Table Of ContentLecture Notes ni
Mathematics
Edited by .A Dold and .B Eckmann
843
lanoitcnuF ,sisylanA
,yhpromoloH dna
noitamixorppA Theory
Proceedings of the Seminario de
Ana.lise Funcional, Holomorfia e
Teoria da Aproxima~.o, Universidade
Federal do Rio de Janeiro, Brazil,
August 7 - ,11 1978
Edited by Silvio Machado
galreV-regnirpS
Berlin Heidelberg New York 1891
Editor
Silvio Machado
Instituto de Matem,~tica
Universidade Federal do Rio de Janeiro
Caixa Postal 1835
21910 Rio de Janeiro JR
Brazil
AMS Subject Classifications (1980): 32-XX, 41-XX, 46-XX
ISBN 3-540-10560-3 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-10560-3 Springer-Verlag New York Heidelberg Berlin
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FOREWORD
This volume contains the proceedings of the Semin~rio de An~-
lise Funoional, Holomorfia e Teoria da Aproxima~o held at Instituto
de Matem~tica, Universidade Federal do Rio de Janeiro (UFRJ) in Au-
gust 7-11, 1978. It includes pape~ either of a research or of an
advanced expository nature. Some of them could not be actually pre-
sented during the Seminar, and are being included here by invitation.
The participant mathematicians came from Belgium, Brazil, Chile,
France, Germany, Ireland, Spain, United States and Uruguay.
The members of the organizing committee were J.A. Barroso, S.
Machado (coordinator), M.C. Matos, J. Mujica, L. Nachbin, D. Pisanelli,
J.B. Prolla and G. Zapata. For direct financial support thanks are
due to Conselho Nacional de Desenvolvimento Cient~fico e Tecnol~gico
(CNPq); to Conselho de Ensino para Graduados e Pesquisa (CEPG) of
UFRJ, and here we also thank particularly Dr. Sergio Neves Monteiro
for his understanding; to IBM of Brazil, with special thanks to
Dr. Jos6 Paulo Schiffini. Travelling grants for some participants
were individually provided by Fundaq~o de Amparo A Pesquisa do Estado
de S~o Paulo (FAPBSP) and Universidade de Campinas (UNICAMP), S~o
Paulo, Brazil.
Professor Radiwal Alves Pereira, then Director of the Institu-
to de Matem~tica, collaborated beyond the line of his duty in the
organizational details of the meeting; the Coordinator emphasizes
his heartfelt thanks to him. Professor Paulo Emidio Barbosa made
available the facilities of the Centro de CiSncias Matem~ticas e da
Natureza (CCMN) of UFRJ, of which he is Dean and to which belongs the
Instituto de Matem~tica; it is a pleasure to offer our thanks to him.
A special word of appreciation on the part of the Coordinator goes to
iV
Professor Leopoldo Nachbin for making available his experience and
unfailing moral support. We also thank Wilson G~es for a competent
typing job.
Rio de Janeiro, August 1978
Silvio Machado
CONTENTS
JoM° Ansemil and An Example of a Quasi-Normable Fr~chet 1
So Ponte Space which is not a Schwartz Space
Aboubakr Bayoumi The Levi Problem and the Radius of 9
Convergence of Holomorphic Functions
on Metric Vector Spaces
Edward Beckenstein and Extending Nonarchimedean Norms on 33
Lawrence Narici Algebras
Ehrhard Behrends M-Structure in Tensor Products of 41
Banach Spaces
Mauro Bianchini Silva-Holomorphy Types, Borel Trans- 55
forms and Partial Differential
Operators
Klaus-D. Bierstedt The Approximation-Theoretic Localiza- 93
tion of Schwartz's Approximation
Property for Weighted Locally Convex
Function Spaces and some Examples
Bruno Brosowski An Application of Korovkin's Theorem 150
to Certain Partial Differential
Equations
J.F. Colombeau and The Fourier-Borel Transform in Infi- 163
B. Perrot nitely Many Dimensions and Applica-
tions
J.F. Colombeau, B. Perrot On the Solvability of Differential 187
and T.A.W. Dwyer, III Equations of Infinite Order in Non-
Metrizable Spaces
JoF. Colombeau and C~-Functions on Locally Convex and 195
Reinhold Meise on Bornolo~ical Vector Spaces
J.B. Cooper and Uniform Measures and Cosaks Spaces 217
Wo Schachermayer
Se~n Dineen Holomorphic Germs on Compact Sub- 247
sets of Locally Convex Spaces
G~rard G. Emch Some Mathematical Problems in Non- 264
Equilibrium Statistical Mechanics
Benno Fuchssteiner Generalized Hewitt-Nachbin Spaces 296
Arising in State-Space Completions
Ludger Kaup On the Topology of Compact Complex 319
Surfaces
Wilhelm Kaup Jordan Algebras and Holomorphy 341
Christer O° Kiselman How to Recognize Supports from the 366
Growth of Functional Transforms in
Real and Complex Analysis
IV
Paul Kr6e Linear Differential Operators on 373
Vector Spaces
Bernard Lascar Solutions Faibles et Solutions Fortes 4O5
du Probl~me 5u = f ou f est une
Fonction ~ Croissanee Polynomiale sur
un Espace de Hilbert
M~rio C. Matos and Silva-Holomorphy Types 437
Leopoldo Nachbin
Luiza A. Moraes Envelopes for Types of Holomorphy 488
Jorge Mujica Domains of Holomorphy in (DFC)-Spaces 5o0
Olympia Nieodemi Homomorphisms of Algebras of Germs 534
of Holomorphic Functions
Jo~o B. Prolla On the Spectra of Non-Archimedean 547
Function Algebras
Jean Schmets An Example of the Barrelled Space 561
Associated to C(X;E)
Manuel Valdivia On Suprabarrelled Spaces 572
Maria Carmelina F. Zaine Envelopes of Silva-Holomorphy 581
Guido Zapata Dense Subalgebras in Topological 615
Algebras of Differentiable Functions
AN EXAMPLE OF A QUASI-NORMABLE FR~CHET FUNCTION SPACE WHICH
IS NOT A SCHWARTZ SPACE
J.M. Ansemil and S. Ponte
Departamento de Teor{a de Funciones
Facultad de Matem~ticas
Universidad de Santiago de Compostela
Spain
I. INTRODUCTION AND PRELIMINARIES
Let E and F be complex Banach spaces, U an open subset
of E and (Zb(U;E),Tb) the vector space of the mappings f: U ~ F
which are holomorphic of bounded type on U endowed with its natural
topology T b. It is clear that (Zb(U;F),Tb) is a Fr6chet space.
In [6 3 it has been shown that when U is balanced then the topolo-
gical dual of (~b(U;F),Tb) is isomorphic to a certain space of se-
quences S(U;F). Moreover, assuming that U is, either all of E,
or else a bounded balanced convex open subset of E, then S(U;F)
has been endowed with a natural topology 6 T which has been shown
to be finer than the strong topolo6~y T on the dual space. We shall
8
now show that if U is the open ball B(O,R), O < R { ~ (the case
R = m corresponds to U = E), then the above isomorphism is
topological, and that (Zb(U;F),Tb) is a quasi-normable Fr6chet
space which is not a Schwartz space unless dim E < ~ and dim F < ~.
Finally, we want to acknowled@e Profs. J.M. Isidro, J. Mujica
and L, Nachbin for their help and suggestions while preparing this
paper.
2. THE DISTINGUISHED CHARACTER OF (Zb(U;F),rb).
Definition. For each r, 0 < r < R, we define S (U;F) to be the
r
co
~anaoh space o~ seque~oes U = (Un) C 17 ~(%;F)' for whioh
n=O
there is a constant C z 0 such that
II~nll, n c r
for all n 6 N, endowed with the norm
lln~II
rII~ll : sup n ' ~ = (Un) C Sr(U;F).
nEN r
We define S(U;F) as the vector space U Sr(U;F), and
O<r<R
T6 is defined to be the corresponding inductive limit topolo~ ~ on
The following theorem is Propositions 8 and 9 of [6].
Theorem i, For each ~ = (~n) E S(U;F) the mapping
(*) ~: f. <f,~> = z <~ anf(o),~n ,> f ~ ~b(U;~)
n=O
defines an element ~ 6 (ZD(U;F),~D)' • Conversely, if
6 (~b(U;F),Tb)' , then the sequence (Ks) , ~n = ~ (hE;F) of its
9
restrictions to the subspaces e(ne;F) c ~b(U;F) defines a~ element
or S(U~F) whose associated f..otio~a~ is ~ by (~).
Moreover, if we identify the dual space (Zb(U;F),Tb)' with
S(U;F) by means of the (algebraic) isomorphism given above, we have
([63, Proposition 12)that T% is finer than the strong topology B T
We shall repeat here for further use the proof of Proposition
13, [6].
Lemma Ix For each rB-bounded subset X of S(U;F) there is an r,
0 < r < R, such that X is contained and bounded in the Banach space
Sr(U;F),
Proof. Since (Zb(U;F),~b) is a barreled space, each ~B-bounded
subset of the dual space S(U;F) is equicontinuous. Hence, given X,
there is a neighbourhood W of the origin in (Zb(U;F),Tb) such that
sup ~ I {~ ~nf(e),~n ~ ~ 1.
(Zn)EX, fEW n=O
We may assume that
W-- [f 6 ~b(U;F)" sup Hf(t)II ~ ~]
II r~llt
for some r~ O < r < R, and some 6 > O. Now, for each n E ~[ and
each P ~ ~ (nEfF) with IIPll -- 1 we have
--PE w
n
r
therefore
(~)
r
for all n ~ ~, a1~ PC ~(n~',~F) with IIPII = I and all (~n) ~ ~"
From (*) it is easy to see that X is contained and bounded in
sr(~).
Proposition i. 8 T and T~ induce the same topology on each
~8-bounded subset of S(U;F).
Proof. Let X be a T -bounded subset of S(U;F). Without loss of
8
generality we may assume that X is b~lanced and convex. Because
of Lemma i, X is contained and bounded in Sr(U;F ) for a suitable
r, O < r < R. Take 0 such that r < p < R and let (~)~EA c X
be a net in X which is Ts-convergent to zero. Then (~)~EA con-
verges to zero in Sp(U;F)o Indeed, since (U~)~EA is bounded in
Sr(U;F), there is C ~ 0 such that
for all n E N, and all ~ 6 A, hence
n K C(~
for all n E ~ and all ~ E i. Therefore, given any ¢ > O, there
is an index M 6 q~ such that
sup - - K
n
n~M D
for all ~ E A. Since the closed unit ball of @(nE;F)~ n E ~, is a
bounded subset of (~b(UiF),~b) and (~X)~SA is
cB-convergent to zero, we have
~n 4 0
for each n 6 ~. Hence, there is a k E A such that
o
3I
P
for all X ~ ~o and n = O,l,...,M so that we have
sup - - ~ ¢
n
nEN
for all ~ ~ ~o" This sho~s that (~X)~EA converges to zero in
O S (U;F). Now, we can apply the result of Orothendieck ([5] , p.lOS,
Lemme 5), to conclude that the topology induced on X by ~8 is
finer than the one induced by S (U;F). The converse is obvious and
this completes the proof.
Lemma 1 and the proof of Proposition i proves the following.
Corollary i. The inductive limit
(S(U;F),~t) = lim Sr(U;F )
0<r<R
is boundedly retractive, that is, every bounded set X is contained
and bounded in some Sr(U;F), 0 < r < R and S(U;F) and Sr(U;F )
induce on X the same topology.
Theorem 2. ~B and ~ coincide on S(U;F). Therefore, (S(U;F),I~)
and ((~b(U;F),¢8)' ,~B) are isomorphic as topological vector spaces.
Proof. It suffices to prove that theidentity mapping
)~( (s(~;~),%)
. (s(~;F),~)
is continuous. Since (~b(U;F),Tb) is a metrizable space,
