 
{Zl,
,
+ B(O,
where "cvx" denotes convex hull; as the sum of two convex sets V~ is then convex. However, the vector sum of convex set and a polynomially convex set need not be polynomially convex. This difficulty explains the difference between our proof and that in locally convex case and the role played by the mappings Pt ( see Schottenloher [196]).
9.1.3
B o u n d i n g Sets in Locally P s e u d o c o n v e x Spaces
Let us recall that a subset A of a topological vector space is b o u n d i n g if
II f IIA sup If (x) l < oo xEA
for every holomorphic function f on E. D e f i n i t i o n 61 A topological vector space E is said to be locally p s e u d o c o n v e x if the origin has a fundamental system of pseudoconvex neighborhoods. E is locally p  c o n v e x if the fundamental system is absolutely pconvex, for some (0 < p < 1), Rolewicz [186]. We can see that a locally pseudoconvex topology is determined by a family of pseudonorms while the locally pconvex topology is determined by a family of psemi norms, 0 < p < 1. Let us recall that A p  s e m i n o r m on E i s a m a p p i n g I[[Ip'E~/~+ such that for t E ~ , x , y E E ,
(a) II tx IIpI tiP. II x lip (b) I I x + y l l p <  I l X l l p + Ilyllp If II x IIpr 0 for x r 0, then IIlip is called a p  n o r m . If for some a _ > l , ( b ) is replaced by IIx+yllp
190CHAPTER
9 BOUNDING
AND WEAKLYBOUNDING
SETS
Finally IIlip will be a pseudosemi norm if it is a psemi norm for some p ( O < p < 1). The following theorem characterizes the bounding sets in a class of locally pseudoconvex topological vector spaces which may not be metrizable. T h e o r e m 91 [16](1990) Let E be a complete separable complex locally pseudoconvex space which has the bounded approximation property, and is such that the closed balls B ~ of E. dealed ~y it~ family of p~~ni ~o~n~ ( III1~ )  ~ / . (0 < p . < 1). are polynomially convex. Then the closed bounding subset of E are compact. P r o o f . Let
{z~} be dense in E and B ~ ( 0 , ~)  {x e E; II x I1~_< ~ }
where ( II x IIc~ )c,e/is the family of pc~semi norms (0 < pa < 1), defining the topology of E. Let Pt   I  Ct, t E T, where ( r is the given equicontinuous family of linear mappings. Assume A C_ E is closed and noncompact. Then it is not precompact; so there exists a sequence (xj) of elements of A such that
II ~y  x ~
II~> 5A > o,
when j 7~ k
for some suitable number A > 0 and some c~ E I. From now on the proof can be given along the same lines as in the above theorem. This completes the proof of the theorem. IH R e m a r k 26 For the quasicomplete locally convex spaces the result is due to Schottenloher [196] and for separable or reflexive Banach spaces it is due to Dineen[86]. Theorem
92 [16](1990)
A subset A of the pseudoconvex Fspace
Zo  n~>o/~ 
x  (x~); II x Iio= sup~~.lXn IP< oo p>O
1
,
(9.23)
9.1 B O U N D I N G
SETS
191
(0 < p < 1), of the intersection of all spaces lp of complex sequences x = (xn) with the Fnorm II x Iio= supp>0 ~  ~ I x~ IP< oc is bounding if and only if A is relatively compact. P r o o f . First the space lo is a complete metrizable pseudoconvex space with respect to the Fnorm, oo
if x i]o= s u p ~~lXn ]P. p>O
(9.24)
1
Hence by Theorem 90, the bounding subset of lo are relatively compact. In fact Io has polynomially convex balls, for the function
f(xl, ...x~) =ll ~,, ~
JIo
is convex for x = (x], ...x~) C ( ~ n n E N, see Corollary 15. This completes the proof, i 9.1.4
Bounding
Sets in Non Locally Pseudoconvex
Spaces
In infinite dimensional locally convex spaces E the size of the bounding subsets has been investigated by Dineen [88] and Josefson[122], who answered the following questions negatively for l ~ : "Does there exist an infinite dimensional normed space E such t h a t t h e u n i t ball BE is b o u n d i n g . ? " In some non locally convex spaces the situation is different as it will be asserted by the following theorem.
T h e o r e m 93 / / 6 ] ( 1 9 9 0 ) A subset A of the non locally pseudoconvex Fspace
z(~/
9
(x~); li x ii ~ l ~
I~n< ~ , P~ ~ 0
(9.25)
1
(0 < p~ < 1), Pn ~ O, of all complex sequences x = (xn) with the Fnorm [I x I[ ~ ] xn ]pn< oo, is bounding if and only if A is bounded. Consequently, the unit ball B l(pn)(0, 1) is bounding. P r o o f . The s p a c e l(pn) , (O < Pn < 1),p~ ~ 0, is Schwartz and hence it is Montel space. It is metrizable but not locally pseudoconvex since there is no countable family of pasemi norms defining its metric topology, see
1 9 2 C H A P T E R 9 B O U N D I N G A N D W E A K L Y  B O U N D I N G SETS nolewicz [[186],p.153]. Now as l(pn), Pn ~ O, is a Montel space then every closed bounded subset A of E is compact and consequently A is bounding. On the other hand the bounding subsets of this space are relatively compact since its balls are polynomially convex, see the theorem above. Since B z(p~)(0, 1) is topologically bounded with respect to the only topology define by the given Fnorm, it is bounding, and the proof of the theorem is complete, m C o n j e c t u r e 1 Are the weakly bounded subsets of l(p~), (0 < p~ < 1),p~ ~ 0, weaklybounding? We have a negative conjecture.
9.2
WEAKLYBOUNDING
9.2.1
(LIMITED)
SETS
W e a k l y  B o u n d i n g Set in Locally B o u n d e d Spaces
Definition 62 A subset A of a topological vector space E, whose dual E ~ separates its points is called. " w e a k l y  b o u n d i n g " ( o r limited) if every sequence (On) C E', which converges pointwise to zero in E, converges uniformly to zero on A. That is,
lim Cn (x)  0, for every x E E
lim
n~oo
II
n   ~ oK)
II Cn IIo,
(9.26)
IIA= sup A Ir
Proposition 3 A subset A of a locally bounded pnormed space E is weaklybounding if and only if II f ]]=sup If ( x ) ] < oo, for all A
xcA
f
~
q5~ E H (E)
(9.27)
1
where ( cpn) C E' . P r o o f . =~)" Assume A is weaklybounding. Then ]] On IIA' 0 whenever r (x) * 0 for every x in E. Let
1
9.2 W E A K L Y  B O U N D I N G
(LIMITED)
SETS
193
Since f  ~ r is analytic, the boundedness of f on A is given from the analyticity of the polynomials c n n C N and the assumption that II Cn IIA~ 0 to have ~ ] 1 II On ]]~< c~. Notice that II ~~.~ Cn n IIA~ EF
II
IIA< oo.
r If ~ r n G H (E) then ~~ (On (x)) n tn/p converges for every x E E and t E (/'. By Generalized CauchyHadamard formula in one variable (see subsection 7.2.3, p.139), limsup 1On (x) n 88 Ip  lim 1r (x)I p  0. n~oe
Now if oO
II
r
11,4<
1
then II Cn IIA+ 0 as n + oo and also as a consequence of the radius of convergence formula Oo
R I (x)   s u p ( r < 0; ~
[I Cn n ]I rn/p < c~)
(9.29)
1
of a pnormed space. This completes the proof of the theorem. I Josefson [122] has introduced the notion of weaklybounding sets in his study of bounding subsets of l~. He has proved that the bounding and weaklybounding subsets of l~ are the same. Dineen's result [ Cor.4.19 p.175] showed that this is obviously true for lp, (1 ~< p < oo). However the situation is totally different when we deal with non locally convex topological vector spaces as it is shown in what follows. Before we deal with the weaklybounding subsets of a locally bounded Fspace E, let us first characterize the duals of certain quotient spaces of E. For Banach spaces, see Rudin [[187] Th. 4.9 (b), p. 92]. T h e o r e m 94 [16](1990) Let E be a pBanach space with a Schauder basis. Let M C E be a closed subspace. Then the dual space of the quotient space E / M is isomorphic to the annihilator M • of M, provided that E / M has a Schauder basis, i.e.
( E / M ) ' ~_ M •
(9.30)
194CHAPTER
9 BOUNDING
Let 7r" E * E / M
Proof. define
AND WEAKLYBOUNDING
be the quotient map.
T (r
SETS
For each 4) E ( E / M ) '
 r
(9.31)
T h u s T is an isomorphism of (E/M)' onto M • : * F i x x * E M • 9 T h e n x* E E I and the r e s t r i c t i o n x i M  O. Let N be the null space of x*, hence M C_ N. It follows t h a t there exists a linear functional v on E / M such t h a t prr  x* as a consequence of the existence of a Schauder basis of E / M . (For example choose ~ such t h a t vlTr(N)  0 and v (Tr (x))  x* (x) outside 7r (N)). T h e null space of v is 7r ( N ) , a closed subspace of E / M
by definition of
q u o t i e n t  t o p o l o g y in E / M and hence v is continuous, t h a t is, v E ( E / M ) ' . T h u s T (v)  vTr  x*. On the other h a n d if x E E and y* E ( E / M ) ' , then 7r (x) E E/M, hence x ~ y* (Tr (x)) is continuous linear functional which vanishes for x E M. T h u s T ( y * ) E M • T h e linearity of T is obvious. This completes the proof. I
We give now the following interesting result which shows to what extend the geometry of the spaces may play a role in Holomorphy.
Remark
Theorem
27
95 [16](1990)
Let E be a separable complex pBanach space, (0 < p < 1). Assume that E is isomorphic to a quotient space of lp which has a Schauder basis. Then a weaklybounding subset of E is not necessarily bounding in E. Moreover it could be an unbounded subset of E. P r o o f . A b o u n d i n g set is obviously weaklybounding. Since E is isomorphic to lp/M for some closed subspace M of lp,(O < p < 1), so it is enough to s t u d y the w e a k l y  b o u n d i n g subsets of lp/M. This can be done t h r o u g h s t u d y i n g first the w e a k l y  b o u n d i n g subsets of lp(O < p < 1). Let
A
l)l/p'
x(n);x(~)  ( ( n
I
(1)n
nthplace 1/p
,0,. ) noN
(9.32) I
9.2 W E A K L Y  B O U N D I N G
(LIMITED) SETS
195
and
B 
x(n);
x (n)  
(1, ( ~1) l / p , , , , ,
(n1  
)l/p
' 0,. ) n c N } o
o
~
9
(9.33)
The space lp (0 < p < 1) is dense as a subspace of 11 and it has the same dual space which is isomorphic to lcr The bounding subsets of lp are bounding when they are considered as a subset of ll. Notice that the restriction of every f E H (/1) to lp is in H(lp). The set A is bounded in lp since for every x (~) E A, n
~~.[xj[ p j=l
    n1  1 . n
Moreover A is a weaklybounding subset of lp for it is relatively compact in 11 where the bounding and weaklybounding sets are the same. Notice that the condition cx:)
lim (sup ~ n,c~
xs A
[xj])  0
(9.34)
j=n
is satisfied. However A is not a bounding subset of lp, for it is not relatively compact. In fact the elements of A lie in different directions on the surfaces of the unit ball Bzp. Consequently A C lp, (0 < p < 1), is an example of a weaklybounding set which is not bounding. The set B is a weaklybounding subset of lp (0 < p < 1) since B is bounding as a subset of/1; it is clear that B is a relatively compact subset of ll. On the other hand, B is not bounded in lp for d (x (n) O)  ~ j1 ~ oc. Therefore the weaklybounding subsets of lp could be unbounded sets. Finally let us study the weaklybounding subsets of a quotient space
lp/M where M C_ lp is a closed subspace and lp/M has a basis. Since the dual space (lp/M)' is isomorphic to the annihilator M • of the subspace M by the preceding theorem, t hen every sequence (r (lp/M)' can be assumed to be in (lp)/and vanish on M. Let
7~ " lp > lp/M
in
196CHAPTER 9 BOUNDING
AND WEAKLYBOUNDING
SETS
be the quotient mapping which maps the Fr6chet space lp onto the Fr6chet space lp/M with respect to the Fnorms IJ 7r (x)I[= inf IJ x  m lip, where meM
II x lip= E F Ixjl p, for x 
(xj) C lp.
For the weaklybounding subset B of lp given above, we claim that the subset 7r (B) of lp/M will satisfy the desired property, that is, it is non bounding but weaklybounding in lp/M" Let (On)C (lp/M)' with (~n (X)~ 0 for every x in lp/M. Then oo
II ~ r 1
(x)
II ~ ~'~ II r 7r(B)
~
(9.35)
1
This is because d/)nOTr is a sequence of continuous linear functionals on E which vanishes on M and B is a weaklybounding subset of E. On the other hand 7r (B) is not bounding in lp/M. Notice that 7r(B) is not bounded in lp/M. This completes the proof. II
9.2.2
T h r e e Different Classes of H o l o m o r p h i c
Functions
The above theorem can be formulated using holomorphic functions as follows.
T h e o r e m 96 [16] Let E be a complex pBanach space with a Schauder basis, (0 < p < 1). Then there exists a holomorphic function f in H (E) and a nonrelatively compact set A in E such that
II f IIA= ~ pointwise in E where (r bounded set in E.
a~d
II On IIA~ 0
if On(X) ~ 0
(9.36)
C_ E'. Moreover A could be an unbounded weakly
P r o o f . Every bounding subset A of E is weaklybounding but the converse is not always true; Remark the sets A , B of the preceding Theorem 95. Hence, not every set A in E satisfy II r ]IA~ 0, whenever Cn(x) ~ 0 pointwise for every (On) C_ E', is relatively compact. Therefore a function f E H (E) may exist and satisfies II / IIA= c~. Of course f will not of the form ~ r with II r IIA~ 0 wheneverr n ~ 0 pointwise since A it is not weakly bounding, in accordance with Proposition 3, given at the begining of this part. Such a set A may be a proper weakly bounded set in E. This completes the proof. II
9.2 W E A K L Y  B O U N D I N G
( L I M I T E D ) SETS
197
R e m a r k 28 This last situation is almost similar to the finite dimensional
case(T n in which every f r H ((Tn) is unbounded on unbounded sets o f t ~. Theorem 95 shows the remarkable relevance of the concepts of bounding and weaklybounding sets. We may consider them as basic concepts in holomorphy. To show this, let E be a locally bounded Fspace and let By (E) and WBg (E) denote the classes of bounding subsets and weaklybounding subsets of E respectively. According toTheorem 95 if E is not locally convex space with basis, one may obtain
Bg (E) ~ WBg (E) .
(9.37)
Moreover the class WBg (E) consists of two subclasses: (1) The bounded weaklybounding subsets of E, (WbBg (E)), and (2) The unbounded weaklybounding subsets of E, (W~Bg (E)). Therefore we have precisely the following relations
Bg(E) WbBg (E
(E)
Bg (E) C WbBg (E) C WWBg (E) . Figure 5: R e l a t i o n b e t w e e n t h e d i f f e r e n t classes of b o u n d i n g sets Consequently we will have the following result.
198CHAPTER
9 BOUNDING
AND WEAKLYBOUNDING
SETS
Corollary 17 [16](1990).Let E be a locally bounded space with a Schauder basis. Let us denote by H b (E) the class of holomorphic functions of type Cn n, (r C E' which are bounded on all bounded sets in E. i.e.
H(E) = I
f=Er
nell(E);
(r
V bounded sets B in E
J C_
and by H b (E) the class of holomorphic functions of type E r E', which are bounded on all weakly bounded sets in E, i.e.
H (E) = I
f  ~ Cn n e H (E) ; (r
C E', II f lie< ~ ,
V weakly bounded set C in E
J
Then
H b (E) C H b (E) C H (E).
(9.38)
That is, the class of holomorphic functions H (E) on E contains properly two other different classes.
Proof.
See the proof of Theorem 95. m
(E)
H(E)
H b(E) C H D(E) C H ( E ) Figure6" Relation between the different classes H b ( E ) , H b (E)
and H (E) of holomorphic functions
199
9.2 W E A K L Y  B O U N D I N G (LIMITED) SETS
Remark 29 We notice that these above three classes of holomorphic functions coincide for many locally convex spaces, e.g. if E shown by Josefson [122] and Dineen [82].
lp (c~ >~p >~ 1) as
Remark 30 As we have seen,
the theorems of the two preceding parts explore a new class of bounding sets which appear in the study of holomorphy in the non locally convex spaces E. It is the class of bounded weaklybounding sets ( B  W B g ( E ) or bWBg (E) for short).So recall that we have so far the following relation between the three classes of bounding sets in some non locally convex spaces : Bg(E) ~ B
9.2.3
WBg(E) ~ WBg(E).
(9.39)
E x a m p l e s of H o l o m o r p h i c Functions
We introduce some of the remarkable topological vector spaces which satisfy the hypotheses of the theorems of the preceding subsections 9.2.1 and 9.2.2. We will also construct certain holomorphic functions and non bounding weakly bounding sets in some spaces to explain the performance and effects of these concepts in Complex and Functional analysis.
Construction of Certain Holomorphic Functions ( O n L o c a l l y B o u n d e d Sequence Spaces) Example 31 Let E  lp, (0 < p < 1). According to the above comment the relation
means that not every A C_ E satisfying lim II r
IIA 0 i f
lim
Cn (X)  0 , e v e r y
x e E
(9.40)
is relatively compact. Equivalently, there is A C_ E which may be unbounded satisfying II r IIA~ 0 i r e s ( x ) ~ 0, x E E, (r C_ E', and a function f = E r f C H (E) exists with II f IIA< c~. This is because weakly bounding subsets are not necessarily relatively compact by Theorem 96.
200CHAPTER
9 BOUNDING
AND WEAKLYBOUNDING
SETS
To show this, consider the following set given in the proof of Theorem 95.
B  { x(n);
x (n)  
1 ( 1 ) :/p (1, (~)l/p,..., ,0,...),
n:g
}
.
We have proved that B e WBg (E) /Bg (E) and B is unbounded in E. Now the following function (:x:)
1
:<x>
+niX>
~
~
o
n
I
will be the required one, i.e. II f liB< oo. Obviously (r
(1)
1)1//) __
1{~n {{B ( 1 ) 1( 11 )
"(n a s Tt~ cx).
1
(9.41)
X n
c E' and hence
1
~(1;)
1
,0
Now oo
1 )l_~.Xnn e H (E)
(9.42)
1 and it is bounded on B"
m(1)
II fllB=sup ~ xmgB
m
1
!n P
n
=
1+~
E
1 But the function g(X) g(X)

 E1
Er~
( n1) 1 p: "xn

~ n ( X ) n i8 a n o t h e r
Er~
n1 + (X) a8 m
having
ful~ction
%(x) (~1)•
n
1
_l/n
.:15 n
and does not satisfy (9.40). It is not lin'ear, at
~ (x).
This is because
9.2 W E A K L Y  B O U N D I N G (LIMITED) SETS
201
Construction of Certain Holomorphic Functions(On Locally bounded Fspace) Example 32 A weaklybounding subset may be unbounded by. Theorem 95. That is an unbounded set D may exist such that all holomorphic functions of type O0
1
will satisfy II E ~ r liD< ~, (r c_ E* To show this, let E  l(pn), infpn ~ 0, be a non locally pseudoconvex space, see Rolewicz [[186],p.153]. We now give an example of a function g (x)  ~
r
E H (E)
and a nonbounded weaklybounding subset D of bounded. Set
E
such that gilD
is
Cn (X)  X n
and let D 
1
x ( n ) ; x (n)  (0, ( 2 1 o g 2 2 ) 1 / P , . . . ,
withp=~ p ( 1 
log log n ]~ and 0 < p < 1
(
1 )l/p 0 . . . ) , n e N } n log2n ' '
Obviously p~ p ~ 0, i.e.p~ ~ O. n
Now D C E is unbounded: J
II ~(J)II d(x(J), o)  ~
I ~(n'~)l~"
n1
1 l'p]p(1 loglog. ) 1
~
(nl~g2n) ~e n1
a8 j ~(x).

2
n
lo~n 21oglogn)
e ( log log n
C~
202CHAPTER 9 BOUNDING
AND WEAKLYBOUNDING
But
J cn (X)I~(~)
11
n=l
n1
SETS
n/p < c~
(9.43)
nlog 2 n
D. Thus g = E cn < cx~ on the unbounded set D. On the other hand D is a weaklybounding subset of l(p,~) since it is O0 compact in 11 " lim sup ~~j=~ Izjl  o.A for every x
(Xn) e
n~C~x~ D
P r o b l e m 2. Let E be a topological vector space whose continuous dual space E ' separates the points of E. W h a t are the geometric and linear properties of E so t h a t all its bounding sets are relatively compact? P r o b l e m 3. Does there exist a non locally convex space E with a rich dual space E* whose bounding and weaklybounding sets are the same? ( 0 b vious t h a t the space LP[0, 1] (0 < p < 1) satisfies this property but (LP) ' = {0} is a trivial dual). Could we construct such a space?
Image of B a n a c h Spaces Example
33 / o  S p a c e .
The space l o  A lp
(9.44)
p>o
is a complete locally pseudoconvex Fspace with respect to the sup F  n o r m
II x
IIo=supll
x
p>o
lip
wh~ II x I1~ E j =n ~ Ixjl p x  (xj) e lp. The bounding subsets of lo are relatively compact since lo has polynomially convex balls. Notice that the funct i o n f ( x l , . . . , x n ) [I ( e ~ , . .  , e n ) I[lo is convex f o r x  ( X l , . . . , x ~ ) e r n , n e N, see the author[J15], p. 17] or Corollary 15, p. 185. The weaklybounding subsets of lo could be unbounded. For example the set A  {x (n)} where, x (~)

(1 ,
~
'
...
'
n
' 0,...)
'
is weaklybounding in lo for it is weaklybounding in each lp. Notice that lo is dense in each lp (0 < p < 1). Moreover A is not bounded in lo " n ]l x(n) I l l o  S U p E ~=1
(1)I/pP m
~ 
1  +oo
m,1
m
203
9.2 WEAKLYBOUNDING (LIMITED) SETS
as n ~ oo. Consequently a weaklybounding set in lo may not be bounding. The importance of this Fspace lo is that every separable Banach space E is isomorphic to a quotient space of lo; i.e. for a subspace M of lo,we have
(9.45)
E ~_ l o / M . A
Mapping with Rapidly Decreasing Sequences(FSpaces) Example 34 Consider the space L ( E ) of all continuous linear mappings of a Banach space E. Let L ( E ) have the linear mappings norm. On To E L ( E ) defines r th approximation numbers by (~(To) 
inf {fiT  Toll;T e L ( E ) with d i m T ( E ) <_ r } .
ff lP(E) 
{
To C
L(E); ~ [c~(To)] p < oo r=l
then on the intersection S(E)

(9.46)
}
e >ol (E)
(9.47)
a metrizable non locally convex vector topology is generated by the sets
Bp(To, ~) 
{
T C
S(E); ~ [ar(T To)] p < c r=l
}
,p>O,E>O.
S ( E ) is complete and it has the bounded approximation property if E is complete, as it is shown by Pietsch[[18~],p.139]. If the balls defined by the metrics oo
dp(T, O)  ~
[C~r(T)]p ,
(9.48)
r:l
are polynomially convex then the bounding subsets of S ( E ) are relatively compact by Theorem 89. If S ( E ) has a basis for a normed space E we can apply Theorem 95 which considers weaklybounding sets. Notice that L ( E ) is not a Fr~chet space if E is not Banach.
204CHAPTER
9 BOUNDING
AND WEAKLYBOUNDING
SETS
R e m a r k 31 We could have another types of non locally convex Fr~chet spaces with mappings with rapidly decreasing approximation number as follows: Let Eq be a qBanach space, (0 < q < 1) say with a Shauder basis and construct L(Eq) as above with the linear mappings qnorm:
IITI[ = sup IIT(x)II. x6BE
(9.49)
We shall then obtain different values for c~ and hence different spaces lP(Eq) and S(Eq).
Locally Pseudoconvex
Fspaces
E x a m p l e 35 Let E be a complete locally pseudoconvex space whose topology is defined by a sequence of pnhomogenous pseudonorms {llxll~;n e N } . Then, without loss of generality we can assume that IIxll, <_ llxll: <_ ...
Let E ~ = {x e E; IlxJl~ = 0}. Then ( E ~
(9.50)
is a decreasing sequence of vector subspaces of E; i.e. E~ n E~ n ....
Let E~
be the quotient space E~ E ~ i.e. En  E l E ~
(9.51)
The pseudono,ms, IlXlln, n e N induces pnhomogeneous pseudonorms Ilxll'n, n e N on the spaces En and consequently En with Ilxll'n, n e N are locally bounded spaces. Now if E has a Schauder basis, then the sequence of the cosets
{[ei]n} ~ {el + E~
i E g}n
(9.52)
generated by the basis {en} in the completion E of E, i8 a basis of norm o~,(~h~ ~ o,~it all t h o ~ {[~]n} ~ h ~ h a ~ ~q~al to z ~ o ) to th~ ~ ~o,~plete locally pesudoconvex spaces whose topology is defined by the sequence of pnho,~ogenous pseudonor,~s lixll'n, n e N , see R o Z ~ ~ z [[186],p.239.].
9.2 W E A K L Y  B O U N D I N G
(LIMITED)
205
SETS
In fact this can be metrized by the metric
i
]lxll'n
(9.53)
d(x, y)  ~ 1 ~n 1 + Ilxll'n " If the balls
un(o, r )  {x C E; I[x]l'n_~ r},
(9.54)
are polynomially convex in En, n E N , then Theorem 91 can be applied and bounding sets turns out to be relatively compact.A
Product of Fr6chet Spaces. Example
36
Let E
be the product of Fr~chet spaces E j by the metric d.j(x,
y) 
(9.55)
II~Ej such that the topology of E j
is given
ylJj
Iix 
where x j , y j C E j , and II.IIEj, J C N is a homogeneous pseudonorm and such that the balls defined by them are polynomially convex. It is easy to see that E has the bounded approximation property if each E j, j E N , has the bounded approximation property, see the author[J16], Example ~]. Let
r RN __. [0, §
(9.56)
be convex and homogeneous of degree 1; then an Fnorm II'llE can be defined on E by IIIIE  r IIx2IIE~,...), x (Xl,X2, ...). For example we can take
O(IIxlIIE1, IIx211E~, ...)  (~~
IIxjIIEj) l/q,
(9.57)
j=l
wh~e
1 <_ q < ~ , t o h a v ~ O
< 1/q <_ 1 ,
a~d
IIxjllEj, j e N,
i~
p j  h o m o g e n e o u s with py ~ oo as j ~ c~. Now Theorem 90, p.186, can be applied and hence the bounding subsets of E are relatively compact. For example take
E11
• lll2 •
• ....
(9.58)
206CHAPTER 9 BOUNDING AND WEAKLYBOUNDING
9.3
PROPERTIES OF BOUNDING LIMITED SETS
SETS
AND
In this part we apply the results of the preceding two parts of this chapter to the study of the properties of the classes of bounding and weaklybounding sets in some non locally convex topological vector spaces. 9.3.1
Bounding Sets and Compact Linear Maps
In this section we obtain some fundamental and interesting properties of the bounding subsets of certain non locally Convex topological vector spaces. They fit well with such spaces although some of them do not exist for locally convex spaces.
Definition 63 Let T:
Ep~ Eq
be a continuous linear mapping between two locally bounded Fspaces Ep and Eq (0 < p, q < 1). If for every bounded set B of Ep, the image T ( B ) is relatively compact, i.e. T ( B )  is compact. (9.59) then T is said to be compact.
T h e o r e m 97 [17](1989) If E is a normed space and T : E ~ lp,
(9.60)
(0 < p < 1), is a continuous linear mapping, then T ( B E ) , the image of the unit ball of E, is bounding in lp. Consequently, a domain of holomorphy of a normed space E, may have continuous linear image by T which is not a domain of holomorphy in lp.
Proof. We notice that every continuous linear mapping of a normed space E into lp is compact. This is proved by Stile[[201],Theorem 4]. Hence T ( B E )  is compact and consequently is bounding. Now if U is a pseudoconvex domain of a Levi space E, then it is not bounding. That is, it is a natural domain of existence of a holomorphic function. However T(U) may not have this property since it may be bounding as a compact subset of lp. This completes the proof, n
9.3 P R O P E R T I E S
OF BOUNDING
AND
LIMITED
SETS
207
R e m a r k 32 ( 1 ) T h e property given by the preceding theorem means that : " T h e r e is n o c o n t i n u o u s l i n e a r m a p p i n g o f a B a n a c h space E i n t o lp, (0 < p < 1), t h a t c a n h a v e a n i n f i n i t e d i m e n s i o n a l b o u n d i n g closed range". (2) Also this theorem implies that : "A c o n t i n u o u s l i n e a r m a p p i n g T : Eq ~ lp o f a q  n o r m e d s p a c e Eq i n t o lp h a s a c l o s e d b o u n d i n g r a n g e i f O < p < q < 1 ". We have
of St l [[202], The linear completion (or hull) of a pnormed space E is called the B a n a c h e n v e l o p e / ~ . The following theorem studies the bounding sets in
T h e o r e m 98 [17](1989) If E is a pBanach space with a Schauder basis, then a bounding subset of its Banach envelope E need not be bounding in E.
P r o o f . The space E is locally bounded. Thus the bounding subsets of E are relatively compact and are not the the same as the weaklybounding subsets as we have shown in the preceding part. However on the linear completion /~ of E, the bounding and the weaklybounding sets are the same because E is a separable Banach space. Therefore we choose a subset A of E which is bounding (or equivalently weaklybounding)as a subset of /~ but not relatively compact in E. This completes the proof, m In what follows we give another property of bounding sets in some topological vector spaces. L e m m a 23 Let E be either metrizable or locally pseudoconvex space, with bounded approximation property, and polynomially convex balls. Then the vector sum of finitely many bounding subsets of E is bounding in E.
Proof. The sum of two compact subsets of a topological vector space is compact, Rudin [[187],p.37], and the bounding sets in such spaces E are relatively compact,by the author's result[16], (see section 8.1 on bounding sets). So the lemma is proved, m
208CHAPTER
9 BOUNDING
AND WEAKLYBOUNDING
SETS
R e m a r k 33 The property given by the above lernma implies that: if U1 +U2 is not bounding then either U1 or U2 is not bounding. This means that if U =U~ +U2, is the vector sum of two domains, then U may be a domain of existence of a function f C H(U') for an open domain U' containing U, U1, U2 then it may happen that either U1 or U2 is a domain of existence of the function
f. For metrizable locally convex spaces, the result is due to Dineen[[82], p176]. The following theorem deals with some holomorphic properties of the pconvex hull of bounding sets, see Chapter 3, section 3.1 T h e o r e m 99 [17](1989) If E is a locally bounded pnormed space with a Schauder basis, then the absolutely pconvex hull of a bounding set in E is bounding. P r o o f . Assume that A C E is bounding. Then it is relatively compact by Corollary 16, p.181. Since the absolutely pconvex hull CpA is contained in
)~A + ... + )~2A provided that ~]~' I Ai [P_< 1, then CpA is also compact. Notice that B(0,
= B(0,
(9.61)
Of course the convex hull of a bounding set in a non locally convex space may be unbounded and hence it may not be bounding. This completes the proof. II R e m a r k 34 For the nonseparable normed space l~o, the preceding theorem is proved by Josefson [122]. It is proved for a balanced hull of a bounding subset A of a balanced open set U in a locally convex space by D ~ ~ [ [ 8 2 ] , 174]. Of course the property given in this theorem has its significance when we deal with analytic continuation and some topological problems. For example, it implies the analytic continuation froin a bounding set to its absolutely pconvex hull.
9.3 P R O P E R T I E S
OF BOUNDING
9.3.2
of The Different WeaklyBounding
Properties
AND LIMITED SETS
209
Sets
The following theorem will show some of the properties of weaklybounding subsets of locally bounded sequence spaces. It explores also the appearance of a n e w weaklybounding class (denoted by B  W B g or bWBg) lies entirely between the class of bounding sets (denoted by Bg) and the class of weaklybounding sets(denoted by W B g ) , This class may be also called Bayoumi weaklybounding, B  WBg; see figure 7 drawn after the following theorem. T h e o r e m 100 [19](1989) Let E be a complete locally bounded Fspaces with a Schauder basis. Then we have
(1) The vector sum of finitely many bounded weaklybounding subsets of E is bounded weaklybounding. That is, AI,..., An E B  W B g ~ A1 q... + An C B  W B g (2) The classes of weaklybounding subsets and bounded subsets of E are different and their intersection contains properly the class of bounding subsets of E. That is, Bg Bdd n w s g wBg (9.62) where Bdd denotes the class of topologically bounded subsets of E. (3) Every weaklybounding subsets of E is bounded with respect to the weak topology of E, but the converse is not true. That is a weaklybounded set may not be weaklybounding, i.e. WBg ~ WBdd.
(9.63)
P r o o f . P r o p e r t y (1)" follows directly from the fact that the sum of two bounded set E is bounded, by Rudin[187] and the properties of weaklybounding sets. Notice that sup xcAI,x2CA2
for any r E
Ir
~X2)l< sup I r xcA1
I + sup [ r
[
x2cA2
El and any sets A1, A2 in E.
To s h o w (2)" it is sumces to deal with it for the space lp, (0 < p < 1), since E is a continuous image of lp, by a continuous mapping T, i.e.E = T(lp), see Theorem 88, p.183.
210CHAPTER
9 BOUNDING
AND WEAKLYBOUNDING
SETS
The set
{
x(n);
X(n)


1 (1, (~)l/p
~'''~(n
1) 1/p 0,.
.
o
)
neN}
is weaklybounding for it is bounding in ll, but B is not bounded: n
cl(x (n), O)  ~
1
+OO as 1 fn
n+ oo.
On the other hand the set D{ei;
icN}
(9.64)
of unit vectors in lp is bounded but not a weaklybounding subset of lp, for D is non relatively compact in 11. Thus the classes of bounded and weaklybounding sets are not the same. Now the set A 
I X(n);x(n)  ((n1 )l/p '
1/p, O, nthplace (1) n
.) noN
I
is lying in the intersection of these two classes, for it is bounded and weaklybounding set in lp at the same time. However A is not bounding. So the class of intersection is larger than the class Bg of bounding subsets of lp. P r o p e r t y (3)" is obvious since the elements of the continuous dual E' are in H(E). In fact we have
E'c H (E)
(9.65)
where Hb(E) is the class of holomorphic (continuous and G~teaux analytic )H(E), whose elements have the form ~ r and bounded on weakly bounded sets. Finally the set D of unit vectors is weakly bounded in lp, for it is bounded in l l, where boundedness are the same for the original and the weak topologies. But D is not weaklybounding of lp for it is not relatively compact in 11. This completes the proof of theorem. II
9.3 P R O P E R T I E S
OF BOUNDING
AND
LIMITED
SETS
211
Figure '7: R e l a t i o n b e t w e e n different classes of b o u n d i n g sets a n d classes of b o u n d e d a n d weakly b o u n d e d sets. Here, the given symbols mean : Bg: bounding sets in lp; WBg: the weaklybounding sets in lp. B  WBg: bounded weaklybounding sets in/p.(Or Bayoumi limited sets). Bdd: bounded sets in lp; WBdd:weakly bounded sets in lp. 9.3.3
W e a k * C o n v e r g e n c e in t h e D u a l of a p  B a n a c h S p a c e As Different From Norm Convergence
For infinite dimensional Banach spaces E the weak* topology and norm topology in E ~ differ. This follows from Alaoglu's and F.Riez theorem ( see Rudin[187]).Can they have the same convergent sequences? The answer is "no!". This was proved independently by Josefson [121] and A.Nissenzweig. They proved that : "Weak sequential convergence in the dual of a Banach space does not imply norm convergence . Therefore we recall one of our concepts used in [[15], [16]], that is, the weaklybounding sets as follows : A subset A of a pBanach space E is said to be limited if every weak null sequence in E ~ tends to zero uniformly on A. That is, "limn Cn(X)  0, uniformly for x E A if (r is weak*null sequence in E ~'' So limited sets may be regarded as the linear analogy of bounding sets, as we have treated them before. For this reason we have called them in [[15], [16]] weaklybounding sets. By a bounding set A of E we mean a set which is bounded by all holomorphic (G~teauxanalytic and continuous) functions. That is, f(A) is a bounded subset of C for every Cvalued holomorphic f on E.
212CHAPTER
9 BOUNDING
AND WEAKLYBOUNDING
SETS
Generalized Josefson Nissenzeig theorem
The fact t h a t the unit ball of an infinite dimensional Banach space is not limited (not weaklybounding) is known also as Josefson Nissenzeig theorem: Weak sequential convergence in the dual of a Banach space does not imply norm convergence. In what follows we extend their result to sequence non locally convex pBanach spaces.That is to prove: "If E is an i n f i n i t e d i m e n s i o n a l p  B a n a c h s p a c e w i t h a S c h a u d e r b a s i s (0 < p < 1), t h e n t h e r e e x i s t s a w e a k * n u l l s e q u e n c e o f n o n z e r o p  n o r m v e c t o r s in E' ". 101 ( G e n e r a l i z e d J o s e f s o n  N i s s e n z e i g t h e o r e m ) Weak sequential convergence in the dual of a sequence pBanach space (0 < p < 1)
Theorem
does not imply norm convergence. P r o o f . It is enough to consider the space E = lp. This is because E is a continuous image of lp, by a continuous linear mapping T, t h a t is, E  T(lp), and E' ~_ ( l p / M ) ' ~_ M • (seeTheorem 96) In fact, E _~ lp/M isomorphic to a quotient space of lp, which has also Schauder basis, see[[186],p.65]. T h a t is (lp/M)' are the continuous functionals on lp which vanish on M . We claim that there exists a set D in E such that lim Cn(x) = 0 pointwise in E
but
n
lim r
:fi 0 uniformly for x E D
n
Since bounded set may not be limited (i.e. not weaklybounding ), [[13], Theorem3.2] the proof can also be given by constructing a such type of set. Consider the following set of unit vector in lp: D =
j X}.
(9.66)
D is bounded in lp. But it is not weaklybounding in lp because D is not bounding in l l. Remember that bounding and weakly bounding sets in 11 are the same. See[J13], Theorem3.2 and the diagram p.333]. Hence weak* convergence does not imply norm convergence in E'. This completes the proof. II GelfandPhillips spaces A space in which limited sets are relatively compact is known as GelfandPhillips space, see[126]. So far, it seems that such type of spaces d o n o t
9.3 P R O P E R T I E S
OF BOUNDING
AND
LIMITED
SETS
213
e x i s t in our study of the theory of bounding sets in topological vector spaces without convexity condition. The reason for this might be that the concept of bounded and weakly bounded sets are different, contrary to the cases of locally convex spaces. Theorem
102
Let E be a separable pBanach space isomorphic to a quotient space with a Schauder basis. Then E is not a GelfandPhillips space. P r o o f . In [[16],Theorem 3.2], we have proved that, a weaklybounding (limited) set of E may be unbounded. Consequently, our limited sets are not necessarily relatively compact. Thus E is not GelfandPhillips space. I C o r o l l a r y 18 ( I m a g e
of Banach
S p a c e s ) T h e locally pseudoconvex F
space l o  N lp
(9.67)
p>o
is not GelfandPhillips spaces with respect to the sup F  n o r m oo
II x IIo:Supll x lip=sup ~
p>o
where
p>o j=l
Ixjl p
x  ( xj ) E lp.
Proof. The bounding subsets of lo are relatively compact since lo has polynomially convex balls. Notice that the function f ( X l , . . . ,xn) =11 ( e ~ ,  , e ~ ) I1~ois convex for x  ( X l , . . . ,xn) E ~ " , n E N, see the author[[11],
p. 17]. The limited (weaklybounding) subsets of lo could be unbounded. example the set A
{
x(~); x ( ~ )  ( 1 ,
(1) 1/p ~
(1) lip ,...,
For
} ,0,...), nEN
is weaklybounding in lo for it is weaklybounding in each lp. Notice that lo is dense in each lp(O < p < 1). Moreover A is not bounded in lo :
n(_.~)l/p.p
II x(n) Illosup ~
m1
n 1
= ~
  + oo
rn1 m
as n ~ cx~. Consequently a weaklybounding set in lo may not be compact.
214CHAPTER
9 BOUNDING
AND WEAKLYBOUNDING
SETS
35 The importance of this Fspace lo is that every separable Banach space E is isomorphic to a quotient space of lo; i.e. for a subspace M of lo, we have E ~_ lo/M. (9.68)
Remark
In what follows we prove that the Fspaces lp~ which may be locally bounded or not (this depends on (Pn)) are not GelfandPhillips spaces. Theorem
103 (Sequence F  S p a c e s ) T h e spaces oo
< ~ } , (1 > pn > 0)
lpn   { X   (Xn); X n e (~, ~
1
are not GelfandPhillips spaces Proof. F i r s t : I f pn /~0, then each of lpn is metrized by a pnorm and lp~ become sequence locally bounded Fspaces. Consequently limited (weaklybounding) sets in lp~ are not necessarily relatively compact, see [[12], Theorem3. 2] where we have proved that limited set could be unbounded S e c o n d : If pn ~ 0, the spaces lp~ are non locally pseudoconvex Fspaces. We claim that they are not GelfandPhillips spaces neither. We have to choose an unbounded set which is bounding or weaklybounding. In fact, the unit ball as we have seen [12] are bounding. Remember that these spaces are Schwartz and hence Montel spaces. Therefore every bounded closed subset of them is bounding. Let us consider the following set
A 
1
x(~); x ('~)  (1, (~)
1/pl
_1)1/pn, 0,. . .) , n o N } . '""(n
(9.69)
This set is unbounded; notice that
f;x
n
1 + (:~
II1
m
aS Tt+ 00.
However, A is weaklybounding, i.e.
lim On (x)  0, for every x e lp,~ :=~ lim II r n~
n~(x)
IIA o,
(9.70)
9.3 P R O P E R T I E S
OF BOUNDING
AND
LIMITED
SETS
215
This is because the dual of lp~ is loo and A is weakly b o u n d e d in /1; we note t h a t O(3
lim sup ~ Ixjl O. n x C A .J   n This completes the proof. II R e m a r k 36 Referring to the preceding theorems 102 and 103, we have seen a limited set may not be bounded( see [[12], Theorern3.2] ). Recall that the intersection between the classes of limited sets and of bounded sets is bigger than the class of bounding sets; consider for example the set of
B
{
x(~) ; x ( n )  (
/1) 1/p (1) 1/p n
' ... '
n
' 0,...),
}
of lp . Hence our space E  lp will have b o u n d e d and u n b o u n d e d limited sets which are not relatively c o m p a c t with the priority t h a t all its b o u n d i n g sets are relatively compact(see[16]). T h e following example explains the preceding theorems. E x a m p l e 3 7 A limited set may be unbounded by Theorem 103 and [[12],Th.3.2]. That is an unbounded set D may exist such that all holomorphic functions of type 1
will satisfy II r liD< C E*. {Sm}) To show this, let E  l(p~), inf Pn ~ O, be a locally bounded space. We now give an example of a function g (x)  ~ r
C H (E)
such that g lD is bounded.
and a nonbounded Limited subset D of E Set (x)

and let 1 D  { x (~) ;x (n)  (0, (21og22)I/p
~~
(
1 )liP 0 , . . . ) , n o N } n log2n '
216CHAPTER
with P=n p (1 
9 BOUNDING 1 ~ log log n ]
AND
andO
WEAKLYBOUNDING
SETS
Obviously p~ p r 0
1
n
i.e.pn ~ O. N o w D C E is unbounded:
J II
x(J) II d(x(J), O)
l x(nn)lP rt1
1 p( 1  log log n )
nlog 2 n
n=l
~e 2 

n1
as j ~ oo.
log n e ( log log n
2 log log n)
n
+00
But
J
cn
n= 1

1 ] ,~/p n log 2 n
< c<~
(9.71)
f o r every x  (x,~) e D . Thus g  y~ r < (x~ on the unbounded set D. On the other hand D is a weaklybounding subset of l(p~) since it is compact in
11" n<X~ lim sup x ~D
~j~n
[xj[  O.A
Notes and Remarks The following properties of Banach spaces are not similar to those we have already obtained in this paper and in [[15], [16], [17]], for at least, the non locally convex sequence pBanach spaces. It is known in Banach spaces E that, if E is a separable Banach space then limited sets are relatively c o m p a c t . T h i s is contrary to([12],Theorem2.3). Further limited sets in E are b o u n d e d which is contrary to ([13]Theorem2). Now the set {en; n >_ 1} of unit vectors is limited if E = loo, but not if E=co. This set is in fact is limited in lpn only if Pn ~ 0,because it is closed and b o u n d e d in a Montel space and hence it is compact; but not in/p~, p= /  , 0 (0 < Pn < 1)because they are pnormed spaces for some (0 < p < 1), see T h e o r e m 103. Therefore, some m a t h e m a t i c i a n s paid a t t e n t i o n to the nonseparable Banach spaces. For example Josefson [124] has constructed a Banach space
9.4 H O L O M O R P H I C C O M P L E T I O N
217
in which every bounding set is relatively compact but which contains a closed non compact limited set. Independently, T.Schlumprecht[188] has also written a paper entitled: A limited set which is not bounding. Thus it is noteworthy to say that, in general, the theory of infinite dimensional holomorphy in locally and non locally convex spaces is once more different. We may need to invent and discover new criterion and terminology to solve the new problems that may appear during our study of Holomorphy in the absence of convexity condition and non linearity.
9.4
HOLOMORPHIC
9.4.1
COMPLETION
H o l o m o r p h i c C o m p l e t i o n in F  S p a c e s
Let E be a Fspace whose dual be its linear completion (It is also the maximal space t o w h i c h all e x t e n d u n i q u e l y as c o n t i n u o u s
E' separates the points of E. Let /~ called the Banach envelope). That is, continuous linear functionals on E linear functionals.
Kalton [131] has studied the linear completion of some quasiBanach spaces( i.e. for complete quasinormed spaces). In analogy with the case of locally convex spaces we define the holomorphic completion.
Definition 64 Let F, be the linear completion of a topological space E whose dual separates points. The maximal subspace E5 of the linear completion to which all holomorphic functions f E H(E) (continuous and GOteaux analytic) are extended analytically is said to be h o l o m o r p h i c completion. E5 is well defined since any f E H(E) can be extended analytically to an open neighborhood ~ C_ E as can be shown by using Taylor series expansion of f at the points of E, see, Ch.. 7, section 7.2 It follows that each f has a unique m a x i m a l d o m a i n o f e x i s t e n c e ~fC_E. ^
Hence
E~  n { f~:; f E H ( E ) }
(9.72)
and N { f~f; f E H ( E ) } is a vector subspace of E. E is called holomorphically complete if E
E~.
(9.73)
218CHAPTER
9 BOUNDING
Now the linear span F
AND WEAKLYBOUNDING
SETS
of the basis {e~; n C N}
of a Banach space E is proved to be holomorphically complete, by Noveraz
[~80]. In what follows we extend this result to some metrizable spaces to obtain examples of non complete holomorphically complete non locally convex spaces. T h e o r e m 104 Let E be metrizable space with a Schauder basis (en) and plurisubharmonic logarithmic metric. Then the linear span F of ( en ) , i.e.
F  span { en ; n C N }
(9.74)
is holomorphically complete. P r o o f . Assume t h a t log d(x, O) is a plurisubharmonic function on E where d is the metric defining E. Let
7r,~ " E* En the projection of E into the first n basis vectors. Let y E E / F and choose a sequence (en), Cn > 0 such that (x)
E
en log d(Tcny, y) >  c o .
n=l
The function (:x:)
n(~)  ~ ~ log d(..x,
x).
xCE
(9.75)
n=l
is plurisubharmonic and hence
 {x e E; R(x) < R(y)} is a pseudoconvex domain of E. We notice that F C_ ~, since R ( x ) ~   o o , for x ~ U ~ ( E )  F.
9.4 H O L O M O R P H I C C O M P L E T I O N
219
Now apply the Levi problem solution given by the author[12], [13],and [14], (see also Ch.10 ,subsection 10.1.1, p.231). We note ft is a domain of existence of a holomorphic function f E H(E). Hence the holomorphic completion F5 is contained in ft to have in particular y ~ Fs. Since y E E / F is arbitrary we conclude that
FF~. This completes the proof of the theorem. I
Examples of Holomorphically Complete Fspaces Example 38
The complex spaces
l(p~) { x(Xn);~~.lxn IP~< cx~ ,
0 < Pn <_ 1,
(9.76)
1
l o  np>olp
x
(Xn); sup p>0
I xn IP< c~
,
(9.77)
1
lp  Uq<_plq  { x E lp; x E lq for some q _< p},
(9.78)
where (X)
I xj Iq , q _ 1
d(x, 0)  inf dq(x, 0)  inf E q
q
(9.79)
1
and
l+p  nq>_plq  { x e l ;suP lXj
,0 < p < _ 1
(9.80)
q>P 1
are non locally convex spaces which satisfy the hypotheses of the above theorem. Notice that lp C l+ C lp+e.
(9.81)
220CHAPTER
9 BOUNDING
AND WEAKLYBOUNDING
SETS
Relation Between Bounding Sets and Holomorphic Completion In what follows we give the relation between the union of the bounding subsets of a locally convex pseudoconvex Fspace E and its holomorphic completion E.
T h e o r e m 105 [17](1989) If E is a locally pseudoconvex Fspace with the bounded approximation property, then E D  U A E B g ( E ) A 
where A orE.
is the closure of A, and Bg(E)
is the class of bounding subsets
For the proof of the theorem we need the following lemma.
L e m m a 24 Let E be a locally pseudoconvex Fspace, with the bounded approximation property, f E H ( E ) , and A is a bounding subset of E. Then there exists a zero neighborhood of V in E such that
flfllA§ P r o o f . For f E H(E), and x E E
fz(Y) :
(9.82)
< we
define f~ : ~  ~
by
f (x + y).
(9.83)
It is clear that f~ E H(E), for every x in E. Similar to the method of locally convex spaces, see Dineen [[82], proposition 4.22], we can show that the family (fx)~eA is t o  b o u n d e d subset of H(E), where ro is the compact open topology on H(E). In fact, if K is an arbitrary compact in E, then sup IlfyllA =
yEK
sup
yEK, xEA
IIf(x + Y)II = sup I]LIIK < xEA
Since E is metrizable, (fz)zEA is a locally bounded subset of H(E) and hence there exists a neighborhood V of zero, for example a ball B(0,e), such that sup IIfzIiv < oo.
xEA
Hence sup xEA,yEV,
] f ( x + y)I]lfiiA+y < o o . I
9.4 H O L O M O R P H I C
COMPLETION
221
P r o o f o f T h e o r e m . If x C E5 then there exists (xn)~ C E such that x as n ~ oc. Since x C E5, Xn lim f(Xn) exists for every f E H(E), n~ o o
and hence sup I f ( x ~ ) I < oo. n
Thus {xn} is a bounding subset of E
and x E UAEBg(E)A, that is
E5 C UACBg(E ) A.
On the other hand if A is a bounding subset of E and f E H(E), then by the above lemma a pseudovonvex balanced neighborhood V of zero exists in E such that LI/IIA+v < o o . By using the Taylor series expansion about points of A we find that there exists a holomorphic function f ~ on A + W such t h a t f A+W  f A+V
where W is the interior of the closure of V in E. As A C A + W, we will have A C_ E5 and hence E D  UACBg(E)A.
This completes the proof of the theorem. I
9.4.2
Holomorphic Extension Problem
In this section we study the holomorphic extension problem which is purely an infinite dimensional problem in some Fspaces : Let F be a subspace of a locally pseudoconvex metrizable space E, whose dual E' separates the points of E. " W h e n can every h o l o m o r p h i c function on F be e x t e n d e d to a h o l o m o r p h i c f u n c t i o n o n E ?"
222CHAPTER
9 BOUNDING
AND WEAKLYBOUNDING
SETS
Holomorphic Extensions of Dense Subspaces The following theorem gives a counterexample to holomorphic extension problem. More precisely, it supplies us with a dense subspace which cannot be extended to the whole space all holomorphic functions.
Theorem 106 [19](1989) Not every holomorphic function on lo

Np>olp
with the suptopology, that is, the one defined by the Fnorm OG
jixll sup p>0
,xo ,p 1
can be extended to a holomorphic function
on
lp
(1 > p > 0).
P r o o f . lo is a dense subspace of each lp (1 > p > 0), by Stile[[201],p.117]. Hence every bounding subsets of lo is bounding as a subset of lp. Since bounding subsets of lo and of lp are relatively compact, by Theorem 90, p.186, it suffices to pick up a non compact subset of lo which is compact as a subset of lp. For example the set
D
~0,x(1),x(2),...~,
where
(0,. ,0,( n1 )Il/p, 0,...), n E N
(9.84) (9.85)
is compact in lp if (1 > p > 0), but it is not compact in lo. Notice that oo
]lx(n)llo


s u p ~  ~ . l x n Ip p>O 1
= sup(1 )(11lp)p

n 1p + oo
p>0 ?%
as n ~ c~. This completes the proof of the theorem, m As another counter example where the holomorphic extension fails to occur from a non locally convex subspace to a locally convex one we obtain the following result.
9.4 H O L O M O R P H I C
COMPLETION
223
T h e o r e m 107 If f is a holomorphic function on lp (1 > p > 0), then f may not be extended to a holomorphic function on 11. That is, not every holomorphic function on lp can be extended holomorphically to its Banachenvelope 11. P r o o f . The set A 
1 )i/p
x(~); x (~)  ( ( n
is not bounding as a subset of lp, for it is not relatively compact. However it is bounding as a subset of 11. See also proof of T h e o r e m 95. This completes the proof of the theorem, m
R e m a r k 37 It follows from the preceding theorem and the fact that the linear completion l~  1 1 that
lp C lpt . , also that ( lp)5 = lp. Holomorphic HahnBanach Extension Theorem We consider the second type of holomorphic extension problem : " W h e t h e r eve r y h o l o m o r p h i c f u n c t i o n d e f i n e d o n a c l o s e d s u b s p a c e F of a c e r t a i n F  s p a c e E can be e x t e n d e d analytically to E". Since there exists a continuous linear functional on a subspace of lp (1 > p > O) which cannot be linearly extended to lp, by Stiles [201], we will deduce the following theorem.
T h e o r e m 108 Let M be a closed subspace of lp (1 > p > 0) whose unit ball is llprecompact. Then not every holomorphic function f in H ( M ) can be extended analytically to lp. P r o o f . By the result of Stile[[201], p . l l l ] , where under this assumption one can find a continuous linear functional on a subspace of lp (1 > p > 0) which cannot be linearly extended to lp. m Consider now the following normalized sequence (u~) which has disjoint supports:
224CHAPTER
9 BOUNDING
=
u2 
AND WEAKLYBOUNDING
SETS
(1,0,..),
1 1 (0, (~)I/P, (~)l/p, 0, ...), ..
Un(O'""(n
1 )l/p
'
ooo~
(1)l/p, 0,. ) n ~
oo
(9.86)
nthplace Then we obtain the following theorem. T h e o r e m 109 Let M be a closed subspace of ~ which is spanned by the above sequence {u~; n E N } . Then not every holomorphic function on M can be extended analytically to 11. P r o o f . The set {Un;nEN}
is not bounding in the subspace M, for it is not relatively compact. On the other hand it is compact in 11. Hence it is bounding in 11. m R e m a r k 38 The above theorem implies that M~ = M r lp. The locally convex space l~ has the property given by this theorem. This is because the set D = {ej; j C N } o f unit vectors in l ~ is bounding in l~ but not as a subset of Co. That is the holomorphic completion satisfies
(Co)5 5r lc~
(9.87)
Discussion We have studied some problems which arise in infinite dimensional complex analysis. In fact, the study of bounding and weaklybounding subset of non locally convex spaces provides information which is of fundamental interest in Holomorphy. There are essential relations between this study of this chapter and the radius of convergence problem suggested by Kisleman [134]. That is,
9.4 H O L O M O R P H I C
COMPLETION
225
"To c o n s t r u c t a h o l o m o r p h i c f u n c t i o n w i t h p r e s c r i b e d r a d i u s of c o n v e r g e n c e " and the Levi problem, which will be discussed in the next chapter, that is, to answer " W h e t h e r e v e r y p s e u d o c o n v e x d o m a i n is a d o m a i n of h o l o m o r phy". For example we can easily deduce that there are entire functions of H(E) with finite radius of convergence after we have characterized the bounding subsets to be relatively compact, see the examples of this chapter. In addition we can check : " W h e t h e r t h e Levi p r o b l e m will n o t h a v e a s o l u t i o n " if we can find a certain type of bounding subsets domain of E as Josefson [121] has done for the space loo. This chapter indeed explores an important point that : " N o t e v e r y h o l o m o r p h i c p r o p e r t y in locally c o n v e x s p a c e s c a n b e i n h e r i t e d b y n o n locally c o n v e x s p a c e s " . One of the reasons behind that is, for example, that the boundedness with respect to the original and the weak topologies are not the same. This has been explained via the study which has been held for the weaklybounding sets. It is hoped that the present study in this chapter will be helpful in claiming some achievments in pure and applied mathematics. The next chapter discusses the different approaches which are used to solve the Levi problem in seprable topological vector spaces. The following figure explains the Levi spaces among this class of the separable toplogical vector spaces.
Levi Problem in Separable t.v.s. Separable t.v.s.
FSpaces with plurisubharmonic
tp(1 > p > 0)
Locally pseudoconvex spaces(Lps) with b.a.p.
~logarithmic metric with f.d.Schauder decomposition
/.C.8.
Banach spaces with b.a.p.
9
t~
" I p n ~ p r t ""~ 0~
Ii x II=
Inductive spaces Up>tip ,
II 9 II= ~up~>0E~
~,~%~ Ix,, ITM Lps N Fspaces
loo 9is not a Levi space. Lp :(1 > p > 0) : is not aLevi soace. Figure 8 : S o m e L e v i s p a c e s a m o n g s e p a r a b l e t . v . s .
I~
I~
Chapter 10
LEVI PROBLEM IN TOPLOGICAL
SPACES
One of the important and interesting problems in the field of holomorphy is the Levi problem. To solve the Levi problem means to prove that the class of holomorphic functions has a certain type of richness. In one complex variable, given a domain ~ there exists a holomorphic function f in ~ which cannot be continued beyond the boundary of ~ as a holomorphic function. For instance we may prescribe the values f(zk) arbitrarily if (zk) is a discrete sequence in ~. This means that ~ is the natural domain of existence of f . In two complex variables, we have already domains like
no  {z ~r
1 <1 z I< 2},
which are not the natural domain of existence of any holomorphic function. That is, every holomorphic function in ~o can be extended holomorphically to
{z e~,~ I z I< 2
}.
E.E.Levi discovered in 1911 that the boundary of the natural domain of existence of every holomorphic function should satisfy a c e r t a i n c o n v e x i t y condition. He wondered:
W h e t h e r a n y d o m a i n w i t h this c o n v e x i t y p r o p e r t y is a n a t u r a l d o m a i n o f existence. In today's terminology he asked : 227
228CHAPTER
10 L E V I P R O B L E M
IN T O P L O G I C A L
SPACES
W h e t h e r a p s e u d o c o n v e x d o m a i n is a d o m a i n o f e x i s t e n c e ( o r a domain of holomorphy). Oka [182] proved the answer is affirmative in (T2; He, Norguet and B r e m e r m a n n [179] extended the result to (T~. For Hilbert spaces the Levi problem was solved by Gruman [113]; for Banach spaces with basis by Gruman ~Kiselman [114]. Dineen, Noverraz and Schottenloher [83] solved the problem in certain locally convex spaces. The problem has been solved for R i e m a n n domains spread over locally convex spaces with Schauder decomposition by Schottenloher [197]. The levi problem in locally convex nuclear spaces was solved by Coloumbeau ~ Mujica [78], thus exhibiting the first solution in a space which does not have a Schauder decomposition or is a direct subspace of a space with this propery. As for the negative results, Josefson [121] gave a c o u n t e r e x a m p l e : a pseudoconvex domain in the nonseparable Banach space lc~(A) which is not a domain of holomorphy. In this chapter we explain our extensions of these results to many classes of topological vector spaces which are not necessarily locally convex. It is worthy to point out here that : The author has obtained the corresponding property in 1979(or the geometric condition ), corresponding to what Levi has discovered, but for domains of non locally convex Fspaces. He has called it the P B  p r o p e r t y and asked : W h e t h e r a p s e u d o c o n v e x d o m a i n w i t h the P B  p r o p e r t y is a domain of existence (or a domain of holomorphy). These spaces are called P B  s p a c e s cussed here in detail.
(or Bayoumi
spaces). It is dis
Therefore in this chapter, we focus our attention on one of the central and important problems in this field of holomorphy without convexity condition, that is, the L e v i p r o b l e m . This problem will be studied in different classes of non locally convex spaces having finite dimensional Schauder decomposition (f . d. decomposition) or bounded approximation property (b. a.p.). In particular for locally bounded sequence spaces. We also discuss the different approaches that are used to solve it in those cases. That is, "To p r o v e t h a t e v e r y p s e u d o c o n v e x d o m a i n is a d o m a i n o f holomorphy".
10.1 LEVI
10.1
PROBLEM
AND
RADIUS
LEVI PROBLEM CONVERGENCE
OF CONVERGENCE
AND
RADIUS
229
OF
In this section we study the solution of the Levi problem using the radius of convergence approach. If E is a complex metric space and ~t is open in E, the radius of convergence Rf(x) of a holomorphic function f: ~ ~ (T is a positive number which measures the growth of f near x, to be precise
R f ( x )  s u p {r > O; Taylor series of f at x converges uniformly in B(x, r) } If
E
is of finite dimension,
i.e. E  ( ~ ,
the radius of convergence
Rf(x) of any holomorphic function f" ~ ~ ~T is greater t h a n or equal to the distance to the boundary dn, i.e.
R: >_da where inf I I x  Y l l x e Ft. da(x)  ycOa
In particular Rf = c~ for every entire function on ~Tn. If E is of infinite dimension, this is not always the case. Dineen [88] has noticed the existence of entire functions with finite radius of convergence. This fact also follows easily from the study which we have undertaken on bounding and weaklybounding sets in some separable Fspaces E, which are not necessarily locally convex, see the preceding chapter 9. In fact, there always exists an entire function f
on
E such t h a t
sup I f ( x ) I  c~ xEA
for every closed noncompact set A in E, provided that E is of the kind we considered in Ch.9. As an example is the function (X)
f(x)  ~~ x~ 1
which is an entire function and has finite radius of convergence
Rf(x)  1, every x E lp
230CHAPTER
10 L E V I P R O B L E M
IN TOPLOGICAL
SPACES
not only for 1 <__p _< c~, but also for 0 < p < 1. It can even happen t h a t inf R S (x)  0
xEA
for some f E H ( E ) , and some bounded A in E, see Theorem 112,p.238 and examples 41,p.233. In this P a r t we start with the radius of convergence approach to solve the Levi problem. It is then natural to get as a consequence of having a solution to the Levi problem, several other results in Complex and Functional Analysis. Let E denote either a locally pseudoconvex space or an Fspace D e f i n i t i o n 65 .A function f " U ~(T on an open subset U of E is said to be h o l o m o r p h i c if it is continuous and Gdteaux analytic i.e. f is analytic on every complex line of U. D e f i n i t i o n 66 An open subset U of E is said to be p s e u d o c o n v e x if  log 5u (x, a) is plurisubharmonic in U x E,
where 5v(x, a) 
inf{[ A [; x + Aa E OU}, x E U, a E E, A E ~T
(10.1)
= sup {r; x + rDa C U } , x e U, a e E, D unit disc in (/(10.2)
i.e. for every finite dimensional subspace F of E, the open set (U M F) pseudoconvex in F, see Noverraz[181]. The class of these subsets of E denoted by P C X ( E ) .
is is
D e f i n i t i o n 67 A function u " U . [oc, oc], where U is open in E, is said to be p l u r i s u b h a r m o n i c if it is upper semicontinuous and subharmonic on every complex line. The class of these functions is denoted by P S H ( U ) . D e f i n i t i o n 68 An open set U of E is called a d o m a i n o f existence, if there exists an f E H(U) which cannot be continued analytically beyond the boundary OU. D e f i n i t i o n 69 A space E is said to b a L e v i space if the Levi problem has a solution for E.
10.1 L E V I P R O B L E M A N D R A D I U S OF C O N V E R G E N C E 10.1.1
231
PBSpaces
A metric d on a complex metric vector space E is not always h o m o g e n e o u s or even phomogeneous, i.e. we do not necessarily have
d(tx, O)  ItlPd(x, O) (0 < p _~ 1 ) , t C ( ~ , x E E. So we found in 1979 t h a t it was useful to introduce the following concept in h o l o m o r p h y during the s t u d y of the Levi problem 9 70 A complex metric vector space E with a translation invariant metric d is said to have the p s e u d o c o n v e x b o u n d a r y d i s t a n c e p r o p e r t y ( P B  p r o p e r t y for short) if for every pseudoconvex domain f2 in E, it follows that the function
Definition
x ~  l o g dn(x) is plurisubharmonic in ~.
Such a space E is also called a P B  s p a c e , where dn(x) 
inf d(x, y) yEOf2
(10.4)
is the distance to the boundary O~ of ~. Note t h a t  log da(x) is not necessarily plurisubharmonic whenever ~ is pseudoconvex. Example
39 The space ~Tn is a P B  s p a c e for any norm,
see e.g. Hormander [118], where it is only assumed that the distance is measured by a 1homogeneous function which is positive except at the origin. Moreover a pnormed space E is a P B  s p a c e . This is easily seen from the definition of pseudoconvexity using the direction boundary distance defined by d(x, y ) = ] I x  YlIB, where Iltxll = ItlPllxlI.A Example
40 The space
lp  Uq<_plq  {x C lp;x C lq for some q <_p} with the metric (2<3
d(x, 0)  inf dq(x, 0)  inf q
q
Ixl q (q j=l
p
1)
232CHAPTER
10 L E V I P R O B L E M
IN TOPLOGICAL
is a PBspace. Notice that if U is pseudoconvex in lp,
SPACES
then
 logd u ( x )  sup  1Ogqday (x) q
oO
= sup s u p (  l o g ~ Ixj q
Y
yjl q)

j=l
is plurisubharmonic as supremum of plurisubharmonic functions, of course d is not phomogeneous for any p, (0 < p <_ 1). A
The following theorem gives a sufficient condition for a metric d to have the PBproperty. T h e o r e m 110 [12](1979) Let d be a translation invariant metric on a complex metric vector space E such that each of the functions (10.5)
f~ " s +  l o g d ( e * x , 0), s E r
is convex in (F for each fixed x E E. Then (E, d) is a PBspace. In particular, this is the case if d is given by a p homogenous norm ]1. ]]p, i.e. d ( x , v)  IIx  vll .
P r o o f . Assume that f~ is an arbitrary pseudoconvex domain in E. We claim that  l o g d a is plurisubharmonic in f~. Since the function (9.5) is convex and decreasing, it has the Legendre transform satisfying f:r,(s)  s u p ( r s 
~x(r)),
x E E, s E/~,
(10.6)
fz(s)),
x e E, r e
(10.7)
r<0
tgz(r)  s u p ( r s 8
T h a t is,
 l o g d ( e S x , O) 
s u p ( 7  s  ~(~,x)),
x E E , s E/~,
r<0
~ ( r , x)  s u p ( r s + log d(eSx, 0)),
x E E , ' r E 1~.
8
Changing the notation by putting s  log Itl, we have
10.1 LEVI
PROBLEM
AND
d(tx 0) 
RADIUS
OF CONVERGENCE
inf Itl'~(~  x) T>0
x e E , t e(T,
233
(10.8)
~
~(7, x) = sup Itl'd(tx, 0),
x e E , T e ~.
(10.9)
tcr
Note t h a t if d(x, 0) is phomogeneous, i.e. d(tx, O) = ItlPd(x, 0), t e r x E E, 0 < p _ < 1, then we get ~ ( p , x ) = d ( x , 0) w h e n T  = p o r o t h e r w i s e
~(~,
x) = +oo.
Now
da(x) = =
inf d(x,y) =
y60~2
inf
x+tz60~
inf
x+yEOf~
d(y,O)
d(tz, 0)  i n f inf inf It] ~ (~ z) z
t
T>_O
~D
= inf inf inf Irish(w, z)  inf inf ~(~, z)Sa(x z) ~ z
T>0
t
Z
T>0
for 5a(x, y) = inf(ltl; x + ty e Oft). Hence  log dfl(x) = sup s u p [  log ~(~, z)  7 log 5a(x, z)] z
(10.10)
T>0
and since this function is either   c o or else is continuous, it is plurisubharmonic as the s u p r e m u m of a family of plurisubharmonic functions. I R e m a r k 39 In the above theorem we have used the following property of a pseudoconvex domain f~ C E : if ~ C E and E is a topological vector space, then f~ is pseudoconvex if and only if the function
(x,y) H  l o g S a ( x , y )
is plurisubharmonic on ~ x E
w h e ?~e
5a(x, y) = sup(r; x + r D y C ~ ) , x E t~, y E E,
(lo.11)
D = {t C ~ ; Itl _< 1}. In fact, this is equivalent to our definition of pseudoconvexity, i.e. ft N F is pseudoconvex in F for every finitedimensional subspace F of E, see Noverraz [ [181], Lemma 2.1.5]. I As an application of the above theorem we get Example
41
A complex vector space E
d(x, y)
with the metric given by
llxyll  1 + IIx
YlI'
(10.12)
234CHAPTER
10 L E V I P R O B L E M
IN TOPLOGICAL
SPACES
where
I1 II i~ phomogeneous norm, (0 < p < 1) is a PBspace. that the function

log d(etx, O)  log(1 +
Iletxll)
 log
is convex in t ~T for every fixed homogenous.A
Iletxll
 log(1 + ozeltlp),
x C E. Notice that
d
We note 1 OZ'
IIxll
is not p
40 According to the definition of a PBspace E: for every pseudoconvex domain f~ in E,  l o g dn is plurisubharmonic in ft. The converse is also true: Indeed, assuming that  l o g d~ is plurisubharmonic in ft and F is an arbitrary finitedimensional subspace of E, f~ N F is pseudoconvex in view of the classical properties of plurisubharmonic functions of finitely many variables. Hence f~ is pseudoconvex. Consequently if E is a PBspace, then f~ is pseudo convex if and only if  log da is plurisubharmonic in f~.
Remark
10.1.2
Properties
of the Radius
of Convergence
Let us recall that the r a d i u s o f c o n v e r g e n c e R s ( x ) of a function f E H(f~) at a point x E f~ is the least upper bound of all numbers r > 0 such that the Taylor series of f at x converges uniformly in B ( x , r ) , the closed ball of centre x and radius r in E. Also the r a d i u s o f b o u n d e d n e s s Rb(x) of a numerical function
at a point x C f~ is the least upper of all numbers r > 0 bounded above in B ( x , r ) with B ( x , r ) C f~.
such that
u is
For a function f 6 H(ft),

sup(r > O; IlflIB( , )
is finite, B ( x , r ) C ~),
x E ~.
(10.14)
By these definitions of the radius of convergence R I and the radius of boundedness R} for a function f C H(ft), it is easy to prove t h a t
R}  inf(Rs, da)
(10.15)
where da(x)  infyeE/a d(x,y), x C f~, is the distance function on f~ defined by the metric d of E.
10.1 L E V I P R O B L E M
AND
RADIUS
OF CONVERGENCE
235
It may happen that f can be continued analytically beyond the boundary of Ft and hence R} will be less than R I (i.e. RbI <_ RI). It is clear that the radius of boundedness R b for any numerical function u is globally Lipschitzian continuous. However, the radius of convergence is only locally Lipschitz continuous. We formulate these results in what follows. L e m m a 25 If u
is any numerical function defined in a subset f~ o f a metric space E we define u(x)  +oo, Rb(x)  0 for x e E / a . With this convention we have
I n~(x)  n~(y) ,< d(x, y) P r o o f . If R b ( x ) 
Rb(y)
for all x, y C E .
(10.16)
the conclusion is valid; let us assume that
n~(x) > n~(y) > 0. Then u is bounded above in B ( x , r  d ( x , y ) ) C B ( x , r ) for all r < Rb(x). Hence Rb(y) >_ r  d(x, y) and letting r tends to Rb(x) we get
R b (x) > R b (y) > R b (x)  d(x, y). This proves (10.16) in this case, and by s y m m e t r y the estimate holds everywhere, m L e m m a 26 If f
vector space E
is a holomorphic function on a subset f~ of a metric with a translation invariant metric d satisfying
d(tx, O) then
<_ d(x, O)
for all x C E , t C(T, I t I<_ 1
I n~(x) n~(y)I<_ d(x, y), fo~ all x,y e a,
(~o.~7)
and d(x, y) < sup(df~(x), dry(y)). P r o o f . By s y m m e r t r y it is again enough to consider the case
hi(x) > h i ( y ) > o. Let w be the open ball B(x, R f ( x ) ) ~ and denote by g the extension of f l B ( x , da(x)) ~ to w. Then R g ( x ) = R l ( x ) and g is bounded in
236CHAPTER
10 L E V I P R O B L E M
IN TOPLOGICAL
SPACES
B(x, r) for all r < R l ( x ). Now, if R l ( x ) <_ d(x,y) we are done; assume t h a t R l ( x ) > d(x, y). T h e n y C w and if d(x, y) < d~ (x) or d(x, y) < d~ (y) the whole segment between x and y is contained in w A ~ must have Rl(Y )  Rg(y). By the preceding lemma,
so we
R ,b( y ) > R ,b( x )  d(x, y)  R z ( x )  d(x, y). In conclusion we have
> R f ( y )  Rg(y) > R gb ( y )  d(x,y) = Rf(x)  d(x, y)
Rf(x)
t h a t is, (9.16) is proved. This completes the prove of the lemma, m The following result shows t h a t R~ for f E H(f~) admits a certain geometric property which is a consequence of the PBproperty. Theorem
111
Let ~ be a pseudoconvex domain in a PBspace. Then  l o g R b is plurisubharmonic in
(10.18)
if u is plurisubharmonic in Consequently logRbf is plurisubharmonic in
(10.19)
for every f E H(gt). P r o o f . Let u be a plurisubharmonic function in f~. Let
~
 {x e ~; ~(x) < k}, k e N.
(~0.20)
For every k E N, the set f~k is pseudoconvex. This follows from the definition of pseudoconvexity and the fact u is plurisubharmonic in f~. Let
d~k(x )  inf d(x,y),
x e f2k, k C N.
YEO~ k
Hence
R b  supd~ k k
lira d~k, k~
(10.21)
10.1 L E V I P R O B L E M
AND
RADIUS
OF CONVERGENCE
237
that is  log R b  ir~f(  log d~ k).
(10.22)
The functions  l o g d~ k are plurisubharmonic in f~k for every k C N, since ftk is pseudoconvex and E is PBspace. Thus  l o g R b is plurisubharmonic as a decreasing limit of a sequence of plurisubharmonic functions. Since log l f ] is plurisubharmonic in ft for every function f E H(ft), we also get b ]   log R}  log/i~loglf
(10.23)
is plurisubharmonic in Ft. I Let
Ep
be a complex vector space with the phomogeneous norm
II.llp d(x,y) II x  y l l p ,
such that
IItxllp I t lp d(x, 0),
xEE, tE~. The following Lemma gives a formula for the radius of convergence R I for I E H(ft), ft C Ep open, which is well known for normed spaces, i.e. the 1homogeneous spaces (see chapter 7). The given formula is called the pCauchyHadamard formula(or the Generalized CauchyHadamard formula).
(pCauchyHadamard formula)The radius of convergence R S for f c H ( f ~ ) , ft is open inEp (O
Ri(x )  l i m i n f IIP ll , x e a
(10.24)
where Ilrnll sup
Ilxll
I rn(x) l
and Pn is defined by the formula f(x + y )  ~ Pn(y), Proof.
p.139). I
y E Ep, Ilyll is small.
(10.25)
Similar to the one given in Theorem 68 (Ch.7, subsection 7.2.3,
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The following theorem solves the radius of convergence problem stated by Kisleman [135], [136] 9 "Can one construct a holomorphic function with a prescribed radius of convergence?". D e f i n i t i o n 71 A topological vector space E is said to have a f i n i t e dim e n s i o n a l S c h a u d e r d e c o m p o s i t i o n (rid. d e c o m p o s i t i o n ) if there is a sequence of equicontinuous linear projections " E ,
n(E)
such that 7Tn ( E )
~ 00,
7~nOT~m ( x )

71"min(n,m)(x)
and x

lim ~n (x).
(10.26)
T h e o r e m 112 [12](1979) Let E be an infinitedimensional complex metric vector space, with a translation invariant metric d and with a finite dimensional Schauder decomposition (~n ) , such that x ,  log d(x, O) is plurisubharmonic on E. Let R " ft ~ ~+
be defined on a domain f~ c E  logR
with
p lurisub ha rmonic
and R <_ da.
Then, there exists a holomorphic function f E H ( ~ ) convergence
(10.27) with the radius of
(10.28)
R S _< R.
P r o o f . We shall construct, under the hypotheses of the theorem" a holomorphic function f on ~ such t h a t for a suitably chosen sequence (x~), which depends on the metric d, we have
I f ( x n ) I>_n,
nEN,
and R S < R
Here a covering (Vn) on ~ will be defined such t h a t
I[flly~  sup [ f ( X n ) I < xC Vn
for
n e N.
(10.29)
10.1 L E V I P R O B L E M
AND RADIUS
OF CONVERGENCE
239
The proof can be completed as in the forthcoming Theorem 117, p.247, (see also the author [12land Schottenloher [198]). T h e o r e m 113 [12](1979) Let ~ be a domain in an infinitedimensional metric vector space E with a monotone Schauder decomposition (~n) and with a translation invariant metric d such that log d(x,O) is plurisubharmonic function in E. Let R: ~   ~ + be such that
 l o g R plurisubharmonic in E
and R <_ d~
and I R(x)
R(y)]<_ cd(x,y),
for x , y e ~, d ( x , y ) < d~(x)
and some c e ]0,1]. Then there exists f e H ( ~ ) K(c) > 0 depending on c such that
and a constant
m
(10.30)
K R <_R f <_ R. Without assuming (~n)
K
to be monotone we get
r
(10.31)
_< Rf < R,
where r is continuous with r depends on only d and (~n).
r
> 0 , for t > O which
Proof. See the author[[12],Proposition], m (For similar results in Banach spaces, see Schottenloher [198], Coeure [73] and Kiselman [134],[135] ). E x a m p l e 42 Consider the function f ( x )  ~~ x~. 1
Recall that it is entire and has finite radius of convergence R f ( x )  1 for every x E lp
not only for l <_p <_ oo, but also for O < p < l. We show that it can even happen that
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SPACES
inf R S (x)  0
xEA
for some f E H ( E ) and some bounded set A in E (See Theorem 113, p.239 and Aron [~]). In fact, the problem of constructing holomorphic functions with this or similar properties is reduced to the much simpler problem of constructing plurisubharmonic functions with a certain growth by means of the foregoing result, Theorem 110. To illustrate how this result implies Aron's result [4]. We just take e > 0, and define r
= supn(I xn ] e) +
x
=
e
n
Then r is convex, hence plurisubharmonic, on every lp, 0 < p < cx~,and still unbounded in every ball B(x, r) of radius r > e. An entire function f E H(lp) with radius of of convergence R I <_ er has the desired properties. A In a particular case, we have the following interesting result for all Fspaces /(p~)which they are either locally bounded when Pn /+0 or non locally pseudoconvex metric spaces when Pn + O, see examples, p.85, 126. C o r o l l a r y 19 Let ~
and R
E
be as in the above theorem, and
l(p~), (1 > Pn > 0).
Then there always exists an f E H ( ~ )
with
RS = R P r o o f . See the author[[12],Proposition 3.b.1,p.20].
(10.32) I
Regarding the richness of the space of holomorphic functions in some metric space we obtain the following result where To denotes the compact open topology on H(E).
10.1 L E V I P R O B L E M
AND
RADIUS
OF CONVERGENCE
241
T h e o r e m 114 [12](1979) Let E, f~, and R as in Theorem 111. Then the set
TI(p)  { f e H ( E ) ; n I < R.}
(10.33)
is sequentially dense in (H(E), T0). Moreover, if f in addition is locally Lipschitz continuous and the Schauder decomposition is monotone, then
T2(p)  { f e H ( E ) ; K R < R/ < R}
(10.34)
is sequentially dense in (H(E), To) for a constant K. P r o o f . See the author [[12] , proposition 3.c.l,p.29]. II
10.1.3
The Levi Problem
in PBSpaces
In the preceding section we have discussed the problem of constructing a holomorphic function with a given radius of convergence. It has a relation with the Levi problem for some metric vector spaces E which contain in particular locally bounded sequence Fspaces. We have proved, in the preceding subsection 10.1.2, that for an open set U of E the radius of convergence, has the following properties: (i) logRf is plurisubharmonic (if d is a PBmetric) and R/ <_ du. (ii) I R / ( x )  R / ( y ) I < d(x, y) if x, y e U and d(x, y) < sup(du(x), dv(y)). In fact, it is worth to note that condition (i) implies the pseudconvexity of U provided that E is a PBspace. Hence the problem of finding a holomorphic function with a prescribed radius of convergence is closely related to the Levi problem which can be formulated as follows: " G i v e n a p s e u d o c o n v e x d o m a i n U, d o e s t h e r e e x i s t a f u n c t i o n f E H(U) w i t h R / < dv, t h a t is, w i t h U b e i n g t h e d o m a i n of e x i s t e n c e of f ?" As a matter of fact this close relation was one of the reasons to tackle the radius of convergence problem in non locally convex spaces and to obtain the results of subsection 10.1.2, mainly Theorem 112, p.238. Now we give the solution to the Levi problem in PBspaces as a consequence of Theorem 112.
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T h e o r e m 115 (Levi P r o b l e m i n PBspaces)[12](1979) Let E be an infinite dimensional PBspace with f.d. decomposition such that x , logd(x,O) is plurisubharmonic in E. Then every pseudoconvex domain U in E is a domain of holomorphy. P r o o f . Take R = du. Since U is pseudoconvex,  l o g R is plurisubharmonic. Applying Theorem 112, p.238, we find a holomorphic function f C H(U)with R I < R = du. This implies that U is a domain of existence of f, consequently a domain of holomorphy. This completes the proof, m C o r o l l a r y 20 (Levi P r o b l e m i n Locally B o u n d e d Spaces)J13]. Let E be a locally bounded space with f.d. decomposition. Then every pseudoconvex domain U in E is a domain of holomorphy. P r o o f . The topology of any locally bounded space can be defined by a phomogeneous norm for some p (0 < p < 1),see Rolewicz [186]. We can apply the above Theorem 115 noticing that E is a PBspace and x ~ log{{xll is plurisubharmonic by Lelong [169]. This completes the proof of the corollary. E x a m p l e 43 The spaces (X)
1
}
, (0<,o
_<1)
(~o 35)
with the metric d(x, y)  E l lxn [P~ has a Schauder basis. We recall that the linear and topological properties of l(p~) depend on the sequence (p~) we choose. For example, l(p~) is locally bounded if pn /+0 and is not even locally pseudoconvex if p~ ~ O, Rolewicz [[186], p. 155o]. All such spaces have the interesting holomorphic property that the radius of convergence problem has an exact solution, see Corollary 19, p.2~0. When pn /,0, l(p~) are locally bounded spaces which are Levi spaces. E x a m p l e 44 C o u n t e r e x a m p l e to the L e v i space: The space
L~ 
{f; j/~lI f I~< ~ }
(10.36)
of all measurable functions metrized by the metric, d(f, O)  f lo { F: Ip< cc ha~ th~ t~v~al d~al ~p~c~ L~  {r H ~ c ~ a d o ~ n ~ U # L~ ( p ~ d o c o ~ ~ ]
10.2 LEVI P R O B L E M ( G R U M A N  K I S E L M A N A P P R O A C H ) 243 or not) can never be a domain of holomorphy. That is, Lp is not a Levi space. For a counterexample of a locally convex non Levi space, Josefson [121] has given the space lcr P r o b l e m 1: Does there exist a non locally convex space E with a rich dual E' which is not a Levi space ? R e m a r k 41 The class of the P B  s p a c e s E has been discovered in1978 while the author was writing his thesis at Uppsala University. He found the solution of the Levi problem for a big subclass of it to explore the fact that : " T h e r e is a t h e o r y o f i n f i n i t e d i m e n s i o n a l h o l o m o r p h y w i t h o u t any convexity condition". This fact was unbelievable to many mathematicians at that time. The reason for that might be the fact that the dual space of Lp is trivial, that is np  {0}, (0 < p < 1); this was discovered by Day in19~1. Although we have given in this chapter several of the geometric and holomorphic properties of the class of the PBspaces, we still believe that there are many properties of it which have not been discovered yet. The PBspaces are sometimes called Bayoumi spaces by some mathematicians. !
10.2
LEVI PROBLEM (GRUMANKISELMAN APPROACH)
In this part we shall focus our attention on the basic approach given by Grumman and Kisleman [114] which they used to solve the Levi problem in Banach spaces with bases. In fact, this represents the general common method that has been used by several authors, see [113], [114],[100],[101],[197] and [78] to solve the Levi problem in different classes of locally convex spaces. The technique is : To c o n s t r u c t a h o l o m o r p h i c f u n c t i o n on a p s e u d o c o n v e x dom a i n U w h i c h is u n b o u n d e d on s u f f i c i e n t l y m a n y s u b s e q u e n c e s of a s e q u e n c e (x~) in U w i t h t h e p r o p e r t y t h a t t h e a c c u m u l a t i o n p o i n t s (Xn) a r e d e n s e in t h e b o u n d a r y OU of U. We use this method to solve the Levi problem for domains either in metrizable spaces with f.d. Schauder decomposition or in locally pseudoconvex Fspaces with the bounded approximative property (b.a.p.). We start with extending Pelczynski's theorem which deals with locally convex Fr6chet space, to locally pseudoconvex Fspace. This needs the following two lemmas which we state after introducing the following definitions.
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D e f i n i t i o n 72 A closed subspace M of a metrizable topological vector space E is said to be c o m p l e m e n t e d if there exists a continuous linear projection 7~ from E onto M ; hence E  M
 711(0).
(10.37)
D e f i n i t i o n 73 A topological vector space E is said to be a Fr~chet space if it is a complete metrizable space. If moreover E is locally pseudoconvex, that is, E has a fundamental basis of pseudoconvex neighborhoods of zero (Un), then E can be determined by a sequence of pnhomogeneous pseudonorms, see Rolewicz [186]. Notice that not every metric vector space is locally pseudoconvex, e.g. the spaces E =/(p~), p,~ /,0, with the metric d(x, 0)  ~  ~ I xj IPr  (xj) E E, are not locally pseudoconvex, see Rolewicz [[186], p.153]. On the other hand a locally pseudoconvex space is not necessarily metrizable. For example, if its family (U~) of pseudoconvex neighborhoods of zero is not countable, see also Example 3, p.20, of the inductive spaces Ul>_p~>olp, with the qtopology is nonmetrizable locally pseudoconvex space. Pelczynski [[183] ] has succeeded in proving t h a t : A locally c o n v e x F r ~ c h e t s p a c e has t h e b . a . p , if a n d o n l y if it is a c o m p l e m e n t e d s u b s p a c e of a m e t r i z a b l e locally c o n v e x s p a c e w i t h basis. In what follows we extend this fundamental result to all locally pseudoconvex Fr6chet spaces having the b.a.p. We need the following two lemmas: L e m m a 28 ( A u r b a c h ) L e t E be a pnormed space with d i m E  n and with the pnorm IlXllp. Then there exist n points (xi)? e BE(O, 1) of the unit ball of E, and n linear functionals (x*)'~ C E* such that
x~(xy)  5ij
and
IIx~llp sup I x*(x) I< 1
(10.38)
xCBE
where 1 < i , j < n, and with equality when p 
1.
P r o o f . Let V ( y l , ..., Yn) be the determinant of n vectors yi  (a/l, ..., a~) in BE. Assume that V takes its maximum at the points x l , . . . , Xn. Set
10.2 LEVI PROBLEM(GRUMANKISELMAN
x ~, ( x )

V(xl
,...,
,...,x~ x, Xi+l
APPROACH)
245
Xn)
Y ( X l , ...,Xn)
x e E.
,
(10.39)
Since V ( X l , . . . , X , . . . , X n ) ~ V ( X l , ...,Xn) for every x e B, then (x;)~ and (xj)~ satisfy the required properties. This completes the proof of the lemma. i The n points and the n linear functionals determined by the preceding lemma are called the A u r b a c h s y s t e m of E. L e m m a 29 Let E be a p  n o r m e d space with d i m E  n (0 < p <_ 1). Then there exist n 2 continuous linear mappings f r o m E into E with dim u i ( E )  1 such that
n2
x
~
u~(x)
for every
x e E
(10.40)
1
and k I
ui__2
for every
k, (l__k_
(10.41)
P r o o f . Let (xi)~ and (x~)~ be the Aurbach system of E. For i  r n + j , (0 <_ r
.~(x)
x;(x) ~j ,

x C E.
(10.42)
n
Then for krn+j
we obtain, k
rn
1
1
j 1 rn
_< sup (11~ Ilxll <1
1
k
u~(x)ll + II ~ ~+~(x)ll) 1
_< II III § sup ii . 1 Ilzl]
]
k
/.e. i l ~ u i i l 1
_< 1 +
sup i l x i l  < 2  I I iixi]~ 1
n
9 xj(x).~jll,
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R e m a r k 42 Lemma 28 can be extended to any finite dimensional invariant metric space E. In fact, (xi)~ represents the basis of E and (x*)~ represents its orthogonal dual in E' the dual space of E. The proof of Lernma 29 shows that the lemma is valid for any invariant metric vector space E of dimension n with a nondecreasing metric d. This is obvious from the inequality (9.~1). d is called a nondecreasing metric if d(tx, O) <_ d(x, O) for all t C (~ with It ]<_ 1. Note that d' (x, 0) = sup d(tx, 0) (10.43) 0
is a nondecreasing metric if d is invariant, Rolewicz [186]. Let us now state one of the main results of this section:
T h e o r e m 116 [13](1989) Let E be a locally pseudoconvex Fr~chet space. Then E has the bounded approximation property if and only if it is a complemented subspace of a rnetrizable locally pseudoconvex space with basis. P r o o f . Since the topology of E can be determined by a sequence of Pnpseudonorms, ]]xiip~ , it is enough to give the proof in the case of a complete pnormed space. This can be achieved immediately by using the above lemmas and the method used in Pleczynski [18] for a Banach space. I
D e f i n i t i o n 74 A domain U C_ E is said to be d o m a i n of h o l o m o r p h y if there are no domains U1 and U2 such that
r162
uf g ,
(10.44)
with the property that for every f E H(U) there exists f~ E H(U1) such that f IU2 fl IU2" D e f i n i t i o n 75 U is said to be a n a t u r a l d o m a i n o f e x i s t e n c e if there is a function f C H(U) which cannot be continued beyond the boundary OU of U, i.e. there are no domains U1 and U2 with
r and g E H(U1) such that
C_UNU1, U~U1,
10.2 LEVI P R O B L E M ( G R U M A N  K I S E L M A N A P P R O A C H ) 247
f ] u 2  g [u2.
The following theorem extends the results of [84],[114], [197], and [80] for locally convex spaces. T h e o r e m 117 ( L e v i p r o b l e m f o r m e t r i z a b l e spaces)J12](1989) Let E be an infinite dimensional complex metric vector space with a f. d. decomposition (~n). Then every pseudoconvex domain U in E is a domain of holomorphy Proof. We first assume that d is a translation invariant and (Trn) is monotone with respect to d. Since U is pseudoconvex then Un  U N PEn and (10.45)
O'n  Un n Trnl (Un1)
are pseudoconvex in En  7~n(E) for every n C N. Let (zn) be dense in U, z , E Un and such that d(zn, O) + du(zn) ~_ n.
(10.46)
Xn  Zn + )~nen+l
(10.47)
Let
(e, e E , , n , _ l ( e , )
 0 and d(en, 0)  1) and An e (T is chosen such
that d(xn, zn)  du(zn)
tin
2'
n e N
where the sequence (e~) is given by Cl  d u ( Z l ) / 2 , Cn  m i n ( d u ( z n ) , En_l, d u ( x n  1 ) ) / 2 ,
n > 1.
We define a cover (Vn) of the domain U to have 9 Xn C Yn+l/Yn, Xn E
Kn+l
(10.48)
where K~+I is some compact holomorphically convex set in a~+l. In fact the cover is taken as in the author[[12], Th. 3.a.1]. Let Xn{xEU;zrm(X)
EU,
for all m ~ n } ,
(10.49)
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IN TOPLOGICAL
Ln+l  {x Ca_~Tn+l; d(x, O) <_ n, do.n+ 1 (x) >__ en+l },
SPACES
(10.50)
Vn  N~>k>n {x E Xn; du(7rkx) > ~n ]" d("KkX, "KmX), 7Tin(X) E LAIn}
(10.51)
and V1  {Zl}, where L A denotes the holomorphically convex hull of a set L in U and da is the distance to the b o u n d a r y Oa of an open set a in U. Now we have the following properties 9 (1) x . c_ X . + l . ~ m ( x . )  vm a Xn and V. C_ V.+I. r.~ (V.)  Vm a Vnforall m>_n (2) For all x E V and r <_ du(x), there exists n E N with B(x, r) C_ Xn and 7rm(B(x, r)) C_ LAm for m > n. (3) For a l l x E U , t h e r e e x i s t h > 0 andnEN w i t h B ( x , 5) CVn. (4) The sequence (x,,) has the properties t h a t x , C Vn+l/Vn and hence Xn ~ 71"n+l (Yn). The set,
7rn+l(Yn) {x e LnA+I nTrnl(LA);dv(vr,x) >_ s ]d(TrnX,X)}
(10.52)
is not necessarily holomorphically convex since logdu(x) and logd(x, O) m a y not be plurisubharmonic functions in general. Thus we may have xn in 7rn+l(Vn) A although Xn ~ 7rn+l(Vn) as in (4). Fortunately we have the following lemma. L e m m a 30 The previous cover (Vn) of U has the property that, for every fixed n E N, there is a natural number mn <_ n with 7rn+l (Vmn) A C 7rn+l (Vn).
Proof.
(10.53)
S i n c e ~ c_ V2 C_ ... c_ U, it follows for a fixed n, 7rn+l(V1) ~ 7rn+l(V2)~ 7Tn+l(Vn).
Every set 71"n+l(Vn), m < n lies in the element A 1 n "Kn 1(LnA) n ... n 7r~n1 (LAm) Ln+
of the strictly increasing sequence {Zl} C (Ln+IA n 7rn 1 (LnA) n ... n r ~ 1 (L~))C ... C Ln+ 1 A nTrn I (LnA)
10.2 LEVI P R O B L E M ( G R U M A N  K I S E L M A N A P P R O A C H ) 249 of compact holomorphically convex subsets of O'n+l. This sequence covers Zrn+l (Vn) because A 1 A 7rn+l(Vn) C L n + 1 N "zn (Ln+l).
Hence there is an mn <_ n such t h a t 7rn+l(Vm,~)A C_ 7rn+l(Vn). Indeed this l e m m a deals, with the finite dimensional case where the different topologies ( locally and non locally convex) are equivalent. Note t h a t the sets 7rn+l(Vrn) lie in the ball B ( O , m  1) M E n + l , m _< n + 1. I To complete our proof of the t h e o r e m let us put now Kn+l 
7rn+l ( V m ) A
(10.54)
where rn is the biggest mn given by L e m m a 30. T h e n we can deduce from the previous properties and our l e m m a t h a t 9 (5)K~+I is compact holomorphically convex set in a n + l , Xn
~ Kn+l
and 7rn(Kn+l) C_ On.
By (4) and (5),
Xn E Vn+l/gn and xn
r Kn+l, n E N.
(10.55)
Therefore there exists a holomorphic function f~ E H(Vn+I) with
[ f~(Xn) I> 1 > IlfnllKn+~
(10.56)
where Ill.IlK = s u p g I fn(X). Now we apply the extension of G r u m a n and Kiselman's l e m m a [114], which says t h a t " F o r a n y h o l o m o r p h i c f u n c t i o n f o n cr~ = Un N 7r~l(u) a n d a n y e > 0 there exists a holomorphic function g on U with
Ilg
foTrnllv <
e and
[Igllvm < oo for all rn E N "
see Schottenloher [197]. We approximate the function fnOTr~ by a function g,, on V,,+I. Thus there exists a function gn E H(V) with [[giivm < oo, m E N and Ilgn fnOTrn+l[[V,,+l < min(I f , ~ ( x ~ ) I   1 , 1  Ilfnl[K~+l)Since 7r,+l(Vm,) C_ K , + I by (5), we have
Ilgllvmn
IIg'  fnO~~+lllv..~ + IlAoTr~+lllVm
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< 1 IIf~lIK~+l + IIf~lIK~+l ~ 1. Since Xn E V~+I we have
[ gn(X~) I>1 f ~ ( x ~ ) I 
I g~(Xn)  fn(Xn) l> 1.
Thus
I gn(Xn) l> 1 > IIg llv .,
n e N.
(10.57)
Let
hn  kn. g ~ ,
kn > O, O~n C N.
Then the function f = y~, hn is holomorphic in Vm~and by (3) it is holomorphic in U with ] f(xn)I>_ n. Thus U is a natural domain of existence of the function. Consequently U is a domain of holomorphy. Finally, for any metric d there is an equivalent metric which is translation invariant, Rolewicz [[186]. Th.l.l.1]. Also the metric d*(x, 0) Supnd(Trnx, 0) is equivalent to d and (Trn) is monotone with respect to d,. Thus the complete solution is established, m R e m a r k 43 Observe how the Lemma 30 has been used to overcome the absence of the convexity condition in the above Theorem 117, thus departing from the proof in the locally convex case where there is no need of proving it. The next corollary generalizes Schottenloher's results [[197], Cor. 3.4], for locally convex cases. C o r o l l a r y 21 (Levi p r o b l e m in locally p s e u d o c o n v e x Fspaces)J13] Every pseudoconvex domain in a locally pseudoconvex Frdchet space E with the b.a.p, is a domain of holomorphy. P r o o f . According to Theorem 116, E is isomorphic to a complemented subspace of a metrizable space with basis. Hence the corollary follows from Theorem 117. m
10.2 LEVI PROBLEM(GRUMANKISELMAN
APPROACH)
251
E x a m p l e 45 ( M a p p i n g s with R a p i d l y D e c r e a s i n g A p p r o x i m a t i o n Numbers ) Consider the space L(E) of all continuous linear mappings of a Banach space E. Let L(E) have the linear mappings norm. For To E L(E) we define the r th a p p r o x i m a t i o n n u m b e r s by a~(To) = inf {]IT Toll; T C L(E)
(10.58)
with dim T ( E ) < r}.
If lP(E)
{
[a,(To)] p < co
To C L(E); ~ r1
}
(10.59)
,
then on the intersection
(10.60)
S(E) = Mp>olP(E)
a metrizable non locally convex vector topology is generated by the sets
Bp(To, e)


T e S(E); ~
[ar(T

To)]p < e
,p > 0,~ > 0
r1
S ( E ) is complete and it has the bounded approximation property if E is complete, as it is shown by Pietsch[[184],p.139]. It is worthy to point out that the function
log dp(T, 0), T e S(E), is a plurisubharmonic function where oo
0) :
(10.61)
r:l
This is also the case for the function Oo
 log dn (T) 
sup  log ~ ToCO~
[a~ (T  To)]P, T e ~t
r=l
It is plurisubharmonic if ~ is any pseudoconvex domain of S(E). That is, S(E) is a PBspaces. So by Theorems 113, 115, p.,239, 242, it is a Levi space. Note that L(E) is not a Fr~chet space if E is not Banach. A
252CHAPTER
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E x a m p l e 46 ( T h e I n f i n i t e P r o d u c t o f Fr~chet Spaces) We show that the Levi problem can be solved for the product H ~ E j of Fr~chet spaces (complete metric spaces) Ej, when the topology of Ej is given by the metric d(x, y)E~  IIx 
yll~j,
II.ll~j
where is a homogeneous pseudonorm, j E N, and each Ej has the bounded approximation property (b.a.p.), j E N. Let us first show that: the product
E
II~~
(10.62)
has the b.a.p, if each Ej, j C N, has the b.a.p. By Theorem 114, the Fr~chet space Ej, j E N, is a complemented subspace of a space Fj with basis; hence it can be written as Ej  Fj 
(10.63)
Gj.
and there exists a continuous projection IIj
nj.Ej,Fj with H~1 (0)  Gj. Then E  nFEj
 nFFj
@ nF
Gi
Let r 9 ~ N ~ [0,~oo] be convex and homogeneous of degree 1, Fnormllxl] E can be defined for x  (Xl, x2, ...) e E by
]l.llE  r
(10.64) t h e n an
IIx211E=,),
For example we can take oo
r
IIx211E~., ...) 
(~~ IlxjllqEj)1/q j=l
where 1 <_ q <_ co, and II.IIE~, is pjhomogeneous with, e.g. p j ~ O a s j + (X).
Since Hj is continuous,
Ilnj(xjDIIEj ~ Cj IlxjlIEj, Now for x 
(Xl, X2, ...) C E we define
Cj e R,j e N.
10.3 L E V I P R O B L E M ( S U R J E C T I V E
LIMIT
APPROACH)
n(x)  (nl(Xl), n2(x2), )
253
(10.65)
Then [[II(x)[tE
][(IIl (Xl),II2(x2), ...)]lE
 r
IlIh(x2)ll,.) oo
= ( y ~ l l i I j ( x j ) l [ E~ q)l/q j=l oo
j=l (:x)
< (supCj)(y~llxjllq)l/q Ej 

J
'
j=l
i.e. IIn(x)llE ~ CIIxlIE. T h ~ II i~ co~tin~o~ on E  n ~ Ej. Consequently E is a complemented subspace of the space I I ~ F j  H ~ Gj where I I ~ F j has basis. Applying Corollary 21, p.251 we will have that E has a solution to the Levi problem, e.g. take
E 

ll •
• 11/3 •
H~ll/n
(10.66) (10.67)
Of course E is neither locally bounded nor locally convex A
10.3
LEVI PROBLEM (SURJECTIVE LIMIT APPROACH)
In this section we deal with : " T h e p r e s e r v a t i o n of t h e p r o p e r t y of b e i n g a L e v i s p a c e u n d e r surjective limits". Our setting here is the L o c a l l y p s e u d o c o n v e x ( L p s ) which may not be metrizable or locally convex. This will include the special class, the pBanach spaces with the bounded approximation property, (1 > p > 0). We shall first describe some basic properties of surjective limits which can be obtained conveniently as for locally convex spaces.
254CHAPTER
10 L E V I P R O B L E M
D e f i n i t i o n 7 6 A Lps E write
IN TOPLOGICAL
is a s u r j e c t i v e l i m i t of Lps
SPACES
(El)lEA
E = lim Ei
(10.68)
icA
if there exists a continuous linear mapping ui from E ~i : E ~ Ei,
each i E A
and we
onto Ei, i.e.
(10.69)
and the inverse images of the neighborhoods of 0 in Ei , as i ranges over A, form a basis for the neighborhood system at 0 in E. That is, for each neighborhood W of 0 in E, there exists an i C A and a neighborhood V of 0 in Ei such that 7r71(V) C W. E
is called o p e n s u r j e c t i v e l i m i t if 7ri is an open map for each i C A.
The surjective limit approach was used by Hirschowitz [119] in studying the Cartesian product of the complex planes, and by Dineen [84] in characterizing the collection of locally convex spaces (l.c.s) in which certain holomorphic properties are true. The following example has useful consequences. E x a m p l e 47 C P ( E ) denotes the set of all continuous pseudonorms, i.e. p ,  s e m i n o r m s for some p,, (0 < p , < 1) on a Lps E. For each q C C P ( E ) , let Eq denote the vector space E equipped with the topology generated by q and let 7rq denote the canonical map from E onto E / q  l ( o ) . Then E has the surjective representation ( E / q 1 (0), nq) which will be called p s e u d o n o r m
(10.70)
s u r j e c t i v e r e p r e s e n t a t i o n of E. A
In what follows we introduce a big collection of Lps spaces which are closed under the operation of surjective limit. L e m m a 31 [14] The collection of locally pseudoconvex spaces (Ei)icA with the approximation property (a.p.) is closed under the operation of surjective limit. That is, lira Ei has the a.p. if each Ei, i C A, has the a.p.
10.3 LEVI P R O B L E M ( S U R J E C T I V E LIMIT A P P R O A C H )
255
P r o o f . For the proof notice that the a.p. of a topological vector space E does not depend on whether or not E is locally convex. But it does depend on the existence of a family (Tri)icA of finiterank of continuous linear functionals 7li : E~ E which can be used to approximate the identity mapping I. So we may follow here Dineen's method [[84] Example 2.31 for a locally convex space. This completes the proof of lemma, m The following proposition will be helpful to reduce the study of the Levi problem for Lps to spaces which have continuous p~ norms (0 < pa < 1). P r o p o s i t i o n 4 [14]Every Hausdorff Lps E with f.d. decomposition (~i) is a surjective limit of spaces Ei, i E A, where Ei has a continuous p i  n o r m and f. d. decomposition. P r o o f . This can be established in a manner similar to that of Dineed [3, Example 2.4] taking into account Example 47, p.254, m The next result gives a sufficient condition on a pseudoconvex domain U of a Lps E in order to have U  7F1(71(U))
(10.71)
7r" E ~ E / q  l ( o )
(10.72)
where
is the quotient map and q is a continuous pseminorm on E. L e m m a 32 Let U be a pseudoconvex domain of a Lps E. Let q be a continuous pseminorrn on E, (0 < p < i) such that Bq(O, 5) C_ V for 5 > O. Then u = u + {y; q(y) = 0 } . (10.73) P r o o f . The proof is a finite dimensional one which does not depend on whether or not E is Lps. So we can follow Dineen's method [[84], Lemma 1.1] taking into account the result of Hirschowitz's [119]. m The following lemma is of special interest when we deal with the solution of the Levi problem. The proof is analogous to that given by Dineen ([84], Lemma 1.8) using Lemma 31 instead of the corresponding result in normed spaces.
256CHAPTER Lemma
10 L E V I P R O B L E M
IN TOPLOGICAL
SPACES
33 Let U be an open pseudoconvex subset of open surjective limit lim(Ei, 7ri)
(10.74)
lEA
of locally pseudoconvex paces Ei, i C A. Then U is ~iopen for some i C A. That is,for every x E U, there exists Vz open in Ei such that x C 7ril(Yx) C U.
(10.75)
I We are now in a position to introduce a basic theorem, which will enable us to obtain the solution of the Levi problem for Lps spaces. T h e o r e m 118 [1~{1(1989) The collection of Hausdorff Lps spaces with the b.a.p, in which pseudoconvex domains are domains of holomorphy is closed under open surjective limit.
That is, the open surjective limit lim(Ei, ~i) iEA
of the Levi spaces Ei, i C A with the b.a.p, is a Levi space. P r o o f . To establish the proof, we assume that U is a pseudoconvex domain of the surjective limit
lim(Ei, 7ri). lEA
Then there exists i E A such that v

with ~i(U) a pseudoconvex domain, see Lemma 33. By the assumption, ~i(U) is a domain of holomorphy. Now if there exist open connected sets U1, U2 in E such that
u2 c_ u, u2 c_ u n u 1 and for each f E H ( U ) , there is an fl E H(U1) with f [u2 fl Iu2
(10.76)
10.3 LEVI P R O B L E M ( S U R J E C T I V E LIMIT A P P R O A C H )
257
Then, as a consequence of 7ri being open, we obtain 7ri (U1) C_ 7ri(U2). Note that :ri(U) is a domain of holomorphy. Hence U1 C_ 7r~l(Tr(iV))  U, and consequently U is a domain of holomorphy. This completes the proof of the theorem. II We are now in a position to give the solution of the Levi problem in Lps having the b.a.p. (The case of L.c.s.. was considered by Dineen [84] and Schottenloher [197] ). 119 (Levi problem in Lps)[14(1989 ) Let E be a Hausdorff Lps with the b.a.p. Then every pseudoconvex U in E is a domain of holomorphy.
Theorem
P r o o f . Note that every Lps E can be expressed as a surjective limit of p~normed spaces with continuous p~norms q~ (see Example 48, p.254). Hence applying Theorem 116 and the solution of the Levi problem for pnormed spaces with b.a.p. (see Corollary 20, p.242) we obtain the required result and the proof is complete. II The following is an example of a nonmetrizable Lps. E x a m p l e 48 ( T h e inductive space ). Ul>_p>0lp
(10.77)
We recall that we have defined the qtopology on Ul>_p>olp to be the strongest vector topology such that each injection
ip " lp ~ Ulp
(10.78)
is continuous. The space Ulp with this qtopology is a complete separable non locally convex Lps and has the unit vectors (e~) as its symmetric Schauder basis. A sequence converges in Ulp if and only if it is contained in and converges in some lp. A set is compact in Ulp if and only if the set is contained and compact in some lp. Note that no closed infinite dimensional subspace of Ulp is metrizable. Hence Ulp itself is not metrizable, see Stiles
[203]. Since E is Lps with a Schauder basis, Theorem 117 implies that E is a Levi space. A
258CHAPTER
10.4
10 L E V I P R O B L E M
IN TOPLOGICAL
SPACES
LEVI P R O B L E M (QUOTIENT MAP APPROACH)
In this P a r t we solve the Levi problem in some s e p a r a b l e p  B a n a c h s p a c e s , (0 < p _< 1). T h a t is, in a complete locally b o u n d e d space with phomogeneous norm. (The case p = 1 corresponds to Banach spaces). Since every separable Banach space E is isomorphic to a quotient space of
lo = N p > l p , t h a t is,
E ~_ lo/M for a closed subspace M of E, see Stiles [203], and since E is a Levi space, see Corollary 20, p.240, we can obtain the following interesting partial result for the Levi problem. 120 (Levi p r o b l e m in certain separable B a n a c h space)[l~] Let U be a pseudoconvex domain in a Banach space E. Suppose that E is isomorphic to a quotient space of lo, that is,
Theorem
E ~_ lo/M,
(10.79)
and lo/M has the bounded approximation property. Then U is a domain of holomorphy. P r o o f . Since l0 is a Fr~chet space, its quotient space lo/M is a Fr6chet space. Now the space lo/M is assumed to have the b.a.p. Hence a direct application of Corollary 20, p.242, will imply the required result.This completes the proof of the theorem, m Let Ep be a pBanach space. The fact t h a t every separable locally bounded space is isomorphic to a quotient space of lp(O < p <_ 1) will help us to obtain the following consequence which is of special interest. 121 (Levi p r o b l e m in separable p  B a n a c h space)[l~] Let Ep be a separable pBanach space which is isomorphic to a quotient space of lp , that is E "" lp/M (10.80)
Theorem
and lp/M has the bounded approximation property. convex domain in Ep is a domain of holomorphy.
Then every pseudo
10.4 LEVI P R O B L E M ( Q U O T I E N T M A P A P P R O A C H )
259
P r o o f . Ep is isomorphic to lp/M for some closed subspace M of lp. lp/M is a Fr~chet space, and by assumption it has the b.a.p. Then by applying Corollary 20, p.242, we achieve the solution and the proof is established. II R e m a r k 44 The assumption that lp/M
has the b.a.p, cannot be dropped out of the above theorem. We note that Lp[0, 1], (0 < p < 1) does not admit holomorphic functions other than 0, and so each domain (pseudoconvex or not) is not a domain of holomorphy.
P r o b l e m 2.What is the class of separable pBanach spaces each of its elements is isomorphic to a quotient space lp/M with the b.a.p.? C o u l d H a r d y s p a c e Hp(0 < p < 1) be a n e l e m e n t of t h i s class? We note that lp(0 < p < 1) is an element of this class since it has a basis and hence this class is nonempty. We give now more examples of nonlocally convex spaces which have either a Schauder basis or the bounded approximation property. In fact, by the results of this chapter, all pseudoconvex domains in these spaces are domains of holomorphy, that is, they are Levi spaces. E x a m p l e 49 The H a r d y spaces
Hp(1 > p > 0).
The Hardy space Hp of all analytic functions f on the unit disc of (~ is separable, locally bounded, non locally convex space with respect to the pnorm
Ilfll
~~llim/02~ I f @ d ~ Ip dO,
f
H,(r
The Banach envelope of Hp is isomorphic to 11, that is H p ,.o ll
(10.81)
by Kalton [130]. This implies that H p has the b.a.p. Moreover, Hp contains a non locally convex closed subspace Mp of E isomorphic to lp (0 < p < 1), that is
Mp ~_ lp,
(10.82)
see Shapiro [189]. Now every pseudoconvex domain in this complemented subspaces M of
Hp, in H ~ is a domain of holomorphy, by corollary 20, p.242. That is, H p and Mp are Levi spaces. A
260CHAPTER
10 L E V I P R O B L E M
IN TOPLOGICAL
SPACES
R e m a r k 45 ( I m p o r t a n t ) T h e previous results of this chapter can be generalized to nonschlicht domains over a suitable space E. Let us first recall the following concepts which are analogues of those considered by Schottenloher [19~ for locally convex spaces : A Riemann
d o m a i n s p r e a d o v e r a metric vector space E
(ft, q) where ft
is a pair
(10.83)
is a connected Hausdorff space and
q" ~ ~ E
(10.84)
is a local homeomorphism. That is, for every x E f~ there exists a neighborhood w of x such that qw " W ~ E (10.85) is a homeomorphism of w onto q(w). If q is injective, the domain (ft, q) is called s c h l i c h t d o m a i n , and can be identified, via q, with a domain in E. The boundary distance function dn on a fixed domain (ft, q) over E is defined by: dn(x) 
sup(r; there exists a neighborhood U of x s.t.
q I u " U ~ B ( q ( x ) , r ) is a homeomorphism), x E f t
(lo.86) (lO.87)
The ball B ( x , r ) for x E ~, r > dn(x) is just the component of q  l ( B ( q ( x ) , r ) ) which contains x. The plurisubharmonic, the holomorphic or for that matter any locally defined class of functions, can now be defined on (ft, q) using restrictions q[~ of the projection q, see the author[12], [13], and [1~].
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278
Notations
~T: Space of complex numbers. K: Scalar field. (Tn: Space of n complex numbers. C~' 9Space of continuousely mquasidifferentiable maps. ~ : Space of n real numbers. A" Polynomial corresponding to a multilinear map A. a: Constant > 1. E ' : Continuous dual of E. [[T[[ : Norm of T. ElM" Quotient space. PBspace : Bayoumi space. 0U : Boundary of U. Ilxll: Norm(or pNorm ) o f x. FD: Fr~chet Differential. GD: Gateaux Differential. QD: Quasidifferentiable (or Bayoumi differentiable) maps. Dr(a): Quasidifferential (or Bayoumi differential)of f at a. D 2f: Second Quasidifferential. D m f : m th Quasidifferential. dr(a) = D f ( a ) : Bayoumi quasidifferential of at a. d2f(a) e L(2E; F)" Isomorphic to D2f, a bilinear map. deg T : Degree of a mapping T. a~(T): r th approximation numbers. (ELM) ~_ M • 9Annihilator of A. RI (x) : Radius of convergence of f. R} (x). Radius of boundedness of f. Hp(1 > p > 0) : Hardy spaces. I I ~ E j " Infinite product of Ej. NLP: Nonlinear programming. mrnq /P m! : New universal constant. qu(x): Minkowsky functional. E  limieA Ei 9Surjective limit of Ei. d(x, y): Distance between x, y. H ( E ) : Entire functions(continuous and Gateaux analytic). Hbb(E) 9Entire functions of type ~ CX, bounded on bounded sets of E. /)" Linear completion. !
279 Ea : Holomorphic completion.
HBEP: Hahn Banach extension property. HBSP: Hahn Banach separation property. CP(E) : The set of all continuous pseudonorms. (f t, q) : Schlichet domain.
La(E, F): Space of linear mappings. L(E, F): Space of continuous linear mappings. La(mE, F): Algebraic mlinear maps of E into F. L~s(mE, F): Algebraic symmertric mlinear maps of E into F. L(E m, F): Space of continuous mlinear maps of E into F. L~(mE) = Ls(mE,K) : Continuous symmertric mlinear maps. L(E1, .., Em; F): Continuous multilinear maps of E1 x ... x Em into F. P~(mE; F): Algebraic mhomogenous polynomials of E into F. P(mE; F): Continuous mhomogenous polynomials of E into F. p(mE): Continuous mhomogeneous polynomials of E into K. A b  { (1  t)l/pa + tl/pb}, 0 _< t _< 1 "Arc segment between a and b. BE(0, 1): Unit ball of E of center at the origin. Cp(A)" Closed pconvex hull Of a set A. Ep(K) : p  E x t r e m e points of K. d(x, O) =11 x lip: pnorm defined by a phomogeneous metric d. L p ( I )  {f; f : l f
Ip< oo}" Lebsgue integrable functions on [0, 1].
A(x) = f (x)  f (a)  T(x  a): Affine linear functional. QA (U,F): Quasianalytic (Bayoumi analytic)maps from U to F. QH (U; F): Quasiholomorphic(Bayoumi holomorphic)maps fromUto F. QA (U) = QA (U; IK): Quasianalytic( Bayoumi analytic)maps. QH(U) = QH(U,~) : Quasiholomorphic(Bayoumi holomorphic)maps. Qc~ (U, F) : QuasiC~Analytic maps from U to F. ~(a~_l ~) 9Characteristic function on (c~i1, ai). deg(f, M, x ) : Degree of a mapping f On M at x C f(M). det(A) : Determinant of A.
II I llA= sup A JI (x)IT0: Compact open topology. Bdd(E) : Bounded susets of E. WBdd(E) : Weakly bounded subsets of E. Bg (E) : Bounding sets in E. WBg (E) 9Weaklybounding sets(Limited sets) in E. WbBg (E) :Bounded Weaklybounding sets(Bayoumi limited sets) in E. P C X ( E ) : O p e n pseudoconvex sets of E. PSH(U) : Plurisubharmonic functions on U.
280
a.p." Approximation property. b.a.p: Bounded approximation property. [ei]~ 9Cosets of the basis ei. H~ 9Banach envelope of Hp, l~  {x  ( ~ ) ; x~ e r E ~ I~nP < ~ } , ( 0 < p < 1). l o  n ; > 0 1 ;  { x  (x~); sup;>0 E ~ I ~ I;< o o } . l~Uq<_plq{xElp;xElq for some q__p}, where d(x, 0 )  infq dq(x, 0 )  infq E l lXj I~, q s 1 l+  Nq>_plq  {x E/q; supq>__p~}7 I xj Iq< c~ }, (0 < p _< 1). f ~ ( s )  sup~<0(Ts ~ ( T ) ) " Legendre transformation of ~ where, ~ ( ~ )  s u p ~ ( ~  f~(~)), x e S , ~ e ~, ,~ e ~. dmf (a) =re!Am "Isomorphic to D m f (a),using mlinear map Am. dmf (a) =m!P~" Isomorphic to Dmf(a) by mhomogeneous polynomial Pm. L(mE, F) ~ L ( E , L ( m  I E ; F ) " Space of m times repeateadly continuous linear maps. lp(E)" {T E L(E); ~}~1 [a~(T)]P < c~}. Space of rapidly decreasing approximation numbers. 5u(x, a)= inf(]tl; x + ta E OU), x c U, a E E, U C_ E. Distance function to the boundary of U in t.v.s. l ( p n )  {x = (Xn); Xn E(~, ~~c~[xnlP,~ < oo}" Locally bounded Fspaces, if Pn /*0, and non locally pseudoconvex Montel space if P n ~ O. Hb(E) 9Entire functions of type ~ r r e E' bounded on weakly bounded sets of E.
Index Affine maps, 63 Annhilator of subspace of Metric space, 193 Applications to NonConvex analysis to Operations research, 58 Arc segment in Vector space in t.v.s., 50 Baire Category Theorem, 20 Banach Open mapping pBanach space, 23 BanachSteinhaus type Theorem for Polynomials, 46 BanachSteinhaus Theorem for Linear maps Continuous polynomials, 25 Bayoumi Space PBSpace, 231 Bayoumidifferentiability Quasidifferentiability Interchange order with limit, 96 Bolzano IntermediateValue theorem Finite and Infinite spaces, 163
Bounded Linear map Norm, 3 Bounded approximation property (b.a.p.) Approximation property, 186 Bounding sest in Lbs Bounding sest in Fspaces Bounding sets in Lps and t.v.s., 186 Bounding sets (Bg) Weakly bounding sets (WBg),Limited sets BoundedWeakly bounding sets (BWBg),Bayoumi Limited set, 181 Bounding sets and Holomorphic Completions, 220 Bounding sets in Non Lps in t.v.s., 189 Brouwer Fixed point Generalized, 63 9 Cauchy Sequence, 5 CauchyHadamard formula Generalized Radius of Uniform convergece, 139 CauchyHadamard formula Radius of convergence 281
282 Holomorphic function, 237 Compact pExteme set pConvex set, 56 Complemented subspace, 244 Completion Linear Holomorphic, 207 Complex MeanValue theorems MeanValue Inequality, 94 Construction of holomorphic maps on locally bounded spaces on Lps Fspaces, 199 Continuity of Linear Map, 3 Continuous Mapping Derivative, 3 Continuous dual space of E ( E' ), 194 Convex Function Set, 182 Counterexample to Levi space Non Locally convex space, 242 Criterion for continuity, 40 Curvilinear integral of Maps of VectorValued functions, 160 Decreasing Approximation numbers Sequence, 19 Degree of Polynomials
INDEX Degree theory Degree of maps, 124 Derivative Directional, G~teaux, 111 Differential pqDifferential QuasiDifferential (Bayoumi Differential), 77 Dineen, 14 Domain Pseudoconvex of Holomorphy, 228 Rienhardt, Logarithmically convex, 72 Entire function Continuous and Gateaux analytic, 229 Equicontinuous Maps, Family Group, 63 Equivalence relation Vector space of maps of Quasitangency, 78 Examples Non Metrizable space Fspaces, 18 Extension Property Problem, 7 Extension Theorem in Lbs and Lps in t.v.s., 11 A
Fixed points theorem of Brouwer, or of Kakutanis Generalized, 63 Fr~chet spaces Product of
INDEX
Complete metric spaces, 205 Function Regular Bounded, 158 Functional Analysis on pNormed space, 1 Fundamental Theorems in Functional Analysis, 1 Fundamental Theorem of Calculus Riemann Integration Step functions, 157 HahnBanach Extension property(HBEP) Separation property(HBSP), 7 Hardy space Banach envelope of Closed subspace of, 259 Hausdorff Lps, 255 Holomorphic G~teaux Analytic and continuous Quasi, 124 Holomorphic extension problem Dense subspaces Closed subspace, 221 Holomorphic functions Three different classes of, 196 Holomorphic mappings Fixed points of, n non convex domins in ~ , 63 Holomorphically complete Spaces Example of, 217 Homeomorphism between Lbs, 22
283
Homotopy Continuous maps Invariant, 167 HSrmander, 72 Image of separable Banach spaces
(Zo) Image of separable pBanach spaces (Lp), 202 Inductive space Locally pconvex space Locally pseudoconvex spaces, 257 Infinite dimensional Complex analysis Discussion on, 224 Holomorphy case, 1 Infinitely Quasi diffe re nti als Continuousely, 152 Integral MeanValue theorem on Arc segment, 176 Integral domain of QuasiAnalytic maps, 155 Internal point of Arc segment of pExtreme set, 52 Inverse mapping theorem of Quasidifferentiable map, 85 Isometry Natural Isomorphic, 34 Josefson Counterexample to Levi problem, 228 Limited sets, 212 Kspace
284 Threespace problem, 11 Kakutani Fixed point theorem Generalized, 66 Kalton, 14 Theorem Failure of linear extension, 10 Kisleman Approach Radius of convergence, 239 KreinMilman Theorem pConvex hull of pExtreme points, 68 Lemma
Approximation Schwartz, 96 Zorn Aurbach, 10 Levi problem and Surjective Limit Approach in Lps, 257 Grumman and Kisleman Approach, 243 in Metrizable spaces in Locally pseudoconvex spaces, 247 in PBspaces (Bayoumi spaces) in Fspace, 241 in Separable pBanch spaces and Quotient Mapping Approach, 258 Radius of convergence Approach Bayoumi PBspaces, 242 Linear Mapping Functional, 2 Space, 1, 8
INDEX Lipschitzian property Generalized, 80 Local Property Constant map, 153 Locally Bounded space Pseudoconvex space, 8 Locally bounded Space Fspace, 2 m Quasidifferential, 107 mExpansion of maps Finite Uniquness of, 125 mLinear map 2Linear map(Bilinear) Contiuous ...L{,m E,F), 30 mQuasi tangency mquasi differential, 124 Mapping into Product spaces Quasidifferentiable, 82 Martin's theorem, of pNormed spaces, 41 Mathematical Programming Nonlinear, Constraints, 61 MeanValue General Theorem of Integral, 131 MeanValue theorem Lagrange Real, or Complex, 90 Metric Non decreasing by Fnorm, 246 Metric space Translation invariant
INDEX Complete, 229 Milman Generalized theorem of pExtreme points, 50 Multilinear maps, of pBanach space Continuous,, 29 Multinomial Formula Binomial Formula, 36 Natural domin of Existance Levi space, 230 Normalized sequence of disjoint supports, 223 Open mapping theorem of Locally bounded space, 20 p homogeneous norm, quasinorm Polynomial, 24 pConvex combination pConvex hull of Compact set, 58 pConvex hull Absolutely, 208 pConvex set Absolutely, 51 pExtreme Points sets, 50 Path of type Arc segment in r 160 PBproperty Pseudoconvex boundery distance, PBSpace (Bayoumi Space), 231 Pelczynski theorem Fr~chet space with b.a.p.
285 Complemented L.c.s. with basis, 244 Permutation, 35 Physics Thermodynamics Application, 8 Plurisubharmonic Functions Logarithmically...metric, 230 Pointwise Separation property, 16 Polarization Formula of Symmetric mLinear maps, 38 Polynomially convex of Ball of Set, 181 Polynomials mHomogenuous, Continuous, 35 Power series Properties between pNormed spaces, 136 Uniqueness of, Quasidifferential of, 146 Principle of uniform boundedness Equicontinuous family Bounded pointwise, 25 Product of pNormed spaces Cartesian, 29 Properties Radius of convergence Radius of boundedness, 234 Properties of Bounding and Weakly bounding Compact linear maps, 206 Property Finite intersection,
286 PBproperty, 70 Property (p) Extension property Separation property, 11 qBanach Space Separable, 5 qnormed space, 5 Quasi tangent pqTangent of Two maps, 78 QuasiDifferential norm, 2 QuasiAnalytic Principle of, Maps, 151 QuasiAnalytic Continuation Strong form Weak form, 153 QuasiDifferential Higher, G~teaux, 106 Second , Higher, 101 Quasidifferential Properties of, Bayoumi, 79 Quasidifferential (or Bayoumi differential) of Multilinear maps of Polynomials, 83 QuasiHolomorphic (or Bayoumi Holmorphic) Maps Function, 123 Quotient space Quotient map
INDEX Complemented space, 185 Radius of Absolute Convergence Normal Convergence, 144 Radius of convergence Radius of boundedness of Holomorphic function, 229 Relationship between Bayoumi and Frdchet differentials Bayoumi (Quasi) and G ~teaux differentials, 115 Relatively compact sequentially compact, 185 Riemann domain Spread over Metric vector spaces, 260 Rolewicz, 183 Metric Linear Spaces, 2 Schauder Basis Decomposition, 185 Schwartz Symmetric theorem Generalized, 103 Separable Banach spaces Levi Problem in some Separable pBanach spaces, 258 Separation Property Problem, 7 Separation Theorem in Lbs (Locally bounded sp.) in Lps (Locally pseudoconvex sp.), 12 Sequence spaces with Logarithmically convex balls
INDEX with pConvex ball, 73 Space Locally pconvex Locally pseudoconvex (Lpc, 52 of Continuous linear maps L(E,F) of Continuous Polynomials P(E,F), 4 of Convex type Sequence,, 73 Orlicz Montel or Schwartz, 68 Sublinear Functional Absolutely~ 8 Surjective limit of Lps Preservation of Pseudonorm Surjective representation, 254 Taylor formula with Lagrange Remainder with Assymptotic property, 133 Taylor Polynomial Differentiation of Taylor formula, 129 Theorem Generalized Milman, 56 Theorems HahnBanach Mean value, 8 Topology on L(E,F)
Compact open ...on H(U), 5
Uniform Convergence Normal Convergence Absolute Convergence, 138 Universal constant,
287 of mLinear map and Polynomial Generalized, 42 Vector spaces HahnBanach Theorem in, Sublinear functional in,. 8 Vector sum of Bounding sets Weakly bounding ( or Liraited)set, 207 Weakly Bounding Closed, 16 Weakly bounded sets Bounded weaklybounding sets Bounded sets, 210 Weakly bounding sets in Lbs Weakly bounding sets in Fspaces Limited sets in Lbs and Fspaces, 192
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