FUNCTIONAL ANALYSIS, HOLOMORPHY AND APPROXIMATION THEORY
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NORTHHOLLAND MATHEMATICS STUDIES
Notasde Matematica (88) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester
Functional Analysis, Holomorphy and Approximation Theory Proceedings of the Seminario de Analise Functional, Holomorfia e Teoria da AproximaGGo, UniversidadeFederal do Rio de Janeiro, August 48,1980
Edited by
Jorge Alberto BARROSO lnstituto de MatemBtica Universidade Federal do Rio de Janeiro
1982
NORTHOLLAND PUBLISHING COMPANYAMSTERDAM NEW YORK OXFORD
71
NorthHolland Publishing Company, 1982 All rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.
ISBN: 0 444 86527 6
Publishers: NORTHHOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK OXFORD Sole distributorsfor the U.S.A.and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE NEW YORK,N.Y. 10017
Lihrary of Congress Cataloging in Publication Data
Semingrio de Anslise Funcional, Holomorfia e Teoria da Aproxima %o (1980 : Universidade Federal do Rio f de Janeiro) Functional analysis, holomorphy, and approximation theory. (NorthHolland mathematics studies ; 71) (Notas de matematica ; 88) 1. Functional analysisCongresses. 2. Holomorphic functionCongresses. 3. Domains of holomorphyCongresses. 4. Approximation theoryCongresses. I. Barroso, Jorge Alberto. 11. Title. 111. Series. I V . Series: Notas de matematica (NorthHolland Publishing Company) ; 88. QAl.N86 no. 88 [QA320] 510s L515.71 8218908
ISBN 0444865276
PRINTED IN T H E NETHERLANDS
FOREWORD
This volume is the Proceedings of the Semindrio de Andlise Funcional, Holomorfia e Teoria da Aproximapzo, held at the Instituto de MatemBtica, Universidade Federal do Rio de Janeiro (UFRJ) in August 48, 1980.
It includes papers of a research or
advanced expository nature. Seminar.
of an
Some of them were presented at
the
Others are contributions of prospective participants,
that, for one or another reason, could not attend the Seminar. The participant mathematicians are from Brazil, Chile, England, France, Spain, United States, Uruguay, West Germany and Yugoslavia. The members of the organizing committee were (Coordinator), S. Machado, M.C.
Matos,
J.A.
Barroso
L. Nachbin, D . Pisanelli,
J.B. Prolla and G. Zapata. Our warmest thanks are due to the support of the
Conselho
de Ensino para Graduados e Pesquisa (CEPG) of UFRJ, Mainly through its President, Profeseor SQrgio Neves Monteiro,
and to the I.B.M.
do Brasil. We are happy to thank Professor Paulo Emidio de Freitas Barbosa, Dean of the Centro de CiGncias Matemgticas e da Natureza (CCMN) of UFRJ, in whose facilities the Seminar was very comfortably held.
Our gratitude and admiration to our friend Professor Leopoldo Nachbin, whose experience and support made the task of preparing this volume easier. We also tbank Wilson Goes for a competent typing job.
Jorge Albert0 Barroso
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vii
TABLE OF CONTENTS
Rodrigo Arocena and
On a lifting theorem and its rela
Mischa Cotlar
tion to some approximation problems
1
Klaus D. Bierstedt, R.G. Meise and W.H. Summers
Kt)the sets and Kbthe sequence spaces
Bruno Brosowski
Parametric approximation and optimization
M.T. Carrillo and M. de Guzm6n
Maximal convolution operators and
J.F.
Convolution equations in infinite
Colombeau and
Mdrio C. Matos
approximations
27
93 117
dimensions: Brief survey, new results and proofs
131
J.F. Colombeau and Jorge Mujica
Holomorphic and differentiable mappings of uniform bounded type
179
J.F. O.W.
Finitedifference partial differential equations in normed and locally convex spaces
201
Colombeau and Paques
Ed Dubinsky
Approximation properties in nuclear Fr6chet spaces
R6mi Langevin
Geometry of the neighbourhood of a singularity
Pierre Lelong
R.G. Meise and Dietmar Vogt
Reinhard Mennicken and Manfred MOller
215
235
A class of Fr6chet complex spaces in which the bounded sets are Cpolar sets
255
An interpretation of T,, and as normal topologies of sequence spaces
273
Well located subspaces of LFspaces
287
T6
viii
TABLE OF CONTENTS
P.S. Milojevi6
Continuation theory for Aproper and strongly Aclosed mappings and their uniform limits and nonlinear perturbations of Fredholm mappings
V.B.
Moscatelli
New examples of nuclear Fr6chet spaces without bases
Michael 0 ’ Carroll
299
373
A survey of some recent results on the inverse spectral and scattering problems for differential operators
Peter Pflug
379
Various applications of the existence of well growing h o l o morphic functions
391
On the StoneWeierstrass theorem for modules over nonArchimedean valued fields
4 13
Laurent Schwartz
SemiMartingales and measure theory
433
Manuel Valdivia
On semiSuslin spaces and dual metric spaces
G.
Zapata
445
On the approximation of functions in inductive limits
461
Functional Analysis, Holomotphy and Approximation Theory, JA. Barroso led.) 0NorthHollond hblishing Company. 1982
I n memory of A. MONTEIRO, an extraordinary man and teacher
ON A LIFTING THEOREM AND ITS RELATION
TO SOME APPROXIMATION P R O B m M S
Rodrigo Arocena
and
Mischa Cotlar
SUMMARY We point out that there is a close relation between some approximation problems and a lifting theorem studied in previous papers.
A new simplified proof and an improved version of the
theorem, more adequated to our aim, are given.
1.
INTRODUCTION AND NOTATIONS
In this selfcontained paper we continue the study of some questions considered in [2]
and related to a lifting theorem.
We
show that this lifting theorem allows to approach some classical approximation problems.
Conversely, these approximation problems
yield a natural motivation of the lifting and suggest the corresponding theorem.
Thus, we give a new simplified proof and an im
proved version of that lifting theorem, adapted to and motivated by approximation questions. We shall work in the unit circle
T N [0,2rr]
and use the
following notations: en(t) = exp(int), P + = {analytic polynomials, n 1 ckek(t), n ;r 0 3 , P = cke k(t), n=1,2 3 , e JP + = ‘k=O
,...
Ex,,
= {x;+n ckek( t), n n
in
T, P  =~ {
c
t 0)
,
ckek(t), k=1
p = P+
n >
+
P
03,
,
dt = the Lebesgue measure
HP = {f 6 LP(T)
: 2.(n)= 0 ,
R. AROCENA and M. COTLAR
2
Y
n < 01,
f E Hp
so that
and
where
2.
is the Fourier transform of
has an analytic continuation
C(T) = {all continuous functions in
H"
Since
is a subspace of
f(z)
in
{ I z I < 11,
TI. given
Loo,
and p E [ l,m],
f
Lm(T),
g
a clas
sical approximation problem is to characterize the distance of to
H", describe the set of all best approximations of
ments of
g
g
by ele
and give a condition for unicity of the best approxi
H"
mat ion. More generally, given (i)
E Lm(T) and
a condition for the existence of a /(gh((m < d,
(ii)
g
that is
d > 0 we want: h E H" such that
Ig(t)h(t)(
d
S
a.e.;
(1)
a parametrized description of the set 3(g,d) = {h 6
ff
satisfying (I)] ; (iii)
a condition for the unicity of
h E 5(g,d).
Condition (1) is equivalent to saying that the matrix
1
N = Nn = N (ht ) = d is positive definite for almost all
t.
It is easy to see that
this is equivalent to (cfr. (2b) bellow)
J
(Nf,f) = fl?, for all
d dt +[f,?,(gh)dt
f = (fl,f2) E C(T)xC(T),
and say that
N
is positive. 3
Since only matrix
g
and
d
h E H"
+JFlf2(Efi)dt
+(f2f2
d dt
2
and in this case we write
0,
N
2
0
Thus (1) is equivalent to: such that
Nh
5
0.
are given, it is natural to consider the
(la)
ON A LIFTING THEOREM
3
M.
and try to replace (la) by some similar condition on assert that (la) implies M B it is immediate that
0.
However, since
(Mf,f) = (Nf,f),
Y f
implies that
(Mf,f) > 0
M > 0
noted by

M
and we indicate this fact by writting
M
(Mf,f)
and
(Nhf,f)
63,
x
63
.
h
coincide on
is positive on the whole of only on
N.
C(T)
x
M t 0
,
x p
which is de
E H"
p,
C(T),
x
such that the and the form Nh
p
while
M
is positive
We express this property by saying that M.
In other words, if
M > 0
Nh
is
We shall see
implies, and therefore is equivalent to (la),
is the desired condition on
M > 0
,
63
I n particular, (la)
a positive lifting of (the weakly positive form) that
E p, x
is weakly positive.
Thus (la) says that there exists forms
is analytic,
= (fl,f2)
Y f = (fl,f2) E P ,
or by saying that
h
We cannot
so
that
M.
M
then
has a positive lifting
Nh. This fact is a special case of the following general lifting theorem. Consider 2 x 2 matrices M = (mug), a,@ = 1,2, whose elements are (complex) Radon measures in m21 = in
T; we suppose that
i12. With each such matrix M we associate the form (M,*)
C(T)
x
C(T)
(Mf,f) =
defined by
(fbfe
"up '
a ,8=1
It is easy to see that
f
= (flf2)
(Mf,f) = (Nf,f),
E C(T)XC(T)
(2)
V f E C(T)xC(T),
iff
M = N. We write
M

N
if
theorem of FejerRiesz if fl E P,
.
(Mf,f) = (Nf,f), g
E P
and
g
B
0
V f
E p, x
then
g
L
And by classical theorem of F. and M. Riesz,
p
.
flPl
By a where
R. AROCENA and M. COTLAR
4
implies
n12 = m12

h 6 H1.
with
h,
F r o m these facts it follows
easily that

M
N
iff
M
We write (Mf,f)
2
0
V
M z
0
0
2
if
E P+ x P
(Mf,f) 2 0 V f E C(T),
.
If
dm
= gUB(t)dt,
a%
then it is easy to see (by letting
a,% = 1,2, u(t) +
f
= nll, m22 = n22, n12 = m12  h , hcH1.
mll
(l/8)lE(t),
E = (to,to+8))
iff the matrix (g M > 0
Though
M > 0
and g
a%
(2a) if
E L1, f2 = X 2 u ,
fl = X1u,
that, in this case,
(t ) is positive definite, V a.e.t.
M
doesn t imply
2
(2b)
0, however the following
lifting theorem is true:
M > 0
c,
3 N
with
M
N 
and
N
(PI.,.)
That is, if the restriction of
2
to
0.
6, x P 
is
positive then this restriction can be lifted to a positive form on
(N*,)
C(T)
(1"
Y2)> m22
21
for some M > 0
c1
h E H1.
iff
liftings of
3(M)
In other words
x C(T).
Let
f 0,
3(M)
(mil mZ1
m12


hdt
= {h E H1;
and the
h E 3(M)

hdt
m22
h
satisfies
(4)
2 0
(k)].
Then
furnish all the positive
M.
(4a)
This lifting theorem was proved in [ 4 ] detail in [ 2 ] .
and studied im more
In section 2 of the present paper we give a new
simplified proof and a more precise version of (4) which leads to an improved
description of
5(M),
more adecuated to our aims.
Using this version we give in section 3 a condition for the unicity
ON A LIFTING THEOREM
of the lifting (that is for cardinal case where
has the form (lb),
M
valent to (la),
and the set
the set 3(g,d).
3(M)
3(M)
5
= 1).
In the special
M > 0
is, as seen above, equi
is in
11 correspondence with
Therefore the results of sections 2 and 3 furnish
in particular a solution of the above approximation problems (i), (ii),
(iii).
Moreover they also apply to the classical case
d = distance of
g
>.
c;
finition that
to
H".
Z(g,d+e) f Q ,
In fact, in this case we have by deV
e > 0. Hence by the above remark
0, Y E > 0, and letting E + 0 we get M 3 Mo>O. d+E M = By the above lifting result we get that 3(g,d) f 0. We have thus
proved the existence of a best approximation of H"
g
by elements of
(which is well known)and the results of section 3 give a con
dition for the unicity of this best approximation. Moreover, an explicit expression of the unique best approximation
h
is given in terms of its Fourier coefficients, s o that
a condition f o r written down.
h
to belong to a smooth class
Cn+a
can be
These solutions of the above approximation problems
are somewhat different from the well known results due to Adamjan, Arov and Krein. I n section 4 the general results of sectionss 2, 3 , are applied to and motivated by a general approximation problem. Finally in section 6 a similar procedure is used to study the balyage of generalized Carleson measures, whose characterization is also related to approximation questions. We are pleased to acknowledge our gratitude to Prof. Jorge Albert0 Barroso for inviting us to contribute to this volume.
6
AROOENA and M.
R.
2.
3(M)
DESCRIPTION OF
M = (map),
L e t us f i x t h e m a t r i x
are R a d o n m e a s u r e s i n
satisfying:
T
W e associate w i t h
M
a,B = l , 2 , 2
mll
a seminorm
mZ2 2 0 ,
0,
in
p
w h e r e the m
112.
=
mZ1
defined as
C(T),
.
follows
R = {w E C(T):
Let V
COTLAR
t],
so that
E R
w
p(f)
3
implies
a = 1,2,
l/w
1
= i n f f If I (wdmll
It i s c l e a r t h a t if
c > 0
a constant
dm
aa
E R,
+
with
and s e t
1
: w
dmZ2)
= gu ( t ) d t ,
0
(for
E R]
.
(5)
< gu(t) E C(T),
then
(5a) and t h a t i f
"11 = m22
then
(5b)
p(f) = 2
Thus i n b o t h c a s e s i t i s e v i d e n t t h a t
p
is a seminorm.
In order
t o s h o w t h a t t h i s i s t r u e i n t h e g e n e r a l c a s e , i t i s enough t o verify the inequality.
+ Ifldmz2) [ I
(jllllfldmll
2
1
v
p(lfl+lgl),
f
C o n s i d e r f i r s t the case define w , w'
wllfl
by
Since
since
w2/w1
(5c) i s
s i d e of
w
E R.
+ w21g(
+
=
w1/w2
(If 2
2
2 [w(lfl+lgl)dmll
w e have
+
[$
w'
2
1
s o that the l e f t
(Ifl+lgl)dmZ2
The g e n e r a l case i s proved by f i x i n g
wl,
p(lfl+lgl), w2
and
ON A LIFTING THEOREM
applying the result just proved to
+ E + lgl)
P(lfl
Thus w =
p
5
P(lfl+lgl)
and
E
and letting
and lefting
f = flf2
then
p(f)
E
S
[Ifl12
dmll +
0.
(lf212
dm22.
(5d)
I .
in
defined by
f E C(T),
IXMMA 1.
mll
using
a real linear functional
I (f) =  2 Re f dm12 = fdm12 for
b
g,
E + 0+) that
M
We also associate with C(T),
+
is a seminorm, and it is easy to verify (tcking
lf112/(lfl+e) if
If1
7
2
where
C(T)
where
I .
and
p
i12
where
The condition
6f
,
is a matrix measure with
then:
are the functionals associated with
and ( 5 ) , respectively, and PROOF.
(6)
is considered as a real vector space.
If M = (m a B ) , a,p = 1,2,
0, mZ2 2 0, m21 =
  dm21
"f+ = the
M > 0
is the closure of
closure of
by (6)
M
eP+
in
C(T).
can be rewritten as
p*
in
C(T).
Assume that
M > 0,
that is that (#) holds, and let us prove that
Every
I E ef+
can be written as
a finite Blaschke product,
h(O)=O,and
I = Il
Il
E
k3+
h
where
with
h
is
I1(z) f 0
8
R.
for
< 1,
121
2
w = lwll
,
E
w1
m
1
l
Y
~ dmll ~ 1
,
P,
p
l / w E P+
( I Y h i 1;  I
2
.
can be w r i t t e n (by F e j e r  R i e s z )
1
I,(@= ) 2Re
+~
COTLAR
Y E 6;+
=
E R n
w
(#) implies t h a t
0.
Hence
5 w) 1
1
+[& ( @ Idmz2, n
i s dense i n
n]
@ = @1C2
whenever we o b t a i n
= p ( @ ) , wich
I , ( @< ) p(@), V @
This proves ( 7 ) .
m i l a r and much e a s i e r .
f (Ywl)(Yh
[email protected] : w E
Applying ( 5 d ) t o
.
E 6;
Yh(l/wl)
as
E
elP+
and l e t
we o b t a i n t h a t
(#)
(8) i s s i 
The p r o o f of
0
I n what f o l l o w s we c o n s i d e r
C(T)
a s a r e a l v e c t o r space
forms I i n C ( T ) , s u c h t h a t
$1 From
= Io(P),
tf @ f
ep+3

(9)
( 7 ) and ( 8 ) we g e t t h e f o l l o w i n g more p r e c i s e v e r s i o n
of t h e l i f t i n g t h e o r e m . THEOREM 1. ( L i f t i n g t h e o r e m , i n Lemma 1. a)
M
> 0;
c f r . [4] ,[2]).
M , Io, p
Let
be a s
Then t h e f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t s .
b)
3(Io,P)
f Q;
c) 3 N z 0
with
N 
M;
d)
Z(M)f
Moreover t h e r e i s a c a n o n i c a l 11 c o r r e s p o n d e n c e
PROOF.
a ) i m p l i e s b ) : a n immediate c o n s e q u e n c e o f
(7) and the
HahnBanach t h e o r e m .
b) i m p l i e s c ) : n
in
C(T)
Let
I E 3(Io,p),
such t h a t
I = Ren
so that there is a
and
l i n e a r form
Q.
ON A LIFTING THEOREM
~ ( fr:)fIfIdmll
~i.l(f)~
hence
nZ2 = mZ2,
and
f2?,
dnZ2 =
J lf112
I
and that
MII~II
1 n12 = ~
Let
f
n
nZ1 , =
E c(T),
n12,
[ lf2I2
+
N

dmZ2
I(f) s p(f),
Since
I(f) = Io(f),
1(fl?,).
has the same associated seminorm
we obtain from (8) that
f F C(T), V
,
f E elp+
n12  ml2
Riesz theorem that
+
dnZ1
is the linear functional associated with
Since
nll =
(nag), a , @ = 1,2. Then, if
c
dmll
This means that M,
N
I:
= JflFl dnll +fflP2 dn12 +{f,Tl
f = (fl,f2), (Nf,f)
+
JlfIdmZ2
is a Radon measure.
n
= mll,
+
9
+
The implications c) a d)
h E H1,
h,
*
hence
N
N.
N
2
F
it follows from the

as
p
0.
and
M.
M.
0
a) follow from ( 2 ) .
It follows from the preceeding proof that the canonical correspondence. 3(Io,p) 3 I  h
I
t
h:
h(f)
E 3(M)
 21 I(f)
fhdt =
I
is given explicitly by the formulae
+
$ I(if) (.dm12,
(9a)
h + I: I(f) = 2Re [
I E 3(Io,p)
Since h E H1,
we have that
L(n) = h(en)
Let
= 0
for
fZn = emn,
mined by the sequence (9a),
(9b),
{f(n)
of
I
to
elP+
I(en)
= 2Re (einx
Ln
in
dm12
elP+
for
,
and since
n > 0,
n > 0, and by (9a),
f2n+l  ien,
n
{G(n) = h(e_,)],
= I(fn)],
and
I .
coincides with
(9c) that the corresponding
the sequence spanned by
(9b)
fo,
n
...,fn
2
2
0.
n
2
I

Since
h
is deter
0, it follows from
h
is determined by
0. Moreover, if Ln is the subspace
and if
I(n)
is the restriction
then as it follows from the proof of HahnBanach's
R. AROCENA and M. COTLAR
10
where
0 i tn
I(o) = the restriction of
1, and
g
I .
to
elP+
is fixed. Conversely if
(t,)
is any sequence with
0 i tn i 1,
I
these recurrent formulae furnish a real linear form
I(f) s p(f),
such that 3(Io,p)
with the
I(fn)
I
that
so
then
in :Ln
extends to an element in
given by (9d).
We have thus proven the
following theorem. THEOREM la
(Parametrization of
in theorem 1, and let Then for each
T
z(M)).
Let
S = { r = (a +ib )
= (an+ibn)
nz 0
M > 0,
Io, p
be as
a
: an,bn E [0,1]]. n=0 there is a functional
E S
I = I E 3 ( I o , P ) such that the numbers I(fn) = IT(fn) are given r by the recurrent relations (9d), (e), with tZn = a n’ t2n+l = bn* Moreover
r
 IT
S
is a bijection of
onto
;+I
ing this bijection with the correspondence we obtain a bijection of
S
onto
bijection (or parametrization) of
{c; h E S
3(Io,p).
3(M)],
onto
By combin
given by (9c), and hence also a
3(M).
Thus,
and (9c),
s
t
r
4
3(Io,P)3W
Ir

hT
(9e) give a description of all the positive liftings of
M > 0.
If
map
= gae(t)dt
or if
mll = mZ2,
then
p
is given by (5a)
or (5b), and in this case the above recurrent formulae become simplified.
11
ON A LIFTINC; THEOREM
3.
CONDITION FOR THE UNICITY OF THE LIFTING By theorem 1, there is a unique
there is a unique
if and only if
h E 3(M)
I E 3(Io,p).
Thus the problem of the unicity of the lifting reduces to that of the unicity of
I E 3(Io,p).
Consider, more generally, an arbitrary real normed space subspace
Lo c E,
linear form
a continuous seminorm
Lo
Io, in
p
in
E
in
E:
I(f)
E,
a
and a real
such that
Let
and let 3(Io,p)
I(Ep) = Io(lp)
= keal linear forms
for
Ep
f;(f)
B
p(f),
I E 5(Io,p) since
~(f) = I(@)
5
that
implies
+ I(fm)
L
:i
p,
and that
I(f) s :(f), I,(@)
+
V f
p(
[email protected]),
E
E,
( 1% )
v m E L ~ . I E 3(Io,p)
I(e) = p(e). e
p(f),
is a sublinear functional, and in par
Given e E E we cannot assure that there exists
However, given
i
E Lo?.
I t is easy to see that ticular
I
E,
and
T o prove (lob), (lOc), it is enough to show that
with
R.
12
LEMMA 2 .
There i s a unique
= $(e),
V
e E E, I =
I n such c a s e PROOF. and
AROCENA and M.
If
E
I
COTLAR
3(Io,p)
G(e)
if and o n l y i f
or e q u i v a l e n t l y i f f
=
i s a linear functional.
G.
p(e) = p(e),
I ( e ) = I(e) z
p(e)
i f t h e r e i s a unique
V
e
E E,
= p(e),
I E 3(Io,p),
then ( l o a ) gives
I ( e ) r; p ( e )
hence I ( e ) = G ( e ) .
Conversely,
t h e n ( l o b ) and (1Oc) g i v e
? ( e ) = G(e). Let u s o b s e r v e t h a t
since then Hence, span
if
E,
G(fl+f2) s
f;(f,) +
t h e r e i s a sequence then
I E 3(Io,p)
c(f2) = [f,]
C
E
G(fl)

G(f2)
such t h a t
i s unique iff
[f,]
5
G(flf2). and
Lo
ON A LIFTING THEOREM
Let now by
(5).
Since
, I. g i v e n by a = l , i ; n=0,1,... ]
E = C(T),
Lo = elP+
e p +
(ae,
and
:
( 6 ) and span
p C(T)
( c o n s i d e r e d a s a r e a l normed s p a c e ) , t h e u n i c i t y c o n d i t i o n (11) becomes h e r e : inf {Io(@ + )p ( a e  n =inf

( I ~ ( Y ) + p(ae,,
v n = 0,i
,...,a
m)
: 5
Y)
:
E elp+] =
Y E elP+3
,
= l,i,
or e q u i v a l e n t l y
If
I
E
3(Io,p),
then ( l l a ) can be r e w r i t t e n a s
U s i n g t h e b i j e c t i o n between (gb),
we o b t a i n t h a t
h
E
3(I0,p)
3(M)
and
5(M),
g i v e n by
(9a),
i s unique i f f
Thus t h e f o l l o w i n g theorem i s proved. THEOFIEM 2 ( C o n d i t i o n f o r t h e u n i c i t y of t h e l i f t i n g ) . and
Io, p
t h e a s s o c i a t e d f u n c t i o n a l s g i v e n by
Let
M > 0
( 6 ) and ( 5 ) r e s p e c t 
14
R. AROCENA and M. COTLAR
ively.
Then we have the following three equivalent conditions for
the unicity of the lifting: (i)
There is a unique lifting
N
N N M,
2
iff (llb) is
0
satisfied. (ii) (iii)
I E 3(Io,p)
There is a unique h
is the unique element of
Moroover in this case
I =
5,
I
iff
3(M)
satisfies (llc).
iff (lld) is satisfied.
that we have an explicit expres
so
sion for the Fourier coefficients of the unique condition can be written down for ness class REMARK.
h
h E 5(M),
and a
to belong to certain smooth
cn+U.
In the special case where dmue = gae(t)dt,
gae E
f,
and
811 = g22
’
condition (llb) becomes
@
inf {(lI+enlglldt+ ,YEelP+
( 1YenlglldtRe[a(I+Y)g12dt) a
tt n = o,i...;
(lle)
= 0,
= 1 o r i;
while (lld) becomes glldt+(l)P
v n = a I,...; 4.
a
ReCa = 1 or
APPLICATION TO APPROXIMATION PROBLEMS Let
G
Lz = {f E Lm(T)
LOD(T) be a fixed nonnegative function, and set : If(t)l
i
c G(t),
IIflloDG= the least such constant
a.e., C.
&
(n) = cn = g(n) for all n 1 exist 3, f g with gl(n) = cn,
If
Let then
v
for some constant g
E LG, cn = t(n).
gl = g,
n < 0.
c).
OD
but there might
W e consider then the
ON A LIFTING T'HEOREM
following semireduced Markov problem: exists a function
where
d
gl
determine wether there
satisfying
is a fixed constant.
Of course this problem is equivalent
n < 0
to the similar one where the condition
is replaced by n
2
0.
Now (12) is equivalent to
3 h € H"
such that
Ig(t)h(t)l
S
dG(t),
a.e.
(12a)
Thus the above semireduced moment problem is equivalent to the following approximation problem: d 5 0. h
Given
with
For
G
g E Lz
Ighl z; d G
I
a.e., that is
aG(g,d)
H"
= inf (d
n L;
G,O i G E Lm(T)
and
I)ghllmG 4 d.
this is the problem (i) stated in the Introduction.
1
In analogy with the case
dG(g)
we fix
we ask wether there is an analytic function
S
G
E
1, we set
= (h E H"
: h
satisfies (12a)},
0: SG(g,d) f Q] = distance in
m
= HG = closed subspace of
LE
LL
of
g
to
.
We consider the two following questions:
#
a)
determine when
SG(g,d)
b)
determine when
aG(g,dG(g))
$r
and give a description of SG(g,d);
# 0,
that is when
g
has a best
approximation, and when this best approximation is unique. (Thus b) is the problem a) for the special case where d = d = dG(g) + e , c > 0 , then there is an e tisfying (12a), or equivalently such that
If
hence
d = dG(g)).
h = h e Hm
sa
16
R. AROCENA and M . COTLAR
for all
a
>
0.
Letting
E
t
we get that
0
M = M
> 0, and by
the lifting theorem we get, as in section 1, that 3G(g,d)
f
' dG(g)'
69
Moreover by theorem 1 we have a canonical bijection Qg,d)
= 3 (M)
given by the formulae (9a), (9b).
Io,p,

5 (IoP)
Here the associated functionals
to the matrix
Moreover, the results of section 2 furnish a parametrization of ZG(g,d)
by means of the formulae (9c), (9d), (9e):
we assign a function onto
zG(g,d).
hT E HE
and
7
hT
The Fourier coefficients of
to each
is a bijection of hT
7 E S
S
are given explicity
by the recurrent formulae
I(n+l)T(f+fn)= I(n)T(f) + I & )
I(.lT(f)
where
= 2Re [fgdt,
In particular, if
d = dG(g),
Ll = elpl,
this gives a description of the set
of all the best approximations of
norm
11
1lG '
n = 1,2,...
g
by analytic functions in the
ON A LIFTING T H E O m M
Finally the unicity condition (lle) of theorem 2 becomes now: h E ZG(g,d)
There is a unique
iff
V n = 0,l
Similarly,
is the only element in ZG(g,d)
h
iff
r
We have thus proved the following theorem. THEOREM 3 . (i)
Let
0
The set
h
(iv)
E
in the norm
for this
E ZG(g,d)
h
is non empty. give a description of
is the only element of
I = p)
iff (13e) holds. ZG(g,d)
iff (l3f)
w e have explicit expressions
and its Fourier coefficients.
UNIFORM APPROXIMATION BY ANALYTIC FUNCTIONS
If
G
I
1
then
Lz = L"(T),
= Z(g,d) = [h 6 H" : llghllm < d] of
by
g
5G(g,d).
holds; in this case (since
5.
11 llmG ,
There is a unique element in ZG(g,d) h
then,
of the best approximation of
The recurrent formulae (13b)(13d)
all the (iii)
m
g E LG, d 2 dG(g),
G E Lm(T),
ZG[g,dG(g)]
functions h E H" (ii)
S
g
to
tions (i),
H".
(Ti),
and
1) [IrnG
=
dG(g)
I( [Irn , = d(g)
ZG(g,d)
=
= the distance
In this case theorem 3 gives a n answer to the ques(iii) stated in the Introduction, and these results
can be obtained directly from zhe proof of HahnBanach's theorem.
18
M. AROCENA and M. COTLAR
In particular from (l3f) it follows that, given h
v
h E 3(g,d(g)),
is the unique best approximation iff for every
>
E
p = 0
0,
a
1,
or
or
= 1
i,
n > 0
there is a
and
E elp+
@
such
that (l)PRe[a
[(
[email protected])(gh)dt]
)I
Observe that since
2
d(g) (
[email protected] s d(g)(
(e.,
[email protected])(gh)dtl
(14) implies that the three numbers
other, while
d(g)/
and
If(
[email protected])(gh)dtl
len+)dt, condition
(l)PRela((e_,
[email protected])(gh)dtl,
len+ldt
must be very close one to
[5] that a sufficient condition for unicity f E H1
is the existence of an extremal dual function
= 0,
= d(g)?/)f)
(14)
c.
I)gh)lm= d(g).
It is wellknow
?(O)

))flll= 1 hence
Arov and Krein [l] they proved that
and
In this case
d(g) = (gfdt.
{(
[email protected])(gh)dt
gh =
= d(g)(en9)?/)f)dt.
have studied in detail the class 3(g,d(g))
such that
Adamjan,
3(g,d)
and
contains a unique element iff the
constant function 1 does'nt belong to the image of the operator
where
H f = (IP)gf is the Nankel operator associated with
g Thus the three conctaions G I
1)
g.
(15), (14) and (13e) (with d = d(g),
are equivalent.
These authors gave also the following remarkable parametrization of 3(g,d)
(of wich Garnett [ 5 ] gave a simplified proof):
THEOREM (Adamjan, Arov, Krein). 3(g,d) (i)
has more than one element.
g €
Then
LOD,
d > 0
3 ho
E 3(g,d)
There is a unique exterior function
such that (ii)
Let
if
ho = d F / I F ) . x E H"
is defined by
F E H1,
be such that such that:
IIF/I = 28,
ON A LIFTING THEOREM
It seems that (in the notations of section 2 ) specially simple
sequence
parametrization extends to
To
E
ho = hro for a Perhaps the AdamianArovKrein
S.
for a certain class of matrices
3(M)
M > 0.
6. ON A GENERALIZATION OF CARLXSON MEASURES A positive Radon measure
in
p t 0
a Carleson measure if there is a constant c(b)lII
t
p[R(I)]
the Lebesgue measure of 1

IzI s
111/2ll].
and
R(1)
=
E D
[ z
acts as a bounded operator from p E (l,m),
for each
p E (1,m)
C(P U P
'I f l
In particular
is said
aD = T, where
If the Poisson integral
for some
s Kp
I
11
> 0 such that
c(p)
of
h
: z/IzI
is
111
E I and
The following characterization is know (cfr.[7]).
THEOREM (Carleson). z = rei'
I
for every subarc
D = { IzI
then
LP(T,dt)
to
is a Carleson measure;
there is a constant
Lp(T,dt )
Pf(z) = Prf(e),
whenever
p
K P
Lp(D,p),
conversely
such that IIPfll
s
LP(D,Y)
is a Carleson measure.
p t 0 is a Carleson measure iff
[ for some constant
K.
the weaker condition
lPfI2 dc( s K
[ lfI2
(16)
dt,
T
Hence every Carleson measure
c\
satisfies
R. AROCENA and M. COTLAR
20
We say that a complex Radan measure
D
in
p
is a generalized
complex Carleson measure if (16a) holds for some the set of all such measures by give a characterization of
D,
measure in of
w E BMO;
measure
(p =
p
p
is a
p
T, called the balyage
in
pb
b,
other words the set real),
Since
then there is a real Carleson
positive Carleson measures) such that
I n [ Z ] it was proved that if then
dkb = wdt,
{p E GC, p real)
= P(flF2)
b fl E P +
Radon measure in
,
f2 E 63
w E BMO,
1
iff
is a real
u
E G  C ; in
is the preimage of
T
A n = U[An(K) we set
=
I,
7,f2
dPb
,
we write
{w
BMO,
9
(16a) suggests 7
E ALK),
n = O,l,...,
if
tL.
fol
q
is a
satisfying 2 Re
q E An,
and
(Pfl)(PfZ)dP
lowing definition:
We set
dub = w(t)dt
under the balyage.
(Pfl)(Pf2)
whenever
w E BMO
u1*2,pl,kz
( [ S ] ,[6]).
~ $ =4 wdt ~
is a Carleson measure then
moreover if
Radon measure in
if
I n this section we shall
Let us recall that, if
there is a measure
It is known that if
w
G.C.
and denote
such that
k,
with
G.C.
K,
: K 2
lfI2
T flFz dq s K [
01;
in particular
/lqll(n) = min {K
2
0
I
q
E GC
iff
b
E An(K)).
I n the following proposition we identify absolutely continuous
measures with their densities.
Ao;
ON A LIFTING THEOREM
PROPOSITION 1. iff
a ) A complex Radon m e a s u r e
= wdt,
$b
b)
An
c)
if
21
+
w E H1
h E H1]
w = g+p+h,
then
is i n
GC
L".
= [ w d t : w = g + h , g E L", w E A,
5
in
L"
+
,
H1
Il~ll(~),
I(gllm =
n 2 0
P E 6,
,
and
1 h e H .
( P , ~=
m),
)I I/(") H1
PROOF.
V
fl
O S r ,
v
.
+
E p,
f
A,
=
H1],
A.
i s t h e s u b s p a c e of
formed by t h e f u n c t i o n s t h a t h a v e a f i n i t e u n i f o r m d i s t a n c e t o
H1; to
n = 0,1,...
: w = g+f,
The p r o p o s i t i o n 1 s a y s t h a t
COMMENT.
L1
{Ilgll,
)Iw(I ( n ) = min
d)
estimates t h i s distance;
mod p o l y n o m i a l s o f d e g r e e
q F A,.
Let
E
enP+
,
f2
satisfies
,
E 63
f2 E 6
M t 0,
This implies t h a t
measures t h e d i s t a n c e
n.
From ( 1 6 b ) i t f o l l o w s t h a t
.
Equivalently,
c I f l I (11 rlll ( n ) d t )+flT2
fl E p+
11 l l ( n )
.
(endrl
) + f l f 2 ( q G) + I f 2 I 2(11T111(n)dt)l
Therefore the matrix
and b y t h e l i f t i n g t h e o r e m
r)
9
i s absolutely continuous,
3 h
E H1
dq = wdt
such t h a t
and t h a t
R. AROCENA and M. COTLAR
22
m
Clearly
enwh = gl E L
g E
p
,
E P,
p
L",
b
with )Igll), < Ilwll("). 1 hl E H , s o that setting
h
,
A.
this proves also part (a).
with &.elI
/I gillm *
=
mains to show that
I/ g211m 2
implies k = Ilg21I,
IIwIl (n),

en(p2+h2)1
w = engl + p
Since
+
P2 E
g2 E L",
w29
which will also proves (d)
s k
a.e.,
1)
(T
>
and
9
.
+ hl, h2 E H
1 9
Let
= h E H1,
en(p2+h2)
This implies that
0.
fies (16b) and by definition REMARK 1.
iff
e w
k
M =
GC
in order to prove (c) it only re
IIWII (n),
w = g2 + P2
w E
Since
then
9
Ienw hence
g = e,g,
and this proves part (b).
HI,
f
enh = g+h1 ' we get w = g+hI'
wdt = dq
satis
k > I[wlI(").
It is natural to ask wether
GC
contains something more
than combinations of Carleson measures and measures concentrated in
5

D = T.
In [2] it was proved that given
Carleson measure
...,4 )
j = 1,
q
(q = qlq2
+ i(qgqk),
Ub = qb.
such that
E GC,
qj z
o
9
a complex
Carleson,
It was also stated there without
proof that there exists a measure in is positive in
p
GC
wich is not Carleson and
We give the proof of this statement in the
D.
Appendix. REMARK 2.
If
w E A.
then
IIwll(")
is a decreasing sequence of
IIWL/~").
positive numbers and has a limit denoted by sition 1, (a), it: follows that
11 )1,
norm
from
is the distance (in the
IIwII(")
w E H ~ + Lto ~ H~ + c ( T ) .
then
H" + C
is closed and that the distance in
= distance of
w
In particular if
H" + C(T).
w E L"
IIwll(")
F r o m propo
to
L"
to
It is know H" + C
is
always attained [ 7 ] .
( 1 ~ 1 1 =~
Hence IIwJ(m
=
o
iff
min
{ll€Illm
w E H"
: w =
+ c(T).
e + h + 8,
h E H",
P E C(T)],
and
23
ON A LIFTING THEOREM
APPENDIX Let
o
g(r) =
Let h
e0
be the function defined in
g
if
r
c
[0,1/21,
be the function defined in
be the function defined in
h(re i8 ) = g(r) Clearly
h
if
101
e,(r)
5
0, and also
2
h(reie) =
r,‘
[O,l]
by:
g(r) = f r sen 1 ~ ( 1  r ) I ~ F r ~ if ’ rc(F,l]. 1
[O,l)
Let
e o ( r ) =n ( 1  r ) .
D =freie : r E [0,1], 0 E [n,n]] and h(reie) = 0 otherwise.
re0 [i2
h E L1(D)
g(r)rdr
by
by
since
dB = 217
(1r)dr S
sinln(lr)lcr
80
dr
s 2
2
==
JF.
11,2 Hence (i)
dv(reie) = h(reie)rdrde
Let
I be a subarc of
reie
c
R[I]
then
eo(r)
h(reie)rdrde
Therefore
T
is a positive Radon measure in i0 centered at 1 = e , s o that if
s III/2.
v[R(I)]
rdr f1 h(reie)de
=
v[R(I)]/lIl
Therefore
z clIl
1/2
,
c
D.
=
=
> 0, so that (ii)
v
is not
a Carlesson measure.
We shall show now that calculations.
v E GC.
For every integer
1,8/9
1/2 dr
S
* / [ n
This will require some previous n > 0 we have
[
1
rn1 dr+n
1,*8/9 i
with
k > 0.
(lr)’l2
dr s
R . AROCENA and M. COTLAR
24
n
Hence if
is a nonzero integer then
I/. 1
i'
g(r)rl"l+'
sinln(1r) Idr =
,In1
% 2 1r
=
1
rIn1'
= r
(lr)'/'
dr
n
such that
topology on each bounded subset of
En,
E
and
be the inductive
En. If, for each Em induce the same
then a straightforward
application of Grothendieckls theorem on bounded sets in countable 1.c. E
inductive limits of (DF)spaces [ 7 , §29,5.(4)] ind En
2
shows that
is regular, and hence also boundedly retractive.
n1
Moreover, it is clear that, in the terminology of Grothendieck as recalled above, an injective inductive sequence
(En)nEN of Banach
spaces is boundedly retractive if, and only if, it is regular and 1
2.
ind En n+
satisfies the strict Mackey convergence condition.
THE ROLE OF THE SPACE
K
P
IN THE DUALITY THEORY OF ECHELON
AND COECHELON SPACES
In this and the following two sections, we treat a fixed index set I,
KOTHE
KBTHE
SETS AND
SEQUENCE
A = (an)nEN on
a f i x e d Kt3the matrix
39
SPACES
I,
and take
V = (v
)
n€N
t o be t h e corresponding d e c r e a s i n g sequence of s t r i c t l y p o s i t i v e
.
1 
To b e g i n , l e t us c l a r i f y t h e e x a c t r e l a t i o n an of o r d e r p and s h i p between t h e coechelon space 4 = h ( 1 , V ) P P t h e a s s o c i a t e d space Kp = K p ( I , f ) , where f? = f(V) and 1 6 p 5 m
functions
or
=
v
p = 0.
c2.1.
Lemma.
Proof.
1.
For
1 6 p 6
Fixing
("1
x
Ilxllz =
E
c
and
0,
all
i
E
x.
and put
#
G
NOW,
belongs t o
I
on
3
j=1
a
.v.
01.
t
every element of
a j > 0,
where
for
i
i s of t h e form
,...
j = 1,2
jL'
n
I
and an increasing such that
is regularly decreasing and, for given
such that, on each set
vk(i)
inf > 0 iEIo vn(i) For
m E IN,
I . c I
for all
with
k z n'
(and hence
let
is increasing, and, by our choice of
vk (i) S n f n v > 0, k = n+l,n+2
I€Im
choose
SUMMERS
mo E N
with
Conversely, assuming that
5 m
n
and put
n'=
I
..
Then
is an
such that
2n,
then, for each
m z 2n,
K ~ T H E SETS AND
KBTHE
SEQUENCE SPACES
69
At this point, we note that Grothendieck [5,II, p.1021,
following
his discussion of a specific echelon space
with a
KL)the matrix
X1(INXN,Ao)
f o r which the associated sequence
A .
Vo
is defini
tely regularly decreasing (see Example b.11.1 below), claims that his proof of the quasinormability of to show" that a general echelon space able
X1(bIXN,Ao)
"is exactly valid
1, = X1(I,A)
is quasinorm
whenever condition (wG) holds.
Example 3.11 combined with Theorem 3 . 4 show that Grothendieck's claim was erroneous.
However, in view of 3.4 and 3 . 9 , Grothendieck's
statement becomes correct with the related condition (G) in place of (wG),
in which case the converse also holds.
T o conclude Section 3 . , let us show that condition (wG) is nonetheless
strong enough to guarantee the existence of a continuous
norm on
K
P
= K
P
(1,T)
for an arbitrary index set
I.
If condition (wG) holds (a fortiori, if
.12. Remark.
V
is re
then there exists a strictly positive function
Proof.
I
Let
(
as in ( w G ) .
I
~ denote ) ~ an ~ increasing sequence of subsets of
Since, f o r a fixed
such that, for any
io $?
u
n E IN,
there exists
n'> n
Im,
mEFJ
u
I\( Im) must be void; mE N
i.e.,
I =
u &IN
Im. Without loss of gen
K.D. BIERBTED, R.G. MEISE, W.H.
70
erality, we may take $he sequence
SUMMERS
("rn)WN
guaranteed by (wG) to
be strictly increasing, and then, by passing to a subsequence of (v,)
n
,
if necessary, assume that
nm = m,
m = 1,2,...
a l = 1, and inductively choosing am+l 2
Now, for b m = inf i€Im
Vm+l(i) ~ v,(i)> 0,
m = 1,2
,...,
c, a
where
we note that, for each
i € Im,
,...
whence, clearly, akvk z amvm , k = m+l,m+2 At this m'I Im point, taking 7 = inf a v E 7, we have v(i) = min akvk(i) > 0 k€N ksm whenever i E Im, m = 1,2,...; i.e., ? is strictly positive on
I
I.
4.
0
MONTEL AND SCHh'ARTZ ECHELON AND COECHELON SPACES
Again, we start with a definition.
4.1 Definition. (a) The sequence V = ( matrix
A = (an)
)
n€N
no infinite set
I .
In other words,
V
set
I .
(b)
V (or A )
exists
of
( o r the K6the Vn)nEN is said to satisfy condition (M) if there is
C
I
(or A ) satisfies ( M
I and each
m > n
such that, for some
n E N,
no E N,
if, for each infinite sub
there exists
m > n
with
satisfies condition ( S ) if, for each n E IN, vm an converges to 0. such that = V n am
there
K ~ T H E SETS AND K ~ T H E SEQUENCE
SPACES
71
It is clear that (S) implies (M).
4.2 Remark.
I
In the presence of condition (M), the index set
at most countable in which case all echelon spaces
kp,
echelon spaces
1 L p
1
Utilizing
the second
with
vm (i) 1 {i E Io;
2
MN
1 1;3,
we can
v* (i)
kl E W
find
v,1(172 $i1
I1 = { i E Io;
such that
ml
and our choice of
is uncountable,
I1 is a proper subset of
implies that
Io.
Proceeding inductively, we choose a strictly increasing sequence (mn)n
t N
and a sequence vm (i) n
In = {i E Inl;q(g2
,...
Inl, n = 1,2 put
%
2
k.
For
of positive integers such that
1 r n}is uncountable
Now, for any
J = {in; n E W}.
that
(kn)n
Next, fix
n > N,
n E N, k
2
1
and a proper subset of
select
in E Inl\In and
and choose
N E N
such
in E InlC IN, and
we then have
hence N
2
Since
k
min(min rl5N
was arbitrary, we have reached a contradiction to condi
tion (M) in its first form.
0
In this section, we will give new, and quite short, proofs for classical results on echelon and coechelon spaces (of arbitrary order) with condition (M) or (S) (see [ 7 , § 3 0 , 9 . ] ) .
In the process,
72
K.D.
BIERSTEDT, R.G.
MEISE,
W.H.
SUMMERS
we also derive new results, and we demonstrate that the use of the associated spaces
and
KO
lends new insight in this setting.
K,
satisfies (M) if, and only if,
A = (an),

1. I f condition (M) is not satisfied, let
Proof.
I
infinite subset of an (i) +=
inf iEIo y(i) = v
E n > 0, n = no+l,n0+2
(i)
for
n > no
every
we have
i E I.
with
with
,...,
y(i) = 0
no E N, y: I  +R+ by
and define elsewhere.
Since, for
i E Io,
and every
y E Am\ X o .
If there exists
2.
such that, for some
denote an
I.
an (i)ly(i)l
2
I,\ A , ,
y E
we can find
for all
e
i
no E N
and
while
IyI E
where
ak >
(Im)+ = 7 is dominated by some 0,
inf a v kE N Since, for every n > no
k = 1,2,...
I. c I ,
in an infinite set
0
> 0
e


inf
"k,
k E N ak
and every
i E Io,
and hence
114.4.
an (i) inf 2*iEIo n
For
Proposition.
0, condition (M) does not hold.
c > an
1
I
p
L m
or
A,
= A,.
IlMontel space if, and only if,

Proof. 1.
15 p
0 with
of 9
j
I.
A,
is a (semi)
Picking Y E X , \ X o , y(i)l
2
c
for all
Now, fix a sequence (i .) J
where
j'
= l,~,...
0
i .) = y(ij) J Since
and
jEN
K ~ T H E SETS AND K ~ T H E SEQUENCE
SPACES
73
is a bounded sequence in X p which has no j€N convergent subsequence because, for arbitrary k , L E N with k f 6 ,
k = 1,2,...,
(yj)
2. Even though the preceding argument works as well for p =
p = 0
and
we prefer to note a different (direct) proof in these cases.
m,
Supposing, then, that
lo
is a semiMonte1 space, we have that 1,
induces the topology of pointwise convergence on each bounded subset. set
Fixing J
wise.
I by
of
we define
y E A,,
The system
yJ(i) = y(i)
z(1)
YJ = (YJ(i))iEI if
i € J
and
for a finite subyJ(i)
of all finite subsets of
upward with respect to inclusion, and so
since
lyJl
5
IyI
J E 5(I)
for all
and
y E X,,
0
other
I is directed
(YJ)J€Z(I) which converges pointwise to
functions in rp(1) c X o
=
is a net of y.
But, is
(y,)
J€5(I) bounded in X,, and hence also in the topological subspace 10. We thus conclude that ( y J ) J converges in l o , in fact converges to
y,
whence
y € 1,;
3 . We now assume that
1
5
p
0, since
function
an+
7
Xp;
restricted to
?E
7
= (X,)
converges to
lo = 1,.
By Proposi
B = ? .C (I)1 P it is clearly enough to prove that
? E
of pointwise convergence on
and
and that
m
and show that
B
is weaker than the topology
I. Fixing
+
= (lo)+
yo = ?zo E B,
n E W,
by hypothesis, the
0, and we can find a finite
subset
74
BIERSTEM',
K.D.
I1 C I
with
sup
iEI\I1
MEISE, W.H.
R.G.
an(i)?(i)
+. 2 2l p
s
then defines a 7 neighborhood of U
n
B c (y E B; qn(yyo) 5 E } ;
In case
4.
p = 0
or
yo
Putting
which satisfies
in fact, for any
To show that
topology on
yo E B,
than
7,
We note that setting
;(i)
let
By hypothesis,
subset
I1
of
I
G
E A,,
such that
E
V
1,
= (A,)
+
B
in
c > 0
and
n
B
C
an(i)G(i)
an(i)ly(i)yo(i)l
{y E B; qn(yyo) s e ]
5
= A,,
E B]
for all
i E I
with
IyI
for all
?
5
and hence there exists a finite
we have the desired 7neighborhood of
U
xo
B,
be given.
n
such t h a t
78
K.D. BIERSTEDT, R.G.
n E N,
Now, for given
I .
If
condition.
MEISE, W.H.
m > n
fix
SUMMERS
as in the regularly decreasing
is an infinite subset of
I with
tnf
V,(i)
yn(i)>O,
1E I . then we have arbitrary
k 2
vk(i)
inf > 0 iEIo vn(i)
In particular, if
n.
at a contradiction. sets
I .
C
I
for all
Hence V
whereby
V
n
inf iEIo
k 2 m
as in ( * ) , we arrive
k = m
vm(i) V,O~ > 0
converges to
and s o , clearly, f o r
holds only for finite
0
0.
By Proposition 4.8, the class of echelon (resp., coechelon) spaces for which
V
satisfies condition (S) is exactly the intersection
of the quasinormable echelon spaces (resp., coechelon spaces with the strict Mackey convergence condition) with the Monte1 echelon (resp., coechelon) spaces (cf. Section 3 . and 4.7 above).
Of
course, these are just the Schwartz echelon spaces (resp., (DFS) coechelon spaces).
In most books on sequence spaces, one can find a direct proof that
v = ( vn),
satisfies condition (S) if, and only if,
Schwartz space and/or
hp
is a (DFS)space.
1,
is a
(Since these latter
two properties are equivalent by the general theory of ( F S )  and
%
(DFS)spaces, one usually shows, say, that (S) is equivalent to
being a (DFS)space by noting that (S) implies, and is implied by, the compactness of the embeddings given
n E N,
verges to
m > n
Cp(I,vn)
+ Lp(I,vm),
where, for
must be selected in such a way that
vm vn
con
Here, we take a different approach and establish
0.)
equivalences which are interesting in juxtaposition with Theorem
4.7. 4.9. Theorem.
For
15 p s
m
or
p = 0,
the following assertions
are equivalent :
(I) V (or A ) satisfies condition (S), (It) V
is regularly decreasing and satisfies (M);
K ~ T H X SETS AND K ~ T H E SEQUENCE
(11) X p
k
(111)
i s a Schwartz
L
ind nc
P
(and hence a (FS))
(1 S p
(I,vn)
C m),
79
SPACES
space;
resp.,
h0 L
P
ind c o ( I , v n ) , nc
i s a (DFS)space;
(IV)
A0
(V)
Lo
R,
=
(or
ko
r
is a (semi)
k,
t K,);
(V!)
Monte1 s p a c e ,
ko
is (semi)
re
f lexive.
(I) Q (I!) was o b t a i n e d i n 4 . 8 ,
Proof.
3.5.(a),
3.4,
view of
Schwartz s p a c e i f , semiMonte1 s p a c e .
Theorem
Theorem
h
C
4.7.
p = 0
for
and
from 2 . 8 . ( d )
m
But
space i s a
(I)
(IV) i s t r i v i a l ,
=
KO C K,
K O = K,
i n view of
and 2 . 1 .
k,
y i e l d s property
3
h.,
of
S e c t i o n 2.
so that
V
( 4 ) of
make i t o b v i o u s
s a t i s f i e s (M)
(IV), (11) f o r
by d u a l i t y ,
(V)
=
and w e c l a i m t h a t ,
must b e r e g u l a r l y d e c r e a s i n g , w h i l e t h e gen
implies Now,
i m p l i e s (V).
(11)
Next,
V
ho = k,
in
and o n l y i f , i t i s b o t h q u a s i n o r m a b l e and a
3.7 whence
e r a l inclusions that
4.7 b e c a u s e a c o m p l e t e 1.c.
* (It) s i n c e h0
(IV)
conversely,
and
and (It) P (11) h o l d s
i m p l i e s (111)
p = 1
w h i l e (111) f o r
by
p = 0
or
p =
m
(Vl) t r i v i a l l y , and (V') * (IV) f o l l o w s Finally, f o r
1 5 p
0, there
86
K.D.
BIl?RSTEDT, R.G. MEISE, W.H.
F o r any locally convex space
quasinormability of see Grothendieck Proof of 5 . 3 .
E,
E,
property (3) is equivalent to the
as is ( 3 ' ) for metrizable 1.c. spaces
[4, Lemme 6 , Proposition
The equivalence (1)
proof of Theorem 3 . 4 .
SUMMERS
e)
(2)
E;
12, p. 107/108].
follows by modifying the for (1
We next note a direct argument
(3).
=)
If U
= ( Y E ip;
we choose
m > n
w F o r each S
pl/'
e > 0,
evn(i)
(an(i)ly(i)l)P
C
iE I
where
s p],
p
> 0,
according to (wS) and put
ap; z
= EY
iE I
(am(i)Iy(i)l)P
(wS) allows us to find
whenever
G(i) < vm(i).
a,;
is a bounded subset of

E V
11.
such that
By 5.2.(a),
B =
vm(i)
G L (I) P
1
W c O U + B.
we claim
To establish this claim, put
G
L
I1 = { i E I; v,(i)
s ?(i)]
yj = ( ~ ~ ( i ) )by ~
yj(i)
and note
that
If we fix i E I. J
y E W
and
y = y2+yl.
and define
y.(i) = 0 if J The inequality
P
S
shows that
iE I
i
4
Ij,
j = 1,2,
(am(i)lY(i)l)P
1 y2 = e ( x y 2 ) E e U ;
since
?
P
is strictly positive on I1'
1
C
iE I
for
then clearly
(m r
= y(i)
)Ps
1
K6THE SET5 AND K6THE SEQUENCE SPACES
implies
y1 =
1 c c(yyl) E
87
&p(I)l = B.
Conversely, to see that; ( 3 ) implies (l), we remark that, by polarity in the dual system
Uo = ( X p
( 3 ) , for each
wo
=
(Xp,Km) =
(Xp,hm)and
Gp(I,a
) ) " = dm(I,vn)
n l
(1, n P&p(I,am)l) 0 = 1 &m(I,vm)l, where P
m > n,
may take
assuming condition
such that, applying 5.2.(a)
p
,
1
there exists
> 0 and where we
again, for every
e > 0, we can find
with
(G tP(1),)"
=
BO
y1 [ ( e U ) O n
L,(I,V~)~ n
=
EX
sup T(i)Ix(i)l iEI
E K,;
Bo] c (eU+B)O c W o
whereby
Ex E K ~ SUP ; ;(i)lx(i)l iE I
s
01
c
5
2E hm(l,vm) P
Since we are in a locally convex setting now, applying p.1051, we realize that logy on
and
whence, clearly,
&m(I,v,)l
ly retractive.
.Cm(I,vm)
K,
4,
11
.
1
[4, Lemme 5,
induce the same topo
r ind tm(1,vn) nt
We can now obtain (1) from Theorem
is bounded
3.4.
Finally, ( 3 8 ) obviously implies ( 3 ) , while Grothendieck's proof of the converse for metrizable 1.c.
spaces
E
/4, Proposition 12,
p. 1081 only requires Mackey's countability condition which remains valid for arbitrary metrizable topological vector spaces see [ 7 ,
$29, proof of
5.4. Theorem.
For
1.(5)1.
0 < p < 1,
E;
e.g.,
0
the following properties are equi
valent :
(1) V
satisfies condition (M);
(2)
1,
is a semiMonte1 space;
(3)
kp
is a semiMonte1 space.
Proof.
The method used to prove Proposition
(1) e ( 2 ) .
For (1) a ( 3 ) . since
hP L
4.4 also yields
ind 4 (I,vn) nt P
is a regular
88
K.D.
BIERSTEDT, R.G.
inductive limit which equals
K
P
MEISE, W.H.
SUMMERS
algebraically and topologically,
is (relatively) compact in K P' n=1,2,... Now, it is clearly enough to fix n E N and prove that K P induces a topology on 4, (1,v ) that is weaker than pokntwise conP " 1 vergence on I. it suffices to show that
&p(I,vn)l
In the following, we fix
xo E Gp(I,vn)l,
u =
{x E K

c
iE1
P' By 4.6.(att),
nite subset
an J
belongs to of
I
7 E 7,
s 11.
(G(i)lxo(i)x(i)l)p
co(I,G),
such that
and let
and hence there exists a fi
s
T(i)an(i)
&'Ip
f o r all i
J.
Putting
= (x E K p ;

U
G(i)lxo(i)x(i)l
is a neighborhood of
wise convergence on for each
x E
f?
fl tp(I,v
)
n l
E
x
. =
J
IN. 0
e > 0,
...
indices j
v,
il,i2,
Putting
(G(i) lxo(i)x(i)
E
?,
n E N
with
(xj)jcN
for each i E J],
0 n LP(I,v
)
n l
c U
since,
I )'
an E K,\Ko..
and a sequence
I
U P
(M) is not satisfied, Lemma 4.6.(att)
such that
xj = (Xj(iHiEIS
elsewhere,
for each
in
JI
.
F o r the converse, if condition
tells u s that there is
1
21
I which satisfies
c
Go E
()
with respect to the topology of point
xo
iE I
choose
S
I.
of pairwise distinct
Go(i.)an(ij)
where
Hence, we can
J
xj(ij)
2
e
f o r all
= an(ij)
is a bounded subset of
Kp
and since,
SETS AND K 6 T H E SEQUENCE SPACES
89
sup c (G(i)lxj(i j E N iEI But
(xj)j c N with k f & ,
is not precompact in
c (Go(i) I%(i)x&(i) iE I Hence
k
P
'5
K~
')1
K
P
since, for arbitrary
= (+o(ik)an(ik))p
+ (+o(i&)an(iL))p
k,& E N
2
. ' 0 2
is not a semiMonte1 space.
We leave it to the reader to write down the analogs of Theorem 4.9 and Corollary 4.10 for
0
< p < 1.
As we conclude, let us remark that, under the "GrothendieckPietsch condition" (N)
for each an
vm ~a n
_  
m
n E N,
there exists
m > n
such that
is (absolutely) swnmable,
which clearly implies (S) and is equivalent to nuclearity of (see Grothendieck [5,II, p.59]), the corresponding spaces
hp,
all spaces 0 5 p 5 a;
Xp
),,
coincide, as do
for the case
0
< p < 1,
see, for example, [ 6 , Remark €allowing 21.6.2.1. Additional note
In their recent manuscript "A characterization of the quasinormable Fr6chet spaces" (DUsseldorf, December 1981), R . Meise and
D. Vogt

inspired, among other things, by the results in Section
3 . of the present article
 use methods
due to D. Vogt (and M.J.
Wagner) to characterize the class of all (abstract) quasinormable Fre'chet spaces in various ways. Fr6chet space
E
In particular, they prove that a
is quasinormable if, and only if,
tient of a complete tensor product
.L1(I) 6 A(A)
,
E
is a quo
where
index set (one can take I = N whenever E is separable) and = ),,(tN,A)
is a nuclear Kbthe echelon space.
I
is an A(A)
K.D. BIERSTEDT, R.G. MEISE, W.H.
90
SUMMERS
F33FERENCES
c 11
c 21
K.D. Bierstedt and R. Meise, Induktive Limites gewichteter RCLume stetiger und holomorpher Funktionen, J. reine angew. Math. 282 (1976), 186220. K.D. Bierstedt, R. Meise and W.H. Summers, A projective description of weighted inductive limits, Transact. Amer. Math. SOC.
[ 31
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Functional Analysis, Holomorphy and Approxima
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(1981),
247283.
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A.
Grothendieck, Sur les espaces (F) et (DF), Summa Brasil.
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51
2 (1954),
57122.
A. Grothendieck, Produits tensoriels topologiques et espaces nucl6aires, Mem. Amer. Math. SOC. (1955, reprinted 1966).
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c 61
H. Jarchow, Locally convex spaces, B.G. Teubner (1981).
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G. Kt)the, Topological vector spaces I, Springer Grundlehren der math. Wiss.
c 81
(1969).
H. Komatsu, Projective and injective limits of weakly compact sequences of locally convex spaces, J. Math. SOC. Japan 2
(1967), 366383. [ 91
M. Valdivia, Solution of a problem of Grothendieck, J. reine angew. Math.
905 (1979),
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M. Valdivia, Cocientes de espacios escalonados, Rev. Real Acad. Ciencias, Madrid 22 (1979), 169183.
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M. Valdivia, Algunas propiedades de 10s espacios escalonados, Rev. Real Acad. Ciencias, Madrid
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(1979), 389400.
M. Valdivia, On quasinormable echelon spaces, Proc. Edinburgh Math. SOC. & (1981), 7380.
C 131 R. Wagner, Manuscript on inductive limits of weighted LPspaces, Paderborn (1979), unpublished.
KOTHE SETS A N D KOTHE SEQUENCE SPACES
Fachbereich Mathematik
Mathematisches Institut
Gesamthochschule Paderborn
Universitdt DUsseldorf
Paderborn
Dlisseldorf
West Germany
West Germany
Department of Mathematics University of Arkansas Fayetteville, Arkansas 72701 USA
This Page Intentionally Left Blank
Functional Analysis,Holomorphy and Approximation neoty, LA.Barroso (ed.) 0NorthHollandPublishing Company, 1982
PARAMETRIC APPROXIMATION AND OPTIMIZATION
Bruno
B r o s owski
I. INTRODUCTION
T
Let subset of
RN,
be a compact Hausdorffspace, and let
p: U
b
U
be a nonempty open
be a continuous function.
R
For
each pair of continuous mappings
A: T x U + R
and
b: T
b
R
we consider the following minimization problem:
(*)
Minimize
v
p: U
subject to
+ IR
s b(t).
A(t,x)
t€T I n this way we have defined a family of semiinfinite minimization problems with family parameter where
8
meter
0
(A,b)
C(TXU)
is a suitable subset of := (A,b),
and parameter space 8 ,
x
C(T).
For each para
we define the set of feasible points
the minimum value E~
:= inf {p(v)
E R
1
v E zU],
and the set of minimal solutions P
:=
Clearly, the minimum set
cv P
0
E zU
I
P(V) = EJ.
depends on the "matrixtt A
and on
94
B. BROSOWSKI
the "restriction vector"
b.
"continuously" relative to EXAMPLE 1.1.
Let
and
A
b.
T := {1,2
,...,m}
+
+...+
= P
P(X1,X2,"'9XN)
It is natural to ask if
p2x2
Pa
varies
We give some examples.
be a finite set,
with
PN%
U := RN, and
E
P1'P29"'9PN
'
Further we assume, that A(P,X)
E T.
for each
= B(cr).x B: T + RN
Then the mapping
as a (m,N)matrix and the function
b
can be considered
as a vector in
Rm.
With
these assumptions we receive the following minimization problem of type ( * ) : (ML)
Minimize
p(x1,x2,
...,xN)
=
N C
v=1
pyxv
subject to N
C
w =1
BCIvxv5 b W'
= 1,2
,...,m,
i.e. we have a parametric linear optimization problem with variable matrix
B
and variable restriction vector
parameter
a = (B,b)
b.
In this case the
can be considered as an element of
RmXN x Rm.
The question of the continuous dependence of the minimum set has been considered by many authors.
Surveys may be found in
for the case of variable restriction vector and
Nozicka et al. [12]
for the case of a variable matrix.
in Klatte [lo]
Parametric linear
finite optimization has many applications, compare e.g. Lommatzsch [Ill

EXAMPLE 1.2.
X
Let
be a normed linear space and let
unit ball in the continuous dual open subset of F o r any
v
b E X,
:= B(U1)
RN'
X*
of
X.
Let
U1
BX*
be the
be a nonempty
1 b X be a continuous mapping. the set of best approximations to b from the set and let
is defined by
B: U
PARAMETRIC APPROXIMATION AND OPTIMIZATION
95
where d(b,V) The setvalued mapping jection onto and the set
V.
Pv: X + POT(V)
Clearly, the set
V.
0
is a best approximation of
is a minimum point o Minimize
v
E V].
is called the metric pro
PV(b)
depends on the point
b
An element 0
vo = B(X1,X2'
(MA)
I
:= inf{llbvII €
b
from

)&,
* *
V
if and only if
the following minimization problem:
p(xl
subject to V
These inequalities can also be written in the following way:
With the aid of the KreinMilmantheorem the weak*compact set
BX+
can be replaced by the in general not weak*compact set of the extreme points of type (*).
BX*.
Clearly, the minimization problem (MA) is of
The minimum set mapping
P
is connected with the metric
projection by the formula P where
A: BX+
x
(A,b)
= B"(P,(b))
(UIXR) + R
A(x*,xl,x2,.
.
X d(b,V),
is defined by
:=
The problem of dependence of
X*(B(xl,x2, PV(b)
on
V
been considered recently in the paper 1 9 1 .
(for a fixed
b) has
This question seems im
portant since, for example, when approxima ing with spline functions,
B.
96
BROSOWSKI
it is of interest to know how the set of best approximations to a given function depends on the knots which define the splines.
In
practice, the knots cannot be specified exactly, but only up to some error.
It is reasonable to ask if the best approximations
change "continuously" as the error tends to zero.
F o r results com
pare the paper [g].
The question of the dependence of
b
V)
(for a fixed set
on
is essentially the problem of the contin
uity of the metric projection considered by many authors. [ 151 and Singer [ 141
PV(b)
PV
onto
V.
This problem has been
Partial surveys may be found in Vlasov
.
The author (partly in collaboration with Schnatz) extended the methods used for the metric projection to parametric semiinfinite optimization problems [1,2,3,4,5,6,7,8]. In this way, it could be derived not only new results for parametric optimization problems but also it could be given a unified approach to problems of parametric optimization and of the metric projection. In [2,3,4,5,8]
the author resp. Schnatz and the author considered
the linear optimization problem ( * ) and assumed that only the restriction vector was varying.
They gave various conditions f o r the
continuity especially for the lower semicontinuity of the minimum set mapping.
I n [7] the author could extend this result
linear optimization problems with C(T).
A(*,Pb)
to non
convex for each
b
in
In [6] the author investigated also parametric nonlinear mi
nimization problems with certain variations of the matrix. could be shown:
It
If the minimum set mapping is upper semicoiitinuous
then the minimum set contains at least one element, which can be characterized by a certain criterion. In this paper, we continue the investigations begun in [ 6 , 7 ] . We consider only the case of variations of the restriction vector. I n this case, we can assume that
C(T)
is the parameter space.
PARAMETRIC APPROXIMATION AND OPTIMIZATION
97
We prove first an always sufficient criterion f o r a minimal point. This criterion is in general not necessary. pointwise convex optimization problem.
Then we introduce
F o r these optimization
problems the mentioned criterion is always a necessary condition. Many important optimization problems are pointwise convex, e.g. linear, convex, and fractional optimization problems.
Finally we
If the minimum set mapping is upper semicontinuous then at
show:
least one minimum point satisfies the criterion.
11. CHARACTERIZATION O F MINIMAL POINTS
T' := T U {to]
Let to
4
T
be the compact Hausdorffspace with
as an isolated point.
F o r each
b E i2A,p
and each
x E U
we define:
$b(t,x)
:=

A(t,x)
b(t)
+
if
t = t
if
t E T
Eb
and
THEOREM 2.1.
Let
vo
E
U
be an element such that
vo E Pb.
then PROOF.
We introduce the function cpb(x)
Let point
v
:= max
tET'
be an arbitrary element in t E
";,
* vO
$,(t,x). U.
By assumption there is a
such that
(Pb(vo) = eb(t,vo)
$,(t,v)
(Pb(v),
98
B. BROSOWSKI
vo
whence
rpb.
i s a minimal p o i n t o f
a minimal p o i n t of
p
on
Zb.
Each minimal p o i n t i s a l s o
To prove t h i s , l e t
x 6 Zb.
Then
we have
' $b(to,x)
pb(x) If
x
4
Zb,
t l E
then t h e r e e x i s t a point A(tl,x)

' Eb
= P(x)
b(t1)
>
T
such t h a t
0,
which i m p l i e s pb(x) Thus,
' d + ~ ( ~ l =, ~A) ( t l , x )b ( t l )+ Eb ' E b '
e v e r y minimum p o i n t
of
v
pb
s a t i s f i e s the inequality one c o u l d choose a sequence
(vk),
k v
E Zb,
so that
l a r g e one h a s k qb(v
pb vo)
6 Eb
+
0
which i s a b s u r d .
W e g i v e some examples: EXAMPLE 2.2.
Let
B: T + RN
a c o n t i n u o u s mapping,
c o n s i d e r t h e l i n e a r o p t i m i z a t i o n problem,
(LM)
Minimize
...,x N ) :=
p(x1,x2,
N
U = R
,
and
i.e.
N
C
pvxv
v=l
subject t o V
B(t)'x h b ( t ) .
t€T I f we s e t
B(to)
:= p
t h e c r i t e r i o n of Theorem 2 . 1 r e a d s as f o l 
lows:
v V€RN
which i s e q u i v a l e n t t o
min
tEM&vo
B ( t ) ( v o  v ) s 0,
PARAMETRIC APPROXIMATION AND OPTIMIZATION
V
min
B(t).v
i
99
0.
* By the lemma of Farkas, the inequality is equivalent to
and, by a lemma of Caratheodory, there exist real numbers
...,pN+l
2
0
and points
tl,t2
,...,tN+l
E %,vO\(to}
po,pl,
such that
N+ 1
E
pop +
v=l
If in addition the interior of EXAMPLE 2.3.
P,,B(t,,) Zb
= 0.
is nonempty then
po
#
0.
Let be given continuous mappings Bo: T' +
R
and
B: T'
Co: T' + R
and
C: T' + RN.
b
RN
and
The set
is open and convex.
In the following, we assume
U
sider the following fractional optimization problem:
on
U
subject to
V
t€T
A(t,x)
Bo(t) t)
:=
cO(
+ +
B(t)*x c(t).x
b(t).
Then the criterion of Theorem 2.1 reads as follows; V
V€U
min
t€q,v
which is equivalent to
[A(t,vo)A(t,v)] 0
s 0
#
@.
We con
B. BROSOWSKI
100
min
v

[A(t,vo)[Co(t)+C(t).v]
s 0.
[Bo(t)+B(t)*v]]
VERN tEM'
b,vo
Then the lemma of Farkas implies that the last inequality is equivalent to 0
2N+2
E con ([G(t,vo)
E R
' 3). Mb,vo
I
The lemma of Caratheodory implies that there exist points
tl,.. .,t2N+3 E T
and real numbers
Po,P1,.
. .,p2N+3
such that
2N+3 PoG(to,vo) +
If in addition the interior of
c
v= 1
Zb
PwG(ty,vo) = 0 is nonempty, then
p,
f 0.
Assuming this, we can conclude from the last equation the relations:
= o , where
r
v
:= p v / p o
2
0.
The criterion of Theorem 2 . 1 is in general not necessary as the following example shows: EXAMPLE 2 . 4 .
Let
T = (tl,t2]
A(t,u)
:=
I2
with
 l~;1
tl f t2,
U = R,
if
t = t1
if
t = t2
and let
PARAMETRIC APPROXIMATION AND OPTIMIZATION
The feasible point
uo = 1
problem:
p(u)
Minimize
is a minimal point of the minimization subject to
= u V
tET One has
q,uo = [to,tl)
101
A(t,u)
and for
p(uo)

p(u)
A(tl,uo)

A(tl,u)
s b(t).
the inequalities
u = 4
= 1
+ 4
= 3 > 0
and = 1

0
= 1 > 0.
Consequently, the criterion of Theorem 2.1 is not satisfied for the minimal point
uo = 1.
For the investigation of the necessity we introduce the following DEFINITION 2 . 5 . if for each
1>
A: TXU + U7
A mapping
for each pair of elements
0,
each closed subset
is called pointwise convex,
F c T
there exists an element V
tEF
such that
vA
in
A(t,vo)
U

A(t,vo)
v,vo E U
 A(t,v)
>
and f o r 0
on
F
such that
A(t,v
a)
>
0
Then we have THEOREM 2 . 6 .
Let the mapping :$,
be pointwise convex.
T'xU + R
Then an element
vo E U
is a minimal point
102
of
B. BROSOWSKI
p
on
if and only if one has
[p(vo)p(v)
v
vEu PROOF.
Zb
o
i
or
min t€q
(4(t,vo)A(t,v))
nT
01
*
vO
Sufficiency.
a minimal point of
Compare Theorem 2.1. p
on
Zb.
Pb0)
Let
vo
be
I f the criterion were false, one
v E U
could choose an element
Necessity.
so that

P(V) > 0
and
By compactness of real number
and
U
and by the continuity of
r a.
Now define the open set
W
a := {t E T
By compactness of
T\W,
1
A(t,vo)A(t,v)
with a suitable number

M > 0.
b(t)
i
>
If
t
t E Wa,
M < 0
vx
in
0,
A(t,vo) for each
21 .
According to the definition of
pointwise convexity, choose an element
P(V,)
>
one has
V A(t,vo) tET\W

there is a
> 0 such that
p(vo)p(v)
P(V0)
A,
in the closed set

A(t,v,)

W
a ’
then we have the estimate
’
and
0
U
such that
PARAMETRIC APPROXIMATION AND OPTIMIZATION
If t j? W
U'
103
then we have the estimate
Consequently,
zb.
vX E
Since
p(v,)
point, which is a contradiction.
< p(vo),
vo
is not a minimal
W
111. EXAMPLES OF POINTWISE CONVEX MAPPINGS
In this section,
T of
A: TXU + IR
is a continuous mapping, where U
is an compact Hausdorffspace and
is a nonempty open subset
IRN.
EXAMPLE 3.1.
If
uous mapping
A: U + R.
T
only if, for each A(vo)

A(v)
contains only one point, then one has a contin
X > 0
and for each
> 0 there is an element A(vo)
 A(v,)
With a suitable number A(vo) By the continuity of
0 < K(h)
This mapping is pointwise convex, if and
K(X)
A,
>
0
>
0,
Ab,)
vA
in
U
such that
U
such that
IIvavoll < X .
one has
= K(X)(A(v,)A(v)).
one can choose
1
so small, that
< 1. Then, one has ~(v,) = (lK(a))
i.e. for sufficiently small combination of the numbers
A(v~) + K(x)A(v),
X > 0 A(vo)
than one point, then the number point.
and
vo,v
the number and
K(1)
A(v).
A(vh)
If
T
is a convex contains more
depends on the respective
This motivates the notation "pointwise convex".
104
B.
EXAMPLE
3.2.
If
B: T
+
IRN
BROSOWSKI
i s a c o n t i n u o u s mapping, t h e n t h e
mapping
:= B ( t ) . x
A(t,x)
i s p o i n t w i s e convex on v,vo
E IRN
such t h a t
closed subset
F
C
To p r o v e t h i s ,
TxRN.
A(t,vo)
T.

A(t,v)
> 0
l e t be given:
for e a c h
t
> 0,
i n a given
D e f i n e t h e element
with
Then,
IIvoVhll
EXAMPLE 3.3.
Let
U
= X1~IIvovIl < h * open s u b s e t o f
b e a convex, TxU
A:
+
and l e t
IR
b e convex w i t h r e s p e c t t o t h e second v a r i a b l e ,
v
RN
A(t,pv
+ (lP)vo)
[O,l]
and f o r e a c h p a i r
i.e.
PA(t,v) + (lP)A(t,vo)
tET for e a c h A
p
E
v,vo E U.
Then t h e mapping
such t h a t
A(t,vo)A(t,v) > O
i s p o i n t w i s e convex. Let be given:
f o r each
t
X >
0,
Then,
E U
i n a given closed s u b s e t vx :=
with
v,vo
(lh1)Vo
F c T.
+
X1V
D e f i n e t h e element
PARAMETRIC APPROXIMATION AND OPTIMIZATION
for each
105
E F, and
t
EXAMPLE 3 . 4 .
The mapping Bo(t)+B(t).x A(t,x)
:= C0(t)+C(t)'X
defined in Example 2.3 is pointwise convex.
> O,v,vo E U
such that
given closed subset
A(t,vo)
F C T.

A(t,v)
Let be given:
> 0 for each
t
in a
Define the element
:= (1X1)v0 + A,v
with

A(t,v,)

B o (t ) +B (t ) *vo Co(t)+C(t).vo
A(t,vo)
for each
t

B o (t ) +B (t ) 'vx Co(t)+C(t)'vx
E F, since co(t)+c(t).v X1'Co( t)+C(t)
for each
t E T.
EXAMPLE 3.5. {u1,u2, setsof
C(S).
Further we have
Let
...,urn]
'vx ' O
and
S
be a compact Hausdorffspace and let
{v1,v2,
For each
s €
...,vn] be S
two linearly
define the vectors
independent sub
106
B. BROSOWSKI
and
Let
and
If we let T = S 1
u
T
S2,
be the disjoint union of the spaces
S,
i.e.
then, by the considerations of Example 1.2, an element w
c
o
.U
:= ___
ao*v 6
v
is a best approximation (in the sense of Tchebyshev) of from
V,
if and only if
b
E
C(S)
is a minimal point of the
(co,ao,d(b,V))
minimization problem: Minimize
p(c,a,z)
:= z
subject to
where
A(t,(c,a,z))
wz if
tEsl
if
t c s2
b(t)
if
t E S1
b(t)
if
t E S2
:=
and
1

I;(t)
Then the mapping given
A > 0
:=
:,4
(TU[to])xU + R
is pointwise convex.
and
w := (c,a,z),
w
:= (co,ao,zo) E
u
m
:= R X U ~ X R
Let be
PARAMETRIC APPROXIMATION AND OPTIMIZATION
for each If
t
in a closed set
zzo = 0, then
to $? F.
107
F c T U {to]. We define the element
by
with
t E F,
Then, for each
If
z z < 0, then
that for each
X,
to $? F.
There is a real number
one has
E [O,l]
and
where
Then define the element
wX:= (cX,aX,zX) by
cx :=
2
X,C
+
2
2
(1X1)Co* 2
aX z = Xla + (lxl)ao, zx
:= x,z
+
(lX1)Zo
M > 0
such
108
B.
BROSOWSKI
with
Then, f o r e a c h
t E F l7 S1,
For e a c h
n
If
zoz
f o r each
t E F
> 0,
S2,
i t follows s i m i l a r l y , t h a t
t h e n t h e r e e x i s t s a r e a l number
XIE [O,l]
and
Then d e f i n e t h e e l e m e n t
wx
:= ( c , , a x , z X )
L i k e b e f o r e , one c a n p r o v e t h a t
by
M
> 0
such t h a t
109
PARAMETRIC APPROXIMATION AND OPTIMIZATION
EXAMPLE 3 . 6 .
T
Let
be the compact interval
fa,@],
a < 8.
The
mapping A: TXR2N + R defined by A(t,(a,c)) is pointwise convex. (ao,cO),
(a,.)
N c t C aV*e V=l
"
:=
T o prove this, let be given
N
c t a o V * e OV
v =1
t
and
such that
E RZN
C
for each
h > 0
F c
in a closed set
N
 c
v=l
c t a," > O
[a,@]. Then, by a result of
Polya & Szeg8 [ 13, p.481, there is a positive (on F ) function
and
N
v = 1 aoV for each
t E F
e
c
t
N
 c
a
v=l
and
T
XV
c t e 1" = Tw(t)
+
O ( T )
> o
> 0 sufficiently small.
It should be remarked, that the problem of best Tchebyshevapproximation of a function
b
in
C[a,b]
from the set
leads to a pointwise convex optimization problem.
IV. THE UPPER SEMICONTINUITY OF THE MINIMUM SET F o r each pair of continuous mappings
A: TxU t R
and
p: U + R
B. BROSOWSKI
110
we define the solution set z!
4.1.
THEOREM
Pb f
$1.
+ POT(U) be upper semicontinuous and ASP compact valued, and let Qb := A(*,Pb) be convex f o r each b in
n
A,P
Let
I
:= {b E C(T)
A,P
P: 52
*
Then, f o r each in
Pb
there exists an element
1 z
(1) Let 0,
DA,p
bl
and
vo
pbl'
we set b X := A(*,vo)
+
X(blA(',vo)).
for each h E [O,l]. Then PbX Then v T o prove this, let v E Z
ba
whence we conclude
Zb
a minimal point of
p
(2)
v
which satisfies the criterion of Theorem 2.1, i.e.
PROOF. For
,
z!A,p
in
b
Let
bl E
nA,p
x
C
on
and
.
Zbl. Z bl'
satisfies the inequalities
Since
vo
'bl
it follows that
vo E Pbl.
If
then
i.e. the criterion of Theorem 2.1 is fulfilled.
and since vo
'bX*
v
is
PARAMETRIC APPROXIMATION AND OPTIMIZATION
If not, there is an element
Since in
The
v E U
with
is compact and since the mapping
t,
we can choose a real number
111
A
is continuous
a > 0 such that
open set W
contains
U
:= [t E T
I
A(t,vo)A(t,v)
>
Consequently, there exists a real number
Mbl,vo
K > 0
such that
Choose a real number
1
with
It follows that V
tET i.e.
v
zb
h
.
Since
A(t,v)
p(vo)
 bh(t)
> p(v),
< 0,
it follows that
v0
6
Pbx
9
a contradiction.
( 3 ) To prove the theorem, it suffices to show, that for each element
112
bl
B. BROSOWSKI
DA,p
in
there is an element
v
in
Pbl,
which satisfies
condition (A) of (2). In this were false, one could choose an element
bl
in DA
,P'
such that V
CASE 1.
A(v)
2
v E Pb
1
,
there exists
Let the set
be unbounded. pbl
€or each
such that
0
(k(v)
in
V B P b . A
> 0 P,
By the upper semicontinuity of a minimal
3
x
V€Pb 1
I
E n
E
vn E Pbn
9
Then, for each
b'
3
1
there exists an element
N,
vn
such that n E N,
where bn = A(*,vn) By compactness of point
v
maximal
in X(vo).
Pbl,
Pbl.
+
n(blA(*,vn)).
the sequence
F o r the element
For the proof, let
(v,) v
has an accumulation there does not exist a
A > 0
be arbitrary.
Then the
element bX := A(',vo)
+ x(blA(',vo))
is an accumulation point of the sequence
By part (1) of the proof, the element almost all
n E N.
vo
in
.
is contained in
By the upper semicontinuity of
x
Since > 0 pbx does not exist a maximal A(vo). that
vn
was chosen
P,
for gn it follows P
arbitrarily, there
Consequently, we have
113
PARAMETRIC APPROXIMATION AND OPTIMIZATION
CASE 2.
The set
is bounded. Then, let
l o := sup{h(v) By the upper semicontinuity of such that
vo E Pg
,
P,
of
P
g '
P
Then the set g' semicontinuity of P , hood
of
V
g
I
IR
E b'
3.
1
there exists an element
v
in
where we have set
g := A(',Vo)
By compactness of
E
+
Xo(b1A(.,Vo)).
there exists a compact neighborhood
W c U
is also compact. By the upper
con(A(,W))
there exists a bounded and convex neighbor
such that
Ph C W.
V
h€v By the compactness of r
>
0
con(A(.,W)),
we can find a real number
such that the compact and convex set K~
is contained in
V.
:= g
+
Z [ g r+ 1

con(~(.,w))]
The setvalued mapping
is upper semicontinuous and compact and convexvalued. KrC
V,
POT(Kr).
the compact and convex set
Kr
By the fixed point theorem of
is mapped by
h o r there exists an element
which implies
v1
E
h
in
A(h), in
Ph
into
this mapping has
KY FAN,
a fixed point, i.e. there exists an element
A
Since
such that
Kr
with
B. BROSOWSKI
114
Consequently,
g
is sontained in the segment
part (1) of the proof, the element
is contained in
Zh 2 Z
we have elements
and vo E Zg, it follows that g vo, vl, g, h satisfy the relation
t
Now we can determine real numbers Ph
v1
[A(*,vl),h].
and
p
By
Ph.
vo E Ph.
Since The
and an element
7
in
such that
+ t(blA(.,?))
h = A(',;) and A(.,+)
= PA(',V1)
+ (1P)A(.,V0).
The computation yields
By the convexity of Since
Xo(l+r)
? E Pb
1
.
A(*,Ph),
the element
is contained in
Ph,
> X o > 1, by part (1) of the proof, it follows that
Consequently,
1,
is not the maximum of the set
E IR which is a contradiction.
a
I
E b'
3,
1
PARAMETRIC APPROXIMATION AND OPTIMIZATION
115
LITERATURE 1. BROSOWSKI, B.: On parametric linear optimization.
Lecture Notes in Economics and Mathematical System Vol. 157, 1977, 3744.
2. BROSOWSKI, B.:
Zur parametrischen linearen Optimierung: 11.
Eine hinreichende Bedingung far die Unterhalbstetigkeit. Operations Research Verfahren Vol. 31, 1979, 137141.
3. BROSOWSKI, B.: On parametric linear optimization 111. A Necessary condition for lower semicontinuity. Methods of Operations Research 36, 1980, 2130. 4. BROSOWSKI, B.: On parametric linear optimization IV. Differentiable functions. Lecture Notes in Economics and Mathematical Systems Vol. 179, 1980, 3139.
5 . BROSOWSKI, B.: On the continuity of the minimum set in parametric programming. Anais V Congress0 Brasileiro de Engenharia Mecgnica, Vol. D., 1979, 260266.
6. BROSOWSKI, B.: On the continuity of the optimum set in parametric programming. "Col6quio Brasileiro de Matem6tica". Popos de Caldas, MG, Julho 1979.
7. BROSOWSKI, B.: On the continuity of the optimum set in parametric semifinite programming. "Second Symposium on Mathematical Programming with Data Perturbations." Washington D.C.,
May 1980.
8. BROSOWSKI, B., SCHNATZ, K.: Parametric Optimization Differentiable Parameterfunctions. Methods of Operations Research
37, 1980, 99118. 9. BROSOWSKI, B., DEUTSCH, F., NthNBERGER, G.: Parametric Approximation. J. Approx. Theory Vol. 29, p.261277, 1980. 10. KLATTE, D.: Lineare Optimierungsprobleme mit Parametern in der Koeffizientenmatrix der Restriktionen. In Ell], p. 2353.
11. LOMMATZSCH, K. (ed.),
Optimierung.
Anwendungen der linearen parametrischen BirkhauserVerlag, Base1 und Stuttgart 1979.
12. NOZICKA, F. GUDDAT, J., HOLLATZ, H., BANK, B.: Theorie der linearen parametrischen Optimierung. AkademieVerlag, Berlin, 1974.
116
B. BROSOWSKI
13. P6LYA, G.,
S Z E G b , G.: Aufgaben und Lehrskitze aus der Analysis,
Taschenbficher Bd.
74, SpringerVerlag, Berlin, Heidelberg,
New York 1971.
14.
SINGER, I.: The Theory of Best Approximation and Functional Analysis. SIAM, Philadelphia,
1974.
15. VLASOV, L.P.: Approximative properties of sets in normed linear spaces. Russian Math. Surveys 28, 1973, 166.
Johann Wolfgang GoetheUniversitgt Fachbereich Mathematik Robert MayerStr. 610 D6000 Frankfurt
Functional Analysis?Holomorphy and Approximation Theory. JA. Barroso led.) 0 NorthHollnndPLblishingCompany.1982
MAXIMAL CONVOLUTION OPERATORS AND APPROXIMATIONS
M.T.
Carrillo
M.
and
d e Guzmgn
INTRODUCTION
b e a f a m i l y of f u n c t i o n s i n f E Lp(Rn),
1i p
0
~(llfll,/X)~
f o r each
J
and e a c h and
d e f i n e d by (i.e. f E Lp
Kjg(x)
= k.*f. J
K.f
J
I n o r d e r t o f i n d out whether
i s a directed set, e x i s t s a t
f €
LP(Rn).
K*f(x)
= sup IKjf(x) j€ J
if there exists
one h a s
I[x
c
LP(Rn),
then
> 0
: K*f(x)
converges a t almost each
b e l o n g s t o some d e n s e s u b s p a c e o f a.e.
Kj.
For
i t i s very o f t e n s u f f i c i e n t t o look a t the a s s o c i a t 
x
e d maximal o p e r a t o r
f o r each
L1(Rn).
x
I.
If
such t h a t
> A]
I
s
when
Kjf(x)
g
converges
T h i s i s t h e s i t u a t i o n one e n c o u n t e r s
v e r y o f t e n i n F o u r i e r A n a l y s i s and i n A p p r o x i m a t i o n T h e o r y . The p r e s e n t p a p e r h a s two p a r t s .
P a r t I p r e s e n t s some r e s u l t s w h i c h
make e a s i e r t h e t a s k of f i n d i n g o u t a b o u t t h e weak t y p e of
K".
I n P a r t I1 we c o n s i d e r some a p p l i c a t i o n s of these t h e o r e m s t o Approximation Theory.
Most o f the r e s u l t s i n t h i s p a p e r c o n s t i t u t e a n a 
t u r a l e x t e n s i o n o f t h e o n e s c o n t a i n e d i n t h e work o f M.T.
[ 19791.
Carrillo
P r e v i o u s work r e l a t e d t o P a r t I was done b y Moon [ 19741.
M.T.
118
I.
CARRILLO and M .
de G U Z M m
MAXIMAL CONVOLUTION OPERATORS
The f i r s t t h e o r e m , K”
maximal o p e r a t o r
d e a l i n g w i t h t h e weak t y p e
(1,l) of t h e
p e r m i t s a d i s c r e t i z a t i o n and g e o m e t r i z a t i o n
It b e l o n g s
of t h e weak t y p e i n e q u a l i t y t h a t i s o f t e n q u i t e u s e f u l .
t o Guzmgn a n d i t s p r o o f c a n be s e e n i n Guzmen [ 1 9 8 1 ] .
Since t h e
of Theorem 2 i s p a t t e r n e d a f t e r t h a t of Theorem 1, we o m i t
proof
h e r e t h e p r o o f of Theorem 1, THEOREM 1.
Let
m
{k,] j = l
i s o f weak t y p e
Then
K*
type
(1,1)
i f a n only i f
(1,l)
o v e r f i n i t e sums of D i r a c d e l t a s .
>
and o n l y i f t h e r e e x i s t s
c
e a c h f i n i t e s e t of p o i n t s
in
Ifx
L1(Rn).
b e a s e q u e n c e of f u n c t i o n s i n
i s of weak
K*
I n o t h e r words,
such t h a t f o r each
0
R ~ , al,a2,.
., a H
,
),
>
if
and f o r
0
one h a s
H
E Rn
: sup
j
C
kj(xah)
h= 1
For t h e p r o o f of t h i s t h e o r e m we r e f e r t o Guzmen [ 1 9 8 1 , p . 7 5 ] . The i d e a of t h e p r o o f i s s i m i l a r t o t h a t of Theorem 2 b e l o w .
F o r t h e weak t y p e holds,
but
Let
I < p
t h a t f o r each al,a2,
1< p
and f o r e a c h f i n i t e s e t of p o i n t s i n
0
0
such
Rn,
one h a s
c 11 Then
K”
i s of weak t y p e
c a n b e of weak t y p e PROOF.
(p,p)
(p,p).
The c o n v e r s e i s f a l s e , i . e .
w i t h o u t [l] h o l d i n g .
I t i s e a s y t o show t h a t t h e i n e q u a l i t y [ l ] i m p l i e s t h a t ,
K*
119
MAXIMAL CONVOLUTION OPERATORS AND APPROXIMATIONS
for
ch
> 0,
h = 1,2
,...,H ,
one h a s
From [ Z ] we s h a l l now t r y t o deduce t h a t f o r e a c h f u n c t i o n H f of t h e form f = C c where ch > 0 and Ih i s a d y a d i c 1 Ih cube i n 87" so t h a t lIhl i 1, we have, i f we f i x N ,
x
From t h i s f a c t , by a s t a n d a r d argument, one o b t a i n s t h e weak t y p e
(P,P)
of
K*.
To o b t a i n [ 3 ] from [ 2 ] l i j
< N
uous,
I)kjgj)ll < q ,
such t h a t
f i x e d l a t e r on.
we s t a r t by c h o o s i n g
Since t h e
functions
N
p > 0
i f we s e l e c t a n a r b i t r a r y
such t h a t i f
Ixyl
s b ,
W e s h a l l f i x i n a moment
x,y
E
Rn,
p > 0
q
where gj
gj
E Co(lRn),
w i l l be conveniently a r e uniformly contin
b = 6(p) > 0
there exists
we have f o r e a c h
j,
1r j r N ,
c o n v e n i e n t l y and s h a l l t a k e a r e 
H
7. C h 'Ih w i t h ch > 0 , Ih h= 1 and lIhl i 1. with diameter smaller than 6 ,
p r e s e n t a t i o n of
Rn
Take any
f =
a,
0
< a < X.
W e can w r i t e ,
w
= J1
+
J2
d y a d i c cubes i n
choosing
ah
E Ih,
120
C A R R I L L O and M.
M.T.
Since
d e GUZMhJ
w e have
lIhl i 1,
On t h e o t h e r hand
= A1
+
A2
+
A
3
Now
and l i k e w i s e
1
A3 5
Nq
I(flll
On t h e o t h e r hand, we have
Hence, g i v e n small t h a t
s
E/Z.
Since
since
Igj(xy)gj(xah)l
e
A1
+
A3
E
> 0,
s e/2
i p
b(xy,xah) and s o
L
b ( p )
we can f i r s t choose and t h e n
p
for
a
y E Ih,
A2 i 1 p N )Iflll
and
8(p)
gj
with
q
so
so that
SO we g e t
and
a
a r e a r b i t r a r y we o b t a i n
[S]
.
I n o r d e r t o s e e t h a t t h e c o n v e r s e does n o t h o l d we c o n s i d e r t h e f o l l o w i n g example.
MAXIMAL CONVOLUTION OPERATORS AND APPROXIMATIONS
B = B1
Let j =
be the unit ball centered at
121
and let
0
B
j'
be the ball obtained from B by a homothecy of ratio 1 Let k  v X B j . Obviously, since K*f(x) s Mf(x),
2,3...
1 7 . where
M
balls,
is the HardyLittlewood maximal operator with respect to is of weak type
K*
o < ).
0, such that
We can assume
E c (0,m).
E.
If
B(xo,r)
and with radius
r,
we have
IE
n
~(x~,r)l = 1.
< c < 1.
There exists then
n
I
B(xo,ro)
> clB(xO,rO) I .
ro
let
we can find
no
such that if Imn
Obviously
such that
Let
E . J
(xo~o)/(xo+ro).
is the open
IB(X0”)l
and for each
j+l/~ + 1 2
Then we have
E = E(n) c R1,
730
0
(1,l).
be a density point of
interval centered at
Let us choose
E L1(Rn),
m.
Let us assume that
n E N
f
be the maximal operator
= sup lk, .+f(x)I J J
is of weak type
K“
K”
m > no
f
we have
and s o
@
m
m
Let
ES = { e .x : x E E‘] J
U E’.. We shall prove that j=n J
F =
and
0
IF1 >
%en 0
(xo+ro).
i F1E
“0
vals
(xo+ro).
Im
,
Let
Let us take A =
UP

p
so big that
I
m
u
Iml
I:
P+l Im. We can choose, from these inter
m=n, two disjoint sequences,
MAXIMAL CONVOLUTION OPERATORS AND APPROXIMATIONS
EJ1'J2,...,Jq3' so that
say for
Each
6
CK1,K2,".'Kt3
t
9
Im = ( u Ji) U ( (J Ki). m=n i=l i=l 0 Ji, we have
Ji
contains one of the sets
if necessary,
Ji
3
E'
j'
Therefore, for one of them,
Let us write, renaming,
and observe that
ET,
125
lEil
> clJi
.
Therefore
P u Ijl j=no
Now, if
x E E;,
j z no
,
then If_ E E'C E,
2
and so
'j
So we obtain
This inequality, by Theorem 1, proves that
K"
cannot be of the
This proves our theorem.
weak type (1,l).
For the ndimensional case we have the following analogous
result. THEOREM 7 . 0
Let
k E L1(Rn),
> 0 so that e
and
K*f(x)
j)O
and
= sup Ikj+f(x)l. j
unit sphere C
n > 1.
cj+l/cj
Let
+ 1.
(e j]
Let
be a sequence kj(x) =
tin k(x/cj)
Let us define the function
by setting, for
7
H
E C,
H ( y ) = ess sup rn Ik(r7)l rz 0
Assume that
K*
is of weak type
(1,l).
Then there exists a
on the
M.T. C A R R I M and M. de GUZMhi
126
constant
Where
>
C
Let
=
S
{? E C
: H(?)
> I ] .

in the ray of direction
Eo
x E Eo,
such that, if
Let us choose
then c,
yo
Fo
If
0 < c < 1.
yo
on the ray in direction
Yo
E S
there exists a
lEOll > 0
with linear measure
\xln lk(x)l
> X.
Proceeding as in the proof

of the previous theorem, for each xo,
we have
0
C.
measure on
set
>
such that for each
is the outer measure associated to the ordinary Lebesgue
0,
PROOF.
0
yo E S
we can determine a point
enO,
two numbers
and a set
ro
such that
Pol
2
+nO(xO+rO)
and 1x1 Let
Sp =
E
b0+ro),
{YO E
Ike
m
(x)I > 7 A if
x
E
F ~ ,m
z no
1x1
1 S : c ~ ~ ( x ~> + ~ 1 , ~ p) = l,Z,...
5
Then
m
U S and, since S is nondecreasing, P p=l P Therefore it will be sufficient to prove that S =
C
is a constant independent of F o r each
yo
E
sP
p.
we set
We easily find
Therefore, by Theorem 1 we obtain
[: *I
u (S) = lim ae(S ) .
P
oe(Sp) s
x, C
P
where
MAXIMAL CONVOLUTION OPERATORS AND APPROXIMATIONS
yo
On the other hand f o r each
E
S
127
we can determine e
P
PO
such
that 1
PO and so e po+l~xo+~o= )
where
C2 = inf j
c
(xo+ro) >
c
Po
c2
1
P
.
j+l
E j
By the definition of
CYO
we have
and then we get
L **I F r o m the inequalities
with
C
[*]
independent of
and
p.
[**]
we obtain
This proves the Theorem.
When the kernels are continuous there is no difficulty in substituting the sequence ess sup
by the
THEOREM 8.
Let
sup.
rej]
We shall first state such results.
n
k E L1(!R1) K*
f(x)
C(R1) =
R+
Assume that
K*
by a more general family and the
and for
sup Ikc*f(x)I WEER
is of weak type
(1,l).
Then
R+
THEOREM 9 .
Let
k E L1(Rn)
n
C(Rn)
f(x) =
= sup Irnk(r7)l r>O
sup
and for
0, such that for
Then there exists a constant
E
c
:
~ ( 7 >) X I )
C
6
We conclude the paper with a couple of easy consequences, (a)
With the former theorems one can build in a trivial way k
some interesting kernels operator
is not of weak type
K"
R+
L1(R1) ll c(W1)
any function in
=
type
n E
for
JTnT
&.
For example, let
(1,l).
such that
k(x)
The maximal operator
(1.1).
kernel
One can think of extending
k
(b)
[O,l)
one has
cannot be of weak
K"
K+;
k
radially to
cannot be of weak type
R2.
(1,l).
L1(R1)
such that ko j+f + f
with support on
1
k = 1,
[O,l)
then for each
The
Consider an open unbounded set
ing the origin
0.
Let
BG
is
f E L(l+log+L)(R1) o .h 0. Theorem J K" to a
e
(1,l).
G C R2,
IGI
= 1,
contain
the differentiation basis obtained by
taking as differentiating sets for by homothecy of center
k
and nondecreasing
a.e. for each lacunary sequence
lacunary sequence cannot be of weak type
0
all the sets obtained from
0, and, for each
obtained by a translation of the ones of a result of C.P. Calderon [1973], when respect to
The
L~(R~).
6 tells us that the corresponding maximal operator
G
0
x E R2, to
X.
all the sets According to
is starshaped with
0, has a finite number of peaks and satisfies a certain
entropy condition, then the basis ximal operator (191)).
k(n) =
= k(x),
There is a theorem of F. Z o El9761 stating that if
a function in
G
be
is radial, but it cannot be nonincreasing so that at the
same time is in
(c)
k
R+
corresponding operator
in
such that the corresponding maximal
K" R+
BG
corresponding to
differentiates k =
X,
L1
(the ma
is of weak type
MAXIMAL CONVOLUTION OPERATORS AND APPROXIMATIONS
With Theorem
G
set
(1,l). (Equivalently, i f t h e b a s i s
differentiate
> 0
we c a n e a s i l y d e d u c e c e r t a i n f e a t u r e s t h a t a
c a n n o t p r e s e n t i f t h e maximal o p e r a t o r
weak t y p e
u(P)
7
L1(Rn)).
I f for a set
07,
we h a v e t h a t t h e r a y
large distances, n o t of weak t y p e
129
mG
then
P
7
K"
h a s t o b e of
R+
pG
i s going t o
o f p o i n t s of
E P
hits
does n o t d i f f e r e n t i a t e
C
with
a t arbitrarily
G
L1
and
K"
R
is
(1,l).
REFERENCES P.A.
BOO [ 19781
,
N e c e s s a r y c o n d i t i o n s for t h e c o n v e r g e n c e a l m o s t everywhere of c o n v o l u t i o n s w i t h a p p r o x i m a t i o n i d e n t i t i e s o f d i l a t i o n t y p e , Univ.
CALDERdN [ 1979],
P.C.
CARRILLO [
19793,
48.
DE G U Z M h [ 19811,
( 1 9 7 3 ) , 117.
O p e r a d o r e s maximales d e c o n v o l u c i o n ( T e s i s
D o c t o r a l Univ. M.
Complutense d e M a d r i d ,
MOON [1974],
ZO
[1976],
( 1 9 7 4 ) , 148152.
A n o t e on t h e a p p r o x i m a t i o n o f t h e i d e n t i t y ,
Math.
U n i v e r s i d a d Complutense Madrid
4 6 , N o r t h H o l l a n d , 1981).
On r e s t r i c t e d weak t y p e (1,1), P r o c . Amer. Math. S O C . 42
F.
1979).
R e a l V a r i a b l e methods i n F o u r i e r A n a l y s i s
(Mathematics S t u d i e s v o l . K.H.
1978.
D i f f e r e n t i a t i o r i throiigh s t a r l i k e s e t s i n Rm,
S t u d i a Math. M.T.
of Ume;,
55 (1976), 111122.
Studia
This Page Intentionally Left Blank
Functional Analysir, Holomorphy and Approximation Theory, LA.Bmoso led.) 0NorthHolland hblishing Company, 1982
CONVOLUTION EQUATIONS IN INFINITE DIMENSIONS: BRIEF SURVEY, NEW RESULTS AND PROOFS
J.F. Colombeau
and
Mdrio C. Matos
ABSTRACT In the last fifteen years a large amount of results were obtained on convolution equations in normed and locally convex spaces.The aim of this work is to contribute to the improvement and clarification of the theory by presenting new results and connections between previously known theorems.
For convenience and necessity of presen
tation we recall most existing results and give their references, s o that this paper is also a brief survey on the subject.
1.
NOTATIONS AND TERMINOLOGY The notations and the terminology are as classical as possible.
For the general facts about locally convex spaces (C.C.S.) to KOthe [l]
,
Schaefer [l] and Trhves [l]
logical vector spaces (b.v.s.)
.
For the theory of borno
we refer to HogbeNlend El].
a few definitions and results of the theory of b.v.s. understand this paper.
we refer
In fact
are enough to
All the b.v.s. we consider are supposed to
be separated by their duals.
For the main definitions and results on holomorphic functions in infinite dimensional spaces we refer to Nachbin El],
If E
is a complex C.C.S. the space
#(E)
[2] and
[3].
of the (continuous) ho
132
J.F. COLOMBEAU and MARIO C. MATOS
lomorphic functions in logy.
E
If
is equipped with the compactopen topo
E
is a complex b.v.s.
holomorphic functions in
E
the space
ZS(E)
of the Silva
is equipped with the topology of the
uniform convergence on the strictly compact subsets of Colombeau [l], space
e(E)
ColombeauMatos [Z]).
If
E
of the Silva Cs functions in
E
(see
is a real b.v.s.
E
the
is equipped with the
topology of the uniform convergence of the functions and their de
E
rivatives of all orders on the strictly compact subsets o f (see AnsemilColombeau [ 13 and ColombeauGayPerrot [ 11 ) Let
E
be either a L.c.s. o r a b.v.s.
E
of functions on
and
G
.
a linear space
containing all the translations on
E
and
equipped with some structure of either a L.c.s. o r a b.v.s.. usual we define a convolution operator 0" G
G
from
As
as a linear mapping
into itself which commutes with all translations and which
is continuous if
G
is a L.c.s.,
bounded if
is a b.v.s..
G
We recall that in the finite dimensional case, the results considered here were obtained by Ehrempreis [ 11 , [ 21 and [ 31 , Malgrange [ 11 and Martineau [ 11.
2.
(a)
CONVOLUTION EQUATIONS IN SPACES OF POLYNOMIALS If
E
is a complex L.c.s. we denote by
P(%)
the vector
space of the continuous nhomogeneous polynomials on equipped with the equicontinuous bornology. space
p(E)
E,
We define the vector
of the continuous polynomials on
E
as being the
bornological direct sum
where
p(OE)
I
C.
The following result is proved in ColombeauPerrot [ 2 ] .
133
CONVOLUTION EQUATIONS I N INFINITE DIMENSIONS
THEOREM 2.a1.
If
is a complex nuclear &.c.s.,
E
zero convolution operator on
then every non
is surjective.
P(E)
The proof follows from a division result and from the study of the duality for (b)
P (E)
.
Here we obtain new results of existence and approximation for solutions of convolution equations on spaces of polynomials in
If E
a normed space.
(pe(”E),
II.II,)
is a complex normed space we denote by
either the Banach space
nuclear nhomogeneous polynomials on (Pc(%)
,)I ‘Ilc)
of the
o r the Banach space
E
of the continuous nhomogeneous polynomials of com
E.
pact type on
(pN(?E),//*llN)
See Gupta [l] and [2] for the definition and the
basic properties of these spaces.
We set m
and weequip it with the direct sum topology. The proof of the next two results are rather long and they are written in detail in section 7 . THEOREM 2.b1.
If
E
volution operator on THEOREM 2.b2.
P ~ ( E ) is surjective.
If E
volution operator on neous equation
is a complex normed space, any nonzero con
8
is a complex normed space and pe(E),
8u = 0
then any solution
u
is a con
of the homoge
is the limit of solutions of the same equa
tion which are continuous polynomials of finite type. (c)
If E
is a complex k.c.s.,
continuous seminorms in
E.
we set
If
CS(E)
a E CS(E)
the associated normed space and we set
as the set of all we denote by
E
a
J.F. COMMBEAU and MARIO C. MATOS
134
algebraically and topologically (in the sense of locally convex inductive limits).
Now, using the inductive limit technique of
ColombeauMatos [ 13 we obtain THEOREM 2.b1 and THEOREM 2.b2 for any &.c.s.
We refer to these results as Theorem 2.c1 and
E.
Theorem 2. c2. REMARK.
If E
= pC(E)
algebraically (this follows easily from the definitions).
is a complex nuclear &.c.s.,
then
P ( E ) = 6N ( E ) =
The convolution operators on any of these spaces are of the form 8 = T"
T
with
being an element of the topological dual of the A similar proof to that of Proposition
space under consideration.
6.7 in ColombeauMatos [l] characterizes the topological dual of which from a result from ColombeauPerrot [ 2 ]
pN(E),
as the topological dual of on
p(E)
and
PN(E)
Hence the convolution operators
p(E).
are the same.
Thus Theorem 2.c1 generalizes
the result of part (a) with a different proof. = PC(E)
if
E
is a complex Schwarz &.c.s.
fundamental system 0
in
3.
E
with each
(vi)iE I l?
Vi
is the same
Furthermore
p(E)
=
such that there is a
of absolutely convex neighborhoods of
having the approximation property.
CONVOLUTION EQUATIONS IN SPACES OF ENTIRE FUNCTIONS O F EXPONENTIAL TYPE
(a)
If E
is a complex &.c.s.
the space
functions of exponential type in vector space of all
a E CS(E)
f E #(E)
of the entire
is defined as the complex
such that there are
C > 0
and
satisfying lf(x)l
for all
E
Exp(E)
x
E E.
This space
5
Fxp(E)
ce'(x) is equipped with the natural
bornological topology coming up from this definition.
It is proved
135
CONVOLUTfON EQUATIONS IN INFINITE DIMENSIONS
in ColombeauGayPerrot [l] with two different proofs, the following result THEOREM 3.a1.
If
E
is complex nuclear
convolution operator on
Exp(E)
L.c.s., then any nonzero
is surjective.
The classical result of approximation of the solutions for homogeneous equations is also proved in ColombeauPerrot [3].
This
result is refered here as THEOREM 3.a2. In order to get explicit solutions, the following result is proved in ColombeauGayPerrot [l].
If E
THEOREM 3.a3.
is a complex nuclear L . c . s .
convolution operator on
Exp(E),
U
element
T
of
of e'(E') [3#' (E' ) ]'
then every solution in
with and
pU = 0.
pU = 0
Here
EXp(E)
is called the
We note that the in
is that the support of
in the closure of the set of zeroes of
for an
8 = T"
p = t3(T) 6 blS(E' )
characteristic function of the operator 8 . tuitive meaning of
is a
8
6f = 0 is the Fourier transform of
of the homogeneous equation some element
and
U
is contained
p.
The proof of the above theorem uses as tools the division of distributions by holomorphic functions (see ColombeauGayPerrot 113) and the resolution of
C 41
and

a
in
DFN spaces (see ColombeauPerrot
C 5J 1.
The existence and approximation theorems are true for convolution operators in 38' (E')
for a complex Schwarz 4, .c.s.
fundamental system of neighborhoods
of
0
in
E
which
A
is a Banach space with the EVi i E I. See ColombeauPerrot [ 3 ] .
are absolutely convex and such that approximation property, for all
(Vi)icI
with a
E
We refer to these results as Theorem 3.a4 and Theorem 3.a5. (b)
If
E
is a complex normed space we define an entire function
of nuclear exponential type in
E
as being an element
f
of
136
J.F. COLOMBEAU and MARIO C. MATOS
Z"f(0) E p,(%)
such that
B(E)
The complex vector space
for all
ExpN(E)
equipped with a natural structure of a space.
n E IN
and
of all these functions is
L.c.s. which makes it a DF
See section 8 of this paper f o r the details and proofs of
the following new results. THEOREM 3.b1.
E
If
is a complex normed space, then every non
zero convolution operator on THEOREM 3.b2. operator on tion
ExpN(E),
Bf = 0
generated by
If E
E
If
EX~,(E)
is surjective.
is a normed space and
is a convolution
@
then every solution of the homogeneous equa
is in the closure of the vector subspace of { P eCP;P E P N ( % ) ,
ep E E',
B(Peq)
=
ExpN(E)
01.
is a complex .L.c.s. we define Exp (E) = ind lim Exp(E N
a€cs (E
a
)
alge raical y and topologically (in the sense of the locally convex inductive limit).
Now using the inductive limit technique of
ColombeauMatos [l] we obtain the above theorems for any complex 4.c.s..
We refer to these results as THEOREM 3.b3 and THEOREM
3 .b4. It is not too difficult to show that
for a complex Schwarz &.c.s.
for which there is a fundamental system
of absolutely convex neighborhoods of
0
such that
(viIiEI is a Banach space with the approximation property for every
A
E
i'
i E I.
Hence Theorem 3.b3 and Theorem 3.b4 generalize Theorem 3.a4 and Theorem 3.a5 with a proof which seems less complicated.
137
CONVOLUTION EQUATIONS IN INFINITE DIMENSIONS
(c)
If
E
is a real nuclear b.v.s.
3e'(E),
the space
image
e'(E),
through the Fourier transform of the space
is des
cribed in AnsemilColombeau [l] as a space of entire functions on E' + iE'
that, besides the usual inequality, satisfy a technical
condition (which comes from the infinite dimensional case).
Applying
the result of division of distributions by holomorphic functions of ColombeauGayPerrot [l],
with a proof similar to that o f Theorem
3.a1 (in this last paper) one obtains: THEOREM 3.c1.
Let
E
be a complexnuclear b.v.s.,
3e'(E)
zero convolution operator on
which has a Silva holomorphic
characteristic function is surjective. space, E (E)
If
E
then every non
(Clearly if
is a complex
E
is defined by using the real underlying space).
is a real nuclear b.v.s.,
Chansolme [l]
obtains a result
of division of distributions by continuous real polynomials of finite type, more generally by continuous real analytic functions of finite type (defined below) from which Theorem 3.c2 below follows in a staiidard way, and which is a consequence of the finite dimensional results.
f: fl
We say that a map
+ R n
finite type analytic mapping in bounded subset
B
E,
of
or
e
(n
a
open set) is a
TE
if for every convex balanced
the restriction
analytic mapping of finite type, i.e. are an
a:
flnnEB
xo
for every
is locally an
E
EB
> 0, a decomposition of the normed space
logical direct sum
EB = E;
and an analytic map 1 in EB such that
fl
THEOREM 3.c2.
E
Let
convolution operator on
Q
2
EB
(x = x1+x2)
in a neighborhood of
be a real nuclear b.v.s.
ze'(E)
EB
there
in a topo1
dim EB
0 ,
we get A
(P,T*s)
=
(P,?^S>
f o r all
P E P~(E).
A , .
Hence
T*S = T . S . Q.E.D.
PROPOSITION 7.8.
If
P € Pf(E)
and
ever
the quotient in PROOF.
are such that
S,T E p e ( E ) '
S e t (E'
;
then
T+P = 0 ,
S(P) = 0
is divisible by
$
whenwith
1.
B y a result of Dineen [ 2 ]

it is enough to prove that the
m
quotient is in
=
n P(%'). n=O Let ko be the first nonnegative integer such that ('!I?)f 0. k0 Hence ( T ) j = 0 if j = 0 , kol. It follows that T(cpj) = 0 s(E')
...,
for
j < ko.
S(pJ) = 0
Hence
for
T*pJ = 0
j < ko and
( 2 ) .J
for
j
< ko.
= 0 for
j
B y our hypothesis
< ko. Now we want to show
that ( 2 ) is divisible by ('?) with the result in b ( O E ' ) . By a k0 k0 result o f Gupta [ 2 ] it is enough to prove the result on every one dimensional affine subspace where (i)ko f 0. We consider the com
COLOMBEAU and MARIO C. MATOS
J.F.
156
plex functions k g ( t ) = sC(rp+t$) O1 Y
k
f ( t ) = T[(cp+tl)) If
to
i s a z e r o of
0
k
i.
L
S
order
of
i
t E a:
O1
f ,
we h a v e :
f(k)(t,)
= 0
for
Thus
ko(ko1).
..
= 0
(kok+l)T[
for
0 b
k 5 1.
It f o l l o w s t h a t k k
Qk]
T+[(q+t,l))
fo r
= 0
0 b
k 6 i.
Hence kok k S[(cp+to$) ] = 0
for
0 i k s i.
lows t h a t
g
t
Thus
for
i s d i v i s i b l e by
We w r i t e
Ro =
i i
i s a zero of
o f Gupta i m p l y t h a t t h e r e i s
= (')ko'
< k
0
Ro E p ( O E ' )
.
z i
order
such t h a t
E
p(jE')
j = 0,1,..., n
T h i s means t h a t q u o t i e n t of
(2)ko+l
(?)ko+l

such t h a t
= ($)ko+lRo
( G) k o + l R o
g.
It f o l 
('?')ko*Ro
=
Now we s u p p o s e t h a t we h a v e found
(T)ko
R J.
of
As we n o t e d b e f o r e , t h e r e s u l t s
f.
(S)'O
g(k)(to) = 0
and
(')ko+n *
+ ('?)k R1.
n , .
=
jEo
Thus
(T)ko+nj'Rj R1
0
by
('?')ko
and we w r i t e
i s the
*
CONVOLUTION EQUATIONS I N INFINITE DIMENSIONS
157
By repeating this process each time, we get in general:
... Hence each
,
(?)ko+i
R
j = O,l,...,n
j'
0i i
j,
5
and
( 5 )k
can be written in terms of
Ro.
(?)ko+i , ('?)k 0+i , 0 S i s n this polynomial written in this form.
may be written in terms of Ro.
is a zero of
f,
Ittol < r ,
for sider
Pt € bf(E') Q
expression o f
j = O,l,...,n=l,
and
Q
We denote by
Ro
by
1
there is
t
#
to.
r
t f to,
Ittol < r ,
cpoko
,
where
+
1
If
and to
(')ko+j (')ko+
(')ko+j
9
j
cpo E E'
we con
&.
defined in the following way from
we replace
'
> 0 such that
For each
in terms of
+i
As a consequence
and
In the
Ro'
by
is such that
(G)k
Now we consider the following complex valued polynomial.
(ep,) 0
= 1.
J.F. COLOMBEAU and MARIO C. MATOS
158
+
h: {t E C ; Ittol < r]
C
+
t
h(t)
S(Pt)
if
t f to
Q(p+toJI)
if
t = to
=
It is somewhat tiring to check that
T + P ~= Hence
S(P
of
at
h
t
o
for
) = h(t) = 0 Y t f to, to,
Ittol < r.
tfto,
it follows that
Ittol < r.
h(to)
Thus, by continuity
= Q(cp+to$) = 0 .
Hence
Thus we proved that f(to)
t E {t E c ;
where, for
= 0
3
fn+l(to)
= 0
Ittol < r],
I n a similar way we prove that
= 0 whenever
f i : ! ( t o )
Hence
fn+l(t) is a 7
By Gupta's result
(6)ko and
= 0.
polynomial.

(')ko+n+l
the result
= R E $(E')
f(i)(to)
such that
n . . jEo (T)ko+n+lj
Rn+l E p(n+%').
. T'R
=
R J.
is divisible by
Thus we found
= (Rn)m n=0
2. Q.E.D.
8 E Go.
THEOREM 7 . 9 .
Let
generated by
{P; P
of
pg(E)
E pf(E)
Then the \rector subspace and
8 P = O}
in the closed vector subspace
8
of
Pg(E)
is dense for the topology S"({O}).
CONVOLUTION E Q U A T I O N S I N I N F I N I T E DIMENSIONS
pf(E)
Since
PROOF.
sult f o l l o w s trivially. such t h a t
T+ = (9.
Hence,
P E pf(E)
if
Let
X E Pe (E)' T+P = 0, 0
. X
R = S,
such t h a t Now,
w e have
P E S  l ( ( 01 )
if
f
= 0.
X
1
S = 0.
w e have
=
0.
By

X(P)
T+P = 0
s i n c e every t i m e
e
X = T+S
Thus
= S+T.
(X+P) (0) =
and
Hence
X(P) =
T h e o r e m the result f o l l o w s ,
i t vanishes i n
8 ,
vanishes i n
X
7.8
S i n c e t h e r e i s SEP ( E ' )
= T * S = T+S.
B y the HahnBanach
T € pe(E)'
be s u c h t h a t
A , .
w e have
then t h e r e 
There i s
= [ ( S + T ) + P ] ( 0 ) = [ S+(T+P)] ( 0 ) = [ S + O ] ( 0 ) = 0. = (X+P)(O)
= 0
(9
0.
X = T.R.
such t h a t
0.
.
if
8
Now w e a s s u m e
and
R E S e l (E')
there i s
Pe(E),
i s dense i n
159
@l({O}). Q.E.D.
THEOREM 7.10.
p i n g of
t o p o l o g y of If
=
(9,
,
(9
[Pe(E)']
t(9
Pe(E)' T+
E Ge
(9
then
@,
PROOF.
Let
#
If
0.
i s the t r a n s p o s e m a p 
'(9
= [Sl((O])]L
d e f i n e d by
Pe(E).
T E Pe(E)'
and
X =
i s closed f o r the w e a k
t
f o r s o m e R E P (E)',
Q(R)
e
w e have
= R((9P) = 0
X(P)
X E [(91({0])]*,
O n t h e o t h e r hand i f
E
A s i n the proof
(9'(0).
X = S + T = T+S. X(Q)
Hence
X =
t
Y P E Sl(O).
w e have
of 7 . 9 w e f i n d
S
E
X(P)
= 0
Pe(E)'
Q(S)
and
= [ (S*T)+Q]
X E
t
( 0 ) = [S+(T+Q)] ( 0 )
=
Q[Pe(E)'].
We proved t h a t t(9[Pe(E)']
=
IT
= (s'l(E 01 ) 1
(T E Pe(E)';
hence i t is c l o s e d f o r the w e a k t o p o l o g y .
P E
such t h a t
Hence
= (S*T)(Q)
'#
T(P) = O ]
160
COLOMBEAU and M h I 0 C.
J.F.
THEOFZEM
7.11.
If
PROOF.
Since
p8(E)
8 E Ge
f
8
and
pe(E)'
T* = 8 .
t
P E Pe(E).
for all 0
t@(R)
= [R*(TXP)!
(R*T)(P)
.. R =
d e f i n e d by
If
8
i s surjective.
= 0,
Hence
pe(E).
is
i s closed f o r the w e a k
6[Pe(E)']
T E be(E)'
Let
be s u c h
w e have
= R(8P)
= R[T*P]
(0)
'6
W e o n l y have t o p r o v e t h a t
i n j e c t i v e , s i n c e 7.10 says t h a t
that
then
0,
i s a DF s p a c e , w e m a y a p p l y t h e k n o w n t h e o r e m
due t o D i e u d o n n d and S c h w a r t z .
topology of
MATOS
*. R T = R*T
= 0.
= t8(R)(P)
Since
? f
0
= 0 w e get
R = 0.
and
Q.E.D.
a.
SPACES
OF ENTIRE
OF BOUNDED TYPE AND OF ENTIRE
FUNCTIONS
F U N C T I O N S O F NUCLEAR E X P O N E N T I A L T Y P E I N NORMED S P A C E S
( a ) FIRST Let
AND RESULTS
DEFINITIONS
be a c o m p l e x n o r m e d s p a c e .
E
D E F I N I T I O N 8.a1.
v e c t o r s p a c e of a l l n E IN
p > 0,
If
f
w e d e n o t e by
a(E)
such t h a t
B N , p (E)
dnf(0)
the c o m p l e x
for a l l
E P,(%)
and
I t i s easy t o prove t h a t (1) d e f i n e s a n o r m i n
P R O P O S I T I O N 8.a2.
F o r each
p
> 0,
the space
(BN,p (E)
911
'llN,p)
i s a Banach s p a c e . PROOF.
Let
For every
(fn)To,=l 0
> 0
be a C a u c h y sequence i n
there i s
n > 0 e
sueh that
(BN,p(E),II
dIN,p)'
CONVOLUTION EQUATIONS IN INFINITE DIMENSIONS
161
.
n z n It follows that (Gjfn(O))” is a c n= 1 Cauchy sequence in the Banach space pN(JE), hence it converges to for all
m z n
Pj E PN(’E).
an element
for all
and
E
m
2
.
n
If we show that
defines an element of
f E WN,(](E).
and
in
f
This fact and ( 2 ) imply
#(E),
it follows that
Then ( 3 ) implies the convergence of
(BN,p(E),II
In order to show that
(fn)L=l
f E #(E)
to
we note
Hence
and
Hence
since its radius of convergence is
f E #(E)
DEFINITION 8.a3.
u
=
03N,p(E)
We consider the complex vector space
the normed topologies of ExpN(E)
type.
ExpN(E)
=
with the locally convex inductive limit topology of
P>O
of
+m.
b)N,p(E)
for
p E (O,+m).
The elements
are called entire functions of nuclear exponential
J.F. COLOMBEAU and MARIO C. MATOS
162
It is natural to call the elements of
REMARK 8.a4.
ExpN(E)
entire functions of nuclear exponential type because it is quite f 6 #(E)
easy to show that Znf(0) E p N ( % )
is in
n 6
for all
ExpN(E)
if, and only if,
and
(N
1 
lim sup IlZjf(o)llj < jm N PROPOSITION 8.a5. PROOF.
The space
+co
.
is a DF space.
ExpN(E)
This is an immediate consequence of the fact that
ExpN(E
is the inductive limit of the sequence of Banach spaces
DEFINITION 8.a6.
entire functions on
E.
of
On
blb(E)
We denote by
#b(E)
E
the vector space of all
which are bounded on the bounded subsets
we consider the locally convex topology defined
1) *I) p ,
by the seminorms
p
> 0, where
(4) The elements of
PROPOSITION 8.a7. PROOF.
are called entire functions of bounded type.
ab(E)
The space
pleteness of this space. elements of g
,P
is Frbchet.
Mb(E)
It is clear that the topology of
the sequence of seminorms
n
ab(E)
ab(E).
(11 * l / ~ ) ~ = ~ . a
Let
n (fn):=l
Hence for each
may be defined by
Hence we must show the combe a Cauchy sequence of
p
> 0 and
0
> 0 there is
> 0 such that (0
c
(5) for all
j=O
n z n
0
tP
and
I1
2Jfm(0) 
2jfn(0)
II
j!
m > nc
S P
.
This
is a Cauchy sequence in the Ranach space
e
implies that (
P(jE),
;Jfn(0) j!
)Lo
thus it converges
163
CONVOLUTION EQUATIONS I N INFINITE DIMENSIONS
t o an element
E p(JE).
Pj
m
m z m E YP
for a l l
.
1
From
*
( 5 ) imply
T h i s f a c t and
E
( 6 ) we s 0 e t h a t , f o r a l l
+
Ifn
Ilp
0,
'
+
,P
Hence m
d e f i n e s an element converges t o
f
f(x) =
C Pj(x) j=O
f
Xb(E).
of
of
eep,
0
Since t h e Taylor s e r i e s a t
verges t o
f
ExpN(E)
n =1
bib(E).
The v e c t o r s u b s p a c e o f
a l l e n t i r e f u n c t i o n s of t h e f o r m
i n t h e t o p o l o g y of induces on each
E E)
( 6 ) i t follows t h a t (fn)
From
i n t h e t o p o l o g y of
PROPOSITION 8 . a  5 .
PROOF.
(X
ExpN(E)
ep E E ' ,
i s dense i n ExpN(E).
of e a c h
f
ExpN(E)
f"(?E),
E
n
IN,
c l o s e d v e c t o r s u b s p a c e of
g e n e r a t e d by
n E N.
sition
ExpN(E)
con
t h e t o p o l o g y of' t h e
PN(?E)
alL
E
and s i n c e t h e t o p o l o g y
n u c l e a r norm, w e h a v e t o show t h a t ExpN(E)
g e n e r a t e d by
i s c o n t a i n e d i n the
T h i s i s done f o l l o w i n g t h e p r o o f
[e@;
Q
of G u p t a
E
[a],
E']
for
Propo
3, page 4 5 .
REMARK 8 . a  9 .
I t i s e a s y t o show t h a t
f
E H(E)
i s in
Xh(E)
if,
( b ) THE FOURIER BOREL TRANSFORMATION DEFINITION 8.b1. 3T
is d e f i n e d by
If
T E [ExpN(E)]',
ZT(ep) = T ( e 9 )
i t s FourierBore1 transform
for a l l
rp E E ' .
164
COLOMBEAU and MARIO C. MATOS
J.F.
THEOREM 8.b2.
The FourierBore1 transformation is a vector space
isomorphism between If
PROOF.
IT(f)l
T
C(p) > 0
is
E [ExpN(E)]‘, for all
p(f)
S
Hence, if
P
[ExpN(E)]’
E PN(jE),
T . = TI“N(JE).
J
@ T jE P ( j E ’ ) ,
that
for all
such that
p > 0, there
Hence, for all
I T ( f ) l S C(P)l(fl/N,P
for all
f
E ExpN(E).
we have
‘
%l/p/lN
P
By a result of Gupta [2],
defined by
/)BTjl( = l / T j / l .
p E CS(ExpN(E))
there is
I)’(TI
We set
gb(E‘).
f 6 ExpN(E).
such that
(7)
and
there is
for all
@Tj(ep) = Tj(epj)
cp
E E’,
such
By ( 7 ) it follows that
p > 0. Hence we may write
(9)
.
for all
cp E E‘
for all
p > 0.
3T
By (8) we have
Hence
lim sup jtm
11 1 J!
I/ j BT.11 = 0 J
and by 8.a9,
E ab(E‘). I t is clear that
3
By 8.a8, it follows that that
3
is linear from
3
is injective.
is a surjective mapping.
Hence, by 8.a9,
lim sup
1) ~
ExpN(E)
Let
H
into
ab(E‘).
Now we should prove
be an element of ab(E’).
lj!l l ’ j = 0. Thus, for all
p > 0,
j+m
there is
for all
C(p) > 0
j E N.
such that
By Lemma 4, page 59, of Gupta [ 23
,
there is
165
CONVOLUTION EQUATIONS IN INFINITE DIMENSIONS
B T .= ijH(0)
Tj E [PN(JE)]’
such that
Now, for each
f E ExpN(E),
jlTjll = IlA’H(0)ll.
and
J
we define m
TH(f) =
1
C
j=O
T.(T J
J.
Zjf(0)).
We have m
c
ITH(f)l s
(11)
From (10) and the fact that
j=O
TH E [ExpN(E)]’.
ijf (0)
1 1 ~ l 1 ,
IITj(l= IIajH(O)ll,
I
ITH(f Hence
IITjll
‘ ‘(p
)IIfl/N,p
*
(11) gives us
*
It is easy to prove that
(c) CONVOLUTION PRODUCTS IN
3’TH= H.
EX~~(E)
We need the following result in order to define convolution operat or. PROPOSITION 8.c1. (i)
Anf(.)a
Let
a E E
E ExpN(E) =
& i=o C
in the sense of the topology of (ii)
T,f
E ExpN(E)
in the sense of PROOF.
for all
f E ExpN(E).
Then
and m
inf(.)a
and
A.
dl+nf(0).l(a)
ExpN(E).
and
ExpN(E).
(i) We know that
x E E,
the series being convergent in
P(IE).
Hence
166
J.F.
m
iif(x)a
=
C n=0 m
= x E E.
$ dl+nf(0)xn(a)
$ dl+nf(0)al(x)
11 f/l
f E ExpN(E).
N,p
0
N o w we choose
to an
PN(%)
such that
l/fllN,p
< +.
Thus (13) implies lim sup I I P n)m n N and
(T*f)(x)
plies
=
m
1 C n! Pn(x)
n=O IIT+fl/N,p h C(p)
i
p
0
i s bounded, t h e r e e x i s t s C
> 0
such t h a t qCdf(x for a l l
f
E 3
and
hl
+ elhl)hl]
E K1.
It f o l l o w s t h e n f r o m t h e Mean V a l u e
5.1 t h a t
Theorem
q[f(x+clhl)
for a l l Using
C
f
(I),
E 3
and
hl
E
K1,

f(x)] i f l C i
provided
el
1
2
i s s u f f i c i e n t l y small.
( 2 ) follows.
(en)
An i n d u c t i o n p r o c e d u r e y i e l d s a s e q u e n c e
n@
of p o s i 
t i v e numbers. s u c h t h a t x
(3)
+ c lICl
+. . .+
enKn C 0
and
for a l l
f
E 3,
hl
E
K1,.
..,h,
E
Kn
and
n
E
[N.
Then t h e s e t
COJAMBEAU and J . MUJICA
J.F.
190
U =
is a 0neighborhood in
gnKn =
E
such that
nEN
qof(x+h) for all
f E 5
n C
u nCW
C
EjKj j=1
x+Uc 0
and
M + 1
S'
h F U.
and
This shows that the family
is amply bounded.
3
{dmf : f 6 5 1 ,
argument, applied to the set is amply bounded, for every
m E N.
The same
shows that this set
The proof is now complete.
T o prove Theorem 5.2 we will also need the following lemma, which is essentially due to Grothendieck; see [ll, p.168, Th.2, Cor. 11 and [ll, p.167, Prop.2, C o r .
5.4.
(a) If
E
11.
is an infrabarrelled (DF)space and
locally convex space, then each bounded subset o f
is any
F
is
X(%;F)
equicontinuous. (b) If
E
is a (DF)space and
F
is a metrizable locally convex
space, then each equicontinuous subset of
X(?E;F)
is locally
bounded. PROOF OF THEOREM 5.2.
ub From the hypotheses on E
certainly continuous.
e (E;F)
clear that the space
(E;F) C+E(E;F) is
The inclusion mapping E
and
F
it is
is metrizable, in particular borno
logical, and therefore, to complete the proof it is sufficient to show that each bounded subset of bounded in Let family
E ub(E ; F )
3
e(E;F)
.
be a bounded subset of
(dmf : f E
a]
Urn in
E (E;F).
B y Lemma 5.3 the
is amply bounded, f o r every
application of Lemma 3.1 to the family 0neighborhood
is contained and
E,
(dmf : f 6
a bounded subset
Bm
m E N*.
51 of
An
yields a
X(%;F)
and
191
MAPPINGS OF UNIFORM BOUNDED TYPE
a sequence
( r mn ~ lnCN
of positive numbers such that drnf(nUm) c
f E 3
for all
and
n E IN.
umn
Em
I n view of Lemma 5 . 4 we may assume
without loss of generality that for each
m
N*
the set
Em
is
of the form
Em = with
BmC F
E
Since
E S(%;F)
: A(U,
Urn)C Bm]
X...x
bounded (with the obvious interpretation for
in
E
which is absorbed by each
Urn. And since
metrizable we can find a convex, balanced, bounded subset which absorbs every sequence
f E 3, x
E U,
m
F
of
f 6 3,
and
m E N* m E N*
and
E
x . . .x U ) and
N*
C
n
,X N
B (with the obvious in
Then an application of the Mean Value
m = 0).
Theorem 5.1 to the mapping
f E 3
is
It follows that there exists a double
Bm.
dmf(nx)(U
terpretation for
all
B
F
of positive numbers such that
(1) for all
m = 0).
is a (DF)space we can find a convex, balanced Oneigh
U
borhood
{A
d"f
shows that
x,h E E
with
dmf(x+h)
pU(h)
= dmf(x)
for
= 0. I f for each
we define fm: EU + .C(%,;FB)
then
fm
for all that
is welldefined and (1) implies that
f E 3,
x
fo E e(EU;FB)
m 6 N*.
E U,
m E N*
and that
arid
n
E
dmfo = fm
N.
I t follows at once
for all
f E 3
Furthermore, it follows from ( 2 ) that the set
and
{fo : f
E 31
192
J.F. COLOMBEAU and J. MUJICA
Eb(EU;FB).
is contained and bounded in
Eub(E;F)
contained and bounded in
5.5 REMARK.
If F
We conclude that
3
is
and the proof is complete.
E
is not metrizable or if
is not a (DFM)
space then the conclusion in Theorem 5.2 need no longer be true, as the counterexamples in Remark 3 . 2 show.
6. CONVOLUTION EQUATIONS Let
E
be a complex locally convex space and let
E
space of entire functions on This means that (Taf)(x)
Taf E 8
= f(xa).
linear mapping
be a
which is translation invariant.
for every
f E 8
a 6 E,
and
A convolution operator on
where
is a continuous
g
which conimutes with each
B: 8 + 8
8
We recall
Ta.
.
the following classical result (see €or instance Malgrange 1151 )
6.1.
Each nonzero convolution operator Bf = g
jective, i.e. the equation each
g
#(Cn;C)
on
B
has a solution
is sur
f E #(Cn;C)
for
E #(Cn;C). It is yet unknown whether the conclusion in Theorem 6.1
holds when
Cn
normed space
is replaced by an infinite dimensional complex
In order to obtain a version of Theorem 6.1 for
E.
normed spaces, Gupta [12] introduced, for spaces, the space
zNb(E;F)
bounded
E
from
into
We first define the space pings f r o m
Em
into
F
E
and
where
qPjkE E',
F,
which may be described as follows.
XN(%;F)
of all nuclear mlinear map
as the space of all
..,xm) =
b. E F J
complex normed
of all entire mappings of nuclear
A €
can be represented by a series A(xl,.
F
and
2 r p . (x1). J1
. .rp.Jm (x,)
e(%;F)
which
193
MAPPINGS OF UNIFORM BOUNDED TYPE
c. llw
jlll
*
J
Z,(%;F)
The v e c t o r s p a c e
.llep jmlIIIb jll
if
is a EL%pEV By t h e P a l e y  W i e n e r  S c h w a r t z t h e o r e m i n
f o r some
E'(F).
bounded s u b s e t of
€!
1
I;
aQ
E 3F(€!'(F)),
(6 ) = L ( e i g ) = Q , ( g ) ,
t : €! ( F ) e(F),
cp
x e (G)
+ Q:
by
and i t i s immediate
2 08
J.F.
E V
For
( g i v e n by
s u b s e t of
E'(F),
dimension,
therefore
4,
from
e(F)
i s an equicontinuous
{4,cp]
f r o m t h e PaleyWienerSchwartz t h e r e i s a 0neighbourhood
theorem i n f i n i t e W
4,
Gn
eb(G)
172,
GT e b ( G )
E(F)
C.
into
f o l l o w s f r o m T r 6 v e s [ 11 p.
( 3 ) ) then
PAQUES
E(F)
in
,
Since
S c h a e f e r [ 11
theorem 50.1;
= E(F)
6,
i s a nuclear space, i t
e(F)
Eb(G).
(Where
t h e t o p o l o g i e s on t h e t o p o l o g i c a l p r o d u c t ) . t o p o l o g i c a l s u b s p a c e of
e (F) E (F)
E(F)
=
e (F)
e ( F ) @€ E b ( G )
)
.
E
ek(E)
$ = 54,.
3 . 3 REMARK.
The p r o p o s i t i o n 3.2 r e m a i n s o b v i o u s l y t r u e i f
e(E).
4.
A DIVISION RESULT
(hifO, degree
n
For e a c h f i n i t e f a m i l y
on
F&,
Phl
,...,h n ; Y
= hl
i m a g i n a r y e xponen t i a 1po lynomia 1
a f i n i t e sum of
4 . 2 LEMMA.
5 E
ed nonzero is i n
and
E
hl,.
..,hn,
is
Eb(E)
y E F,
hneiY.
@...@
on
F&
of
By d e f i n i t i o n ,
for s h o r t ) i s
(i.e.p.
t e r m s a s above.
Let
for e v e r y
%
4,
w e c o n s i d e r t h e "imaginaryexponentialpolynomial"
i)
Y
Therefore
s e p a r a t e d by i t s d u a l and i f we r e p l a c e
by
4 . 1 DEFINITION.
is a
Theref o r e
=
any S c h w a r t z b . v . s .
denote
By ColombeauMeise [ 11, E ( F ) e E b ( G ) =
E eb(G).
eb(E).
e
and
TT
h a s t h e a p p r o x i m a t i o n p r o p e r t y ( S c h w a r t z [ 11
6, E b ( G )
Corollary 2,
and i s d e n s e i n i t , s i n c e
Eb(G)
Q
,
= E(F,eb(G))
an
such
E W. This implies t h a t i s bounded on W x V , that is cp i s continuous. S o 4, comes from a c o n t i n u o u s l i n e a r mapping 4,
that
4,
COLOMBEAU and O.W.
@
Cn,
i.e.p.
5e'(lRn),
be i n
5e'(Wn)
f o r some on
Cn.
i.e.,
m,
v
l@(g))
(i.e. and
b),
%
L e t us assume there are
I$(~)I < m' ( 1 + / l c l / ) u '

(l+/lg/lV.ebtllmSII
and l e t be i n
v'
and
eb'IIIm'Il,
Y
m',
i
b',
1E
P
be a f i x 
X(Cn), with Cn,
then
FINITEDIFFERENCE
and
v‘
m‘,
and
b‘
209
PARTIAL DIFFERENTIAL EQUATIONS
do n o t depend on
m, v
b u t o n l y on
@,
and
b. T h i s r e s u l t may be c o n s i d e r e d a s a known r e s u l t ; t h e f a c t t h a t
E 3e‘(Rn)
comes from lemma 1, p .
E h r e n p r e i s [ l ] and t h e p r o o f
m, v
o n l y on pg.
and
3eL(E),
if
mapping f r o m
($ * c p ) PROOF.
F&
Let
E, F
eL(G)
into
cp E e b ( G )
acp(6) = @ ( C )  c p ,
G
and
i s a nonzero
P
v’
m’,
and
b‘
depend
4,
13.
i s a n a l y t i c , then If
gives a l s o t h a t
288 i n
T h i s l a s t uniformness r e s u l t i s i n t h .
of E h r e n p r e i s [
290,
4 . 3 PROPOSITION. to
b.
287 and lemma 2 , p .
%
be a s i n $ 3 .
i.e.p.
on
F&
If
,
belongs
@
and i f
$
is a
s u c h t h a t f o r e v e r y cp E e b ( G ) , is in
is fixed,
3ek(E). @cp
if
E
X(F&)
t h e n by h y p o t h e s i s t h e r e a r e
i s d e f i n e d by
“cp’
and
vcp
b
c p ’
such t h a t
o n l y on
“cp9
and
bcp
cp’
v‘ cp
and
bk.
which depend
such t h a t
%E
T h e r e f o r e from p r o p o s i t i o n 3 . 2 ,
5.
m’
there a r e
By t h e p r e v i o u s lemma,
3ek(E).
EXISTENCE AND APPROXIMATION O F SOLUTIONS I N
eb(E)
To e a c h f . d . p . d . o . ,
D =
c
finite
c.D
h;,
(1)
...,h nj , ; y j J
(where x
E
E),
Dhl,.
.., h n i y ( c p ) ( ~ =) c p ( n ) ( x + y ) h l . . . h n ,
we a s s o c i a t e the i . e . p . ,
for
cp
E eb(E)
and
J.F.
2 10
COLOMBEAU and O . W .
. n. c .(i) h i
C
pD =
PAQUES
hi, J
@...@
finite Let
E
E = F @ G.
grange [
11 ,
be t h e v e c t o r s u b s p a c e of E which o c c u r i n ( I ) }C F.
{h;,yj,
of f i n i t e d i m e n s i o n s u c h t h a t Let
F
b e a r e a l Banach s p a c e and
. ei y
The p r o o f s f o l l o w a c l a s s i c a l t e c h n i q u e ,
f o r instance.
5 . 1 PROPOSITION ( A p p r o x i m a t i o n of S o l u t i o n s ) . a s above.
u
Then e a c h e l e m e n t
limit in
eb(E)
of
eb(E)
in
s o l u t i o n s of
C
ci(pi)
C
the form,
ci E
I,
p i E Ek,
6,
a . E Fh
D f 0.
t o consider the case
n u l l on
K e r D.
a fixed
cp
eb(G),
in
the e n t i r e functions
If
),
E C
on
i s a z e r o of
D(q
J
JI E
Then f o r a l l
D(Tl
8.
@.
Therefore
q
e
with
J
$ . E Eb(G). J
It s u f f i c e s
theorem i t s u f 
4
into
5
and
i s a s above,
5.e i s
PD F o r t h i s we c o n s i d e r
ek(G). in
then
F&
,
and we c o n s i d e r
d e f i n e d by
6,
order
0. gives t h a t
F&
fixed
is
which i s n u l l on t h i s s e t i s
F i r s t we show t h a t i f
a w e l l d e f i n e d mapping f r o m
Du = 0
a. J*$j,
eb(E).
From t h e HahnBanach
L E Ek(E)
f i c e s t o prove t h a t any
be
D
these p a r t i c u l a r solutions
for t h e t o p o l o g y i n d u c e d by
Ker D
i s dense i n
and
and
J
We want t o p r o v e t h a t t h e s e t o f
E
P:e
finite
finite
PROOF.
Let
s o l u t i o n of
n. P. =
s e e Mal
n
= 0,
ei('+Aoq))
of
c l a s s i c a l computation
Q,
f o r a11
j
< n.
eb(G),
i(6+Xotl) e
i(5+Xoq)
'$
* $ ) = D(q
Q.
e
i(S+Xoq)
) ' $ = 0.
i s i n t h e above s e t of p a r t i c u l a r s o l u 
t i o n s and t h u s by h y p o t h e s i s ,
& ( q @ je
i (5
+A
1
* $ ) = 0.
Therefore,
FINITEDIFFERENCE
= 0,
F(’)(lo)
if
< n,
j
g r e a t e r than or e q u a l e n t i r e f u n c t i o n on
PARTIAL DIFFEFLENTIAL EQUATIONS
n.
C.
that is,
lo
Hence
i s w e l l d e f i n e d and i s a n
cp E e b ( G ) ,
Fh
and a n a l y t i c on
C
~
i s a z e r o of
t h e map
5 k
eL(G).  i s an e l e m e n t of PD pD(E) = 0 , t h e r e i s a sequence
If
in
Fb
such t h a t
converges t o eb(G).
on
pD(6+hnp)
f
and
KG
for
c,
t h e map
T h i s i s obvious i f pD(g)
(An)
An +
E C,
0
f
0.
and a p o i n t
S i n c e (g+Q).cp 3.e PD we h a v e t h a t c ( 5 ) i s l i n e a r PD i s a 0  n e i g h b o u r h o o d i n eb(E)
0,
for all
3.e ($)cp, if A n + 0 , PD N o w L E V1, where V1
There a r e
order
F&.
3L (c)*q
k
of
i s w e l l d e f i n e d from
Now, we want t o p r o v e t h a t f o r any f i x e d cp
F
Then by p r o p o s i t i o n 13 i n Gupta E l ] ,
every fixed into
211
n.
( c o n v e x b a l a n c e d bounded s u b s e t s of
F and
s u c h t h a t we may assume
G)
E e b ( E ) , s u c h t h a t If(i)(%xKG)(%XKG)il
= [f
V1
5
k , if 0 i i s n}.
Let
E
V = (cp
Then
V
eb(G),
i s a 0neighbourhood
in
eb(G).

PD
m’
,
v’
and
a r e independent of
b’
a€

Q is analytic. PD that i s , t h e r e i s an
F&
and
eb(G).
5
E F&
where
Therefore Then
g ( 5 )
in
R
ep E E b ( G ) ,
c finite
Now, f o r e v e r y f i x e d
From P r o p o s i t i o n 3.2
t(ei5cp)
=
C.
‘4
t h e n by
i s a con
PD
t i n u o u s l i n e a r map on
5 E
a E A.
i s a bounded s u b s e t of
[ ~ ~ ( ~ )  c p u }
3.e
c V,
If
I ~ ( ~ )  i Q m~ ’ I( l + ~ ~ ~ ~ / ) v eb‘I11m5JI, ’
lemma 4 . 2 ,
0 i i r: n } .
I ~ I ( ~ ) ( K ~ ) ( I C , i) ~ k~ , i f
such t h a t
= n.
cj(i)
eL(E)
3.c
cp E e b ( G ) ,
belongs t o PD s u c h t h a t 54, = pD’3R.
seb(E), If
212
J.F.
e(F)
Since
8 eb(G)
4, = R O D
3.1,
e(F),
Q E
B y d e n s i t y , if
5 . 2 LEMMA.
be t h e t r a n s p o s e d of
PROOF.
S E
If
S = tD(u)
S €
S
E
t
3s
= pD*3R,
ek(E)
tD:
P
ek(E)
Then
such t h a t S ( P ) = 0 , f o r a l l
f
E K e r D].
t h e r e i s an e l e m e n t
in
ek(E),
Therefore
ek(E)
and l e t
L ( f ) = 0.
f E K e r D,
be such t h a t
u
S ( f ) = 0.
implies S ( f ) = 0,
and
S = ROD,
R
in
S = tD(R),
that i s ,
with
Con
D f = 0.
whenever
of the previous p r o p o s i t i o n , there i s
F r o m the p r o o f such t h a t
be a s above,
tD(ek(E)),
= UOD. let
versely,
D.
ek(E),
= (S E
tD(e'(E)) b
f E K e r D,
Therefore i f D
and
&(Q.q) = ( R o D ) ( Q  q ) .
e b ( E ) , by t h e a r g u m e n t i n R e m a r k
i s dense i n
E
PAQUES
w e deduce t h a t
ek(E).
in
Let
COLOMBEAU and O.W.
eL(E) i.e.,
D(e(b(E)).
5.3 THEOREM ( E x i s t e n c e of S o l u t i o n s ) . space and l e t D(eb(E))=
D
be any nonz e ro
L e t u s prove t h a t tD(R)
= 0,
soon a s
0.
f .d.p.d.
be a r e a l B a n a c h
E
on
Then
E.
e b ( E )*
B y the previous l e m m a ,
PROOF.
Let
D(ek(E))
i s injective.
by t h e p r e v i o u s l e m m a ,
D f = 0.
S = 0
and t h u s
Remark
3.1.
i s a Fr6chet
tD
t
Then
S
= ROT,
p D * 3 R = 0.
and
Since
i s w e a k l y closed in
R E
If
= S,
tD(R)
3s
ek(E)
D f 0,
i s such t h a t
with
= pD*3R. 3 R = 0,
S ( f ) = 0,
Since thus
tD(R)
as
= 0,
R = 0 , by
The r e s u l t c o m e s c l a s s i c a l y f r o m t h e f a c t t h a t space.
eL(E).
ek(E)
FINITEDIFFERENCE PARTIAL DIFFERENTIAL EQUATIONS
213
REFERENCE S

ABUABARA, T. [l]
A version of the PaleyWienerSchwartz theorem
in infinite dimension

Advances in Holomorphy (Editor J.A.
 NorthHolland, Mathematics
Barroso)

ANSEMIL, J.M.
COLOMBEAU, .J.F. [ 11
isomorphism in nuclear spaces et Appliqubes

ANSEMIL, J.M.



Studies 34 (1979).
The PaleyWienerSchwartz
Revue Roumaine de Math. Pures
Tome XXVI, no 2, p. 169181,Bucarest, (1981).
PERROT, B. [l]

Cmfunctions in infinite dimension
and linear partial differentialdifference cquations with constant coefficientes (to appear).

BOLAND, P.J. DINEEN, S. [l]  Convolution operator on Gholomorphic functions in infinite dimension Trans. of A.M.S.,

vol 190, (1974), P. 313323. COLOMBEAU, J.F.


MATOS, M. [l]

Convolution equations in infinite

dimension Brief Survey and New Results Functional Analysis, Holomorphy and Approximation Theory (Editor J.A. Barroso) North Holland, (to appear). [Z]


Convolution equations in spacesof infinite dimensional
entire functions

Proceedings of the Koninklijke Nederlandse
Academy of Science A83
COLOMBEAU, J.F.

(k), (1980), p. 357389.
MEISE, R. [i]

Cmfunctions on locally convex
and on bornological vector spaces

Functional Analysis, Ho
lomorphy and Approximation Theory (Editor S. Machado), Lecture Notes in Math. SpringerBerlag, 843 (1981), p.195216. COLOMBEAU, J.F.

MUJICA, J. [ 13
uniform bounded type


Holomorphic and Cmfunctions of
Functional Analysis, Holomorphy and
Approximation Theory (Editor J.A. Barroso) (to appear). COLOMBEAU, J.F.

PERROT, B. [l]


North Holland,
The Fourier Bore1 transform in
infinitely many dimensions and applications

Functional
Analysis, Holomorphy and Approximation Theory (Editor S. Machado), Lectures Notes in Math., Springer Verlag, 843 (1981), p. 163186. EHRENPREIS, L. [l]  Solution of some problems of division. Part I1 American Journal of Math., Vol. 77 (1955), p. 287292.

2 14
J.F.
[l] 
A.M.S.
1 6 (1955).
nP
Malgrange Theorem f o r n u c l e a r l y e n t i r e f u n c t i o n s
of bounded t y p e on Banach s p a c e s

Rochester Janeiro,
1966

[l]

[l]



SCHAEFER, H .
SCHWARTZ, L .

Ann.
[l]

[l]
J .F
Comptes Rendus, Acad.
T o p o l o g i c a l V e c t o r Spaces

[ 11



London
(1967).
Talence
FRANCE
O.W.
PAQuES
D e p a r t a m e n t o de Matem6tica IMECC

UNICAMP
U n i v e r s i d a d e E s t a d u a l de Campinas
13.100 BRASIL

T e x t o s de M a t e d 
U n i v e r s i d a d e do

MacMillan s e r i e s ,
J u l y 1981.
T o p o l o g i c a l V e c t o r S p a c e s , D i s t r i b u t i o n s and
New Y o r k ,
d e Mathdmatiques e t d t I n f o r m a t i q u e



P e r s o n n a l Communication,
U n i v e r s i t 6 de Bordeaux I 33405
des Sciences
7 , (1957).
. COLOMBEAU
U.E.R.

Thdorie des d i s t r i b u t i o n s a v a l e u r s v e c t o r i e l l e s
Inst. Fourier,
[l]
Kernels
R i o de
1970.
SCHWERDTFEGEK, K . TMVES, F .

1964.
Brasil,
New Y o r k ,
I

I n s t i t u t o de F i s i c a c MatemLtica
Recife,
IMPA
(19551956), p. 271355.
L e c t u r e s on t h e o r y o f d i s t r i b u t i o n s

U n i v e r s i t y of

( 1 9 4 9 ) , p . 15491551.
de P a r i s 228,
tica
37
Sur l e s algebres denses des fonctions diffe'ren
t i a b l e s s u r une v a r i e ' t 6
[2]

Thesis
E x i s t e n c e e t approximation des s o l u t i o n s
Ann. I n s t . F o u r i e r V I
L.

N o t a s de M a t e d t i c a ,
1968.
M A M R A N G E , B.
NACHBIN,
PAQuES
Produits t e n s o r i e l s topologiques e t espaces
Mem.
nucleaires GUPTA, C . P .

11
[
GROTHENDIECK, A .
COLOMBEAU and O.W.
Campinas

SP
Functional Analysis. Holomorphy and Approximation n t e o y , J A . Barroso led.) 0 NorthHollnnd Publishing Company, I982
APPROXIMATION
PROPERTIES IN NUCLEAR
F ~ C H E TSPACES
Ed Dubinsky
This paper is intended to be a survey of research in nuclear Fr6chet spaces from 1 9 7 4 to the present time.
Our emphasis will be
on problems connected with approximation properties. In 1974 appeared the paper of B . S . [21]
Mitiagin and N.M.
Zobin
solving (in the negative) the problem of existence of bases in
nuclear Frechet spaces.
Since that time there has been a consider
able amount of research into related questions and problems connected with other approximation properties. We give a complete description of all results obtained in this area during this period.
We also s a y something about proofs
although we are far from complete and it is necessary to consult the references given for full details.
What we try to do here re
garding proofs is, f o r a selection of results, to given a n exposition of the main ideas, presented in a somewhat informal, conversational style with a minimum of details.
Hopefully this will be an
aid to understanding the complete proofs which in some cases are rather complicated.
The research was partially supported by the National Science Foundation.
216
E D DUBINSKY
PRE:LIMINARIE S
We denote by
N
the set of positive integers.
A Fr6chet space is a complete, metrizable locally convex space. norms
Its topology is defined by an increasing sequence of semi
(l!*!lk)k
called a fundamental sequence
of
seminorms.
If one
of these seminorms is a norm ( s o all but finitely many are) we say
norm.
that the space admits continuous
Obviously this is independ
(l!.Ilk)k.
ent of the choice of
Another way to describe a Fr6chet space is that it is the projective limit of a sequence of continuous linear operators on Banach spaces,
E =
(X =
(X
seminorms
Ek).
in say
E
Ak: Ek+l+ Ek
): x
= Akxk+l
(11 . I l k ) ,
(k
(k
E
given by
If the operators
Ak
E
Then the space
N).
with fundamental sequence of
N)]
I/x/lk = I/xkl1
(here
)I *I)
is the norm
can be chosen to be compact then we
is a Fr6chetSchwartz Space.
If they can be chosen to be
HilbertSchmidt operators on separable Hilbert space we say that is a Frhchet nuclear space. tinuous norm we can choose
\lllk
E
I n the latter case if
(ll*llk)k
E
admits con
to be Hilbertian s o that each
defines an inner product structure on
E.
The simplest example of a nuclear Fr6chet space is the space w
of all sequences of scalars with the product topology.
include the space valued functions on
Others
Cm(R)
of all infinitely differentiable real
R
with the topology of uniform convergence of
each derivative on each compact set and the space [O,l]
infinitely differentiable functions on
C”[O,l]
of all
(onesided derivatives
at the endpoints) with the topology of uniform convergence of each derivative. A
more extensive class of examples of nuclear Frhchet spaces
is provided by Kathe sequence spaces. termined by an infinite matrix
Such a space k
a = (an)
satisfying
K(a)
is de
APPROXIMATION PROPERTIES I N NUCLEAR F m C H E T SPACES
k 0 S an
Then
5
k+l an
,
sup ak > 0 k n
= {S =
K(a)
(nEN)
(En): 1(5)1k=
pology is determined by
(ll/lk).
, Vk
C 15,1an
3 t
k
n
bnk > 0
and the angle
a fixed line in
, 8
which its major axis makes with
En.
It suffices for the construction that the three values,
0, f
5,
TT:
1(i,j,&) TT
hT + h ,
where
takes on only
but each of these occurs infinitely
often in a rather complicated way. jection
Bnk
Specifically we define a sur
n = [(i,j,&)
is infinite for each
E N3: i < j < t,}
(i,jl&)
E a.
and
Then we take
ED DUBINSKY
220
enk
,
=
or
0,
a c c o r d i n g t o whether
4, < k ,
where
of
one n e e d s a c o n d i t i o n l i k e ,
E,
(i,j,L)
= rr(n).
bnk
n,k+l
i
< k s t,
or
I n order t o assure the nuclearity
C ( r +ankn
k i i,
bn,k+l
)
=. For
Therefore
E
A C A'
x E A
in
xo E G
fn
E
Hence
gA(x) = 0
for
G.
such t h a t
gA(xo) = 0,
l i m f n (x,) n i f necessary, w e o b t a i n
P(G,A)
T a k i n g a subsequence,
C Ifn(xo)l < m
i s empty).
W e have proved
0.
fn
P(G,A).
1'
is strictly Cpolar
then t h e r e e x i s t s a sequence
f n ( x o ) < 0.
13 E
(we recall
A'
g,(xo)
f(x),
C o n v e r s e l y , i f there e x i s t s
.
C
Then w e c o n s i d e r
There e x i s t s
=
in
A'
in
x E A'.
for
f n ( x ) = sup [ l / n
and
is strictly Cpolar
A
i s s t r i c t l y 6polar
A
f E P(G,A)
for
Moreover if
there exists a s t r i c t l y C  p o l a r A C A'
i f and o n l y i f
C
in
x E A.
for
such
g A ( x ) = supf f ( x ) has v a l u e
f ( x ) = 1
(b)
is strictly Cpolar
A C G
xo 6 G
there exists
w i t h conditions
G C E
for
t h e c l a s s of p l u r i s u b h a r m o n i c f u n c 
P(G,A)
such t h a t
=O
The f u n c t i o n s
P(G,A).
S(x) = l i m S ( x ) P w e have
= [ x E G ; S(x) =
belongs t o P ( G ) , P f n ( x ) = 1 and S ( X ) =
m]
is a strictly Cpolar
P. LELONG
260
set in
G.
If (2) holds for tions
yl(A,x)
and
E 3
A
y2(A,x)
and for two different control funcin
then a better control is given
G,
by y(A,x)
= SUP [Yl(A,x),
y2(A,x)]
Hence there exists a best control Let
US
f E P(G,A).
apply ( 2 ) to
A E 3,
for
and
x
E G.
and we have 0 < Y(A,x) 51.
Y(A,x) Because
= 1
f(x)
on
A,
we
obtain f(x) gA(x)
= sup f(x) f
Conversely if f E P(G)
and
(m)lf(x)
f
b
0
5
P(G)
which is (2). = gA(x).
5
for
for all
the constant and
f
E P(G,A).
x E G,
consider for
m = sup f(x) < 0. Then X€A
(m)lf(x),
f1(x) = sup
fl(x)
y(A,x)
gA(x) < 0
belongs to
Therefore
Y(A,X)
11 E P(G,A).
gA(x)
Hence the best control is given taking
y(A,X)
=
We have proved
THEOREM 2.
Given a domain
G
in a
c.t.v.s.
E,
a set
A C G
is
a control set for the class of the negative plurisubharmonic functions in
G
if and only if the upper envelope
of the negative plurisubharmonic functions in on
A
is strictly negative in
G.
with values
1
Then the best control is given
in (1) by
Y(A,x)
G
= gA(x)'
BOUNDED SETS ARE CPOLAR SETS
261
A comparison of Theorem 1 and Theorem 2 gives the following property in a domain THEOREM 3 .
G
A set
of a complex topological vector space A C G,
is a set of control for the class of the
plurisubharmonic functions bounded above in is not a strictly Cpolar set in f E P(G)
with
E:
sup f(x) s M
Then (2) holds for each
G.
for
if and only if it
G
x
G
and
sup f(x) s m
for
x E A. REMARK.
If the control
exists with the same P'(G)
3
P(G)
I
[ a ; y(A,x)]
exists for
and
on an extended class:
y(A,x)
Taking the upper envelope of families
b/
Taking the limit of sequences
2.
then it
which is closed f o r the operations:
a/
the sets
P(G)
A
fn
f,(x)
negative in
G.
which converge uniformly on
E PI.
EXISTENCE OF A SEMICONTINUOUS CONTROL It is important to deduce from (2) a control (perhaps less
precise) given by a semicontinuous function, in order to obtain an upper bound uniform
neighbourhood
of
X.
If such a control
exists it will be given by [U;
y'(A,x)
= lim inf Y(A,y)].
Y+x There exists a semicontinuous control if and only if *from y(A,x)
> 0, we can deduce
y'(A,x)
> 0.
We denote
(4)
gz(x)
= lim SUP gA(y), P
the upper regularizing of is:
g,(x).
X
Then, a consequence of Theorem 2
262
LELONG
P.
THEOREM 4.
There exists for
P(G)
a semicontinuous control
if and only if for each
[ a ; Y'(A,x)]
A E U
gA(x) = SUP f(x); f
(5)
has an upper regularizing A E $3
x F G.
and
gz
f 6
P(G,A)
which is strictly negative for
Then the best semicontinuous control is given
by Y'(A,x)
By a well known result tion in
G.
1 S g*(A,x)
= gI(x); gI(x)
< 0.
is a plurisubharmonic func
Therefore we have only two possibilities:
(I) gI(x) e 0. A semicontinuous control does not exist. (11) gi(x) f 0. for all
Then, by the maximum principle we have
gi(x) t The f u n c t i o n 1 i ~ " ( x )
1
l o g t] and
for
x
E
G
and
u * ( x ) = [g;(x)]l.
~ " ( x ) i s u p p e r semicontinuous
~ " ( x ) = 1,
E
i s a bounded
and we have
t h e n by (16) t h e domain
and we have f o r a l l
x E G
s e t w i t h non empty i n t e r i o r ;
s p a c e i s proved t o b e a Banach s p a c e .
If
4
~"(x)
and
G
itself
a tn 2 tl
t h e Fr6chet 1,
there exists
270
x
P . LELONG
such that
u*(x)
i
2x1
1 +
(17)
= 1
o*(xo)
1 > 0. Then the set w = [x E E;
+ 1,
is a non empty open set in
A(x,t) < $[ (1+2h)log t]
tn z 1,
Then for
w
ly polar in
Ga,
x E
a
.
We have by (16) t > tl > 1.
and
W
the bound
E,
is a bounded set in
proves that
Banach space.
for
G
B
and, if
is not strict
we come back to the situation where
E
is a
Thus Theorem 10 is proved.
The class of complex Fr6chet spaces which are not Banach and which satisfy the property (P) is not empty. complex Frechet space
H(Cn)
of the entire functions in
compact convergence has property (P). A(F,t)
For example, the with
We define
IF(x)I,
z
E en
= suplF(z)l
for
llz/l
= SUP log
Cn
I1 4l 0 there
AN INTERPRETATION
OF
T~
q E P
cp
S
exists
such t h a t
AND r 6 AS NORMAL TOPOLOGIES
q.
275
A(P)
Then t h e K8the s p a c e
is
defined a s
and i s endowed w i t h t h e l o c a l l y convex t o p o l o g y i n d u c e d b y the s e m i norm s y s t e m
I
{rrp
A(P)'
It i s e a s y t o s e e t h a t
P E PI.
= { Y E CN : t h e r e e x i s t s l y n l g pn
W e s h a l l assume t h a t
,
n u c l e a r and c o m p l e t e . compact s e t s i n N
P
i.e.
that
n E N].
for a l l
A(P)
and
A(P)k
are reflexive,
Then a f u n d a m e n t a l s y s t e m f o r t h e r e l a t i v e l y
i s given by t h e system
A(P)k
d e n o t e s t h e normal h u l l of
N~
such t h a t
i s a f u l l y n u c l e a r s p a c e i n the s e n s e
A(P)
of Boland and Dineen [ 2 ]
p E P
p
{Np
I
p E PI,
where
which i s d e f i n e d a s
:= ( y E cN : lynl s pn
n E IN].
for a l l
F o r a s k e t c h of p r o o f o f t h i s remark and some f u r t h e r d e t a i l s see B b r g e n s , Meise and Vogt [ 11 1.1. A c c o r d i n g t o Dineen [ 61
,
a K8the s p a c e
i f t h e r e e x i s t s a sequence m
.
with
i s c a l l e d Anuclear
bn > 1 f o r a l l
n
E
N
and
m
E L
0 such that
for any
f E ~(0).
X€ uk Remark that
(H(R),T6)
is the inductive limit of Fr6chet spaces
and hence ultrabornological.
AN
INTERPmTATION OF
AND
Tw
AS NORMAL TOPOLOGIES
T6
277
(iv) Sequence space representation of spaces of analytic functions Let
be a Kbthe space satisfying the conditions of part (ii).
A(P)
Then it follows from Boland and Dineen [ 2 ] ,
thm. 11, that
is isomorphic to the K6the space
(Hhy(A(P)L),TO)
A(M,P) = r(Xm)mEM where M := Em E
and where for
< x E
m lxmlp
01 c
S C L,
b) In order to show norm in
we take any
7 W
continuous semi
Then it follows from the structure of the compact sets
w.
(indicated in l.(ii))
A(P)L
p E P
exists
C(V)
W(f)
A(P)
, c
6.1.2)
s
V
of
N
P
such that
c(v)
SUP If(x)l.
*
XE v
implies the existence of
p . 0
such that
1" w(zm) s ~ ( +p s) Because of our choice of
Since
(q)'
u E Uq
q, u
for any
and
w,
+ W
the
this implies
( W ( Z ~ ) ) ~ is ~ in
S c L.
c) In order to see that Then there exists
V := N P
m E M.
was arbitrary, this shows that
and hence
: suplyjllwjl
j] f o r j EN.
AN INTERPRETATION OF
AND
,,T,
T6
2.1 and 6.1, it is known that
From Bbrgens, Meise and Vogt [l],
(H((A(P);)~),T~) For
=
7
7
norms on
o,
and
Tu
283
AS NORMAL TOPOLOGIES
A(wQ~).
a fundamental system of T continuous semi
T6
is given by
A(M,Q')
rPx :
,
E AJ
px(
ambm
9
where =
Q*
,
= {x E R y : there exists q C Q such that for all b E A(NxN,Q) with
AT
6
C lbjklqjk c 1 we have C Ibmlxm c BJXN mEw
3,
= Ex E R y : for any j E N there exists q E Q such that
m
xm s q
for all
m E Mj].
For
To
this is just a consequence of the isomorphism noted above.
For
TW
it follows immediately from proposition 2.
For
T6
it is
obtained by the following arguments:
From Bbrgens, Meise and Vogt
[ 13, 6.1, we know that
is the strict inductive limit
(A(M,Q'),T6)
of the topological linear subspaces for all
m
Mj]
(H( (A(P)L)j),To).
of
A(M,QM),
where
nMj AM,
Hence a seminorm on
iff its restriction to
AM.
this implies the result.
3
:= {a
E
A ( M , Q ~ ): am =
o
is isomorphic to
A ( M , Q ' )
is T6continuous,
is continuous.
It is easy to see that
284
R.G.
MEISE and D. VOGT
REFERENCES 1.
BdRGENS, M., R. MEISE, D. VOGT:
Entire functions on nuclear
sequence spaces, J. reine angew. Math. z 2 , 1962213 (1981). 2.
BOLAND, P.J., S. DINEEN: Holomorphic functions on fully nuclear spaces, Bull. SOC. Math. France 106, 311336 (1978).
3.
BOLAND, P.J., S. DINEEN: Duality theory for spaces of germs and holomorphic functions on nuclear spaces, p. 179207 in "Advances in Holomorphy", J . A . Barroso (Ed.) North Holland Math. Studies 34, 1979.
4. BOLAND, P.J.,
DINEEN: Holomorphy on spaces of distributions, Pacific J. Math. p1, 2734 (1981).
5.
DINEEN, S.:
S.
Fonctionelles analytiques et formes sur des espa
ces nucl6aires, C.R.
6. DINEEN, S.:
Acad. Sci., Paris
287, 787789 (1978).
Analytic functionals on fully nuclear spaces,
to appear in Studia Math.
7. DINEEN, S.:
Holomorphic functions on nuclear sequence spaces,
p. 239256 in "Functional Analysis: Surveys and Recent Results II",K.D. Bierstedt, B. Fuchssteiner (Ed.) North Holland Math. Studies 38, 1980.
8.
DINEEN,
S.:
Complex analysis in locally convex spaces, North
Holland Math. Studies 57, 1981.
9. KOTHE, 10.
KBTHE,
G.:
Topological vector spaces I, Springer 1969.
G.:
fiber
nukleare Folgenrfiume, Studia Math.
z,
267271 (1968). 11.
MEISE, R.: A remark on the ported and the compactopen topol o g y for spaces of holomorphic functions on nuclear Frbchet spaces,
to appear in Proc. Roy. Irish Acad.
12.
MEISE, R., D. VOGT: Structure of spaces of holomorphic functions on infinite dimensional polydiscs, to appear in Studia Math.
13.
PIETSCH, A.: Nuclear locally convex spaces, Ergebnisse der Math. 6 6 , Springer 1972.
14.
SCHAEFER, H.H.:
Topological vector spaces, Springer 1971.
AN INTERPRETATION OF
Tu
AND
T6
AS NORMAL TOPOLOGIES
285
15. VALDIVIA, M.U.:
Representaciones de 10s espacios & (n ) y ;Q'(n), Publ. Rev. Real Acad. Sci. Ex. Fis. Nat., Madrid, E , 385414 (1978).
16. VOGT, D.:
Sequence space representations of spaces of test functions and distributions, to appear in Advances in Functional Analysis, Holomorphy and Approximation Theory, G.I. Zappata (Ed.), Marcel Dekker.
Mathematisches Institut der Universitlt Universitatsstr. 1 D4000 DUsseldorf
Gesamthochschule Wuppertal Fachbereich Mathernatik Gaupstr. 20 D5600 Wuppertal
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Functional Analysh, Holomorphy and Approximation Theory, JA. &ROSO (ed.) 0NorthHolland Atblishing Company, 1982
WELL LOCATED SUBSPACES O F LFSPACES
R e i n h a r d Mennicken
and
Manfred M 8 l l e r
1. INTRODUCTION A t t h e I n t e r n a t i o n a l S e m i n a r on F u n c t i o n a l A n a l y s i s , H o l o m o r 
p h y , and A p p r o x i m a t i o n T h e o r y one of t h e a u t h o r s , R.
Mennicken,
gave a l e c t u r e on n o r m a l s o l v a b i l i t y o f c o n t i n u o u s l i n e a r o p e r a t o r s i n d u a l s of L F  s p a c e s .
T h i s a r t i c l e i s a d e t a i l e d v e r s i o n of t h e
f i r s t p a r t of t h a t l e c t u r e .
Let
= 1 . m (Yn,un)
(Y,u)
b e a n LFspace
l i m i t of F r 6 c h e t s p a c e s ) and
R
topology
uR := 1Am u Rn
located i f
,
where
= (R,U
(R,uR)'
R
)'
l o c a t e d s u b s p a c e s a r e known,
be a s u b s p a c e of
oR
c o n s i d e r t h e r e l a t i v e topology
.
(str c t inductive
u
of
:= R
Rn
Y.
we
R
On
and t h e i n d u c t i v e l i m i t
n
Yn.
i s c a l l e d well
R
V a r i o u s e x a m p l e s f o r non w e l l 
see e.g.
Slowikowski
[ l 3 ] , R e t a c h [12],
K a s c i c and R o t h [ 41. Welllocated
s u b s p a c e s of LFspaces
s u r j e c t i v i t y and n o r m a l s o l v a b i l i t y of of LFspaces:
Let
(X,T)
uous l i n e a r o p e r a t o r on where and
To:
(X,T)
+
are closely related t o
l i n e a r operators i n duals
a l s o b e a n LFspace (X,T)
to
(Y,u).
T
and
W e decompose
T = ioT
( R ( T ) , u ~ ( ~ ) c) o i n c i d e s a l g e b r a i c a l l y w i t h
i: ( R ( T ) , u ~ ( ~4) ) ( Y , g )
sequentially closed,
i.e.
i s t h e i n c l u s i o n map.
(R(T),UR(T))
If
i s a n LFspace,
of Dieudonn6 and S c h w a r t z [l] y i e l d s the o p e n n e s s of i n g t o Banach's
be a c o n t i n 
c l o s e d r a n g e theorem ( s e e e . g .
[5]
T
is
R(T)
theorem 1
To.
, (3.1)),
AccordTi
is
288
R. MENNICKEN and M. M(JLLER
normally solvable, i.e. normally solvable iff
R(Tb) = N(T0)'. i'
is surjective.
jective iff the identity map on a weak isomorphism, i.e. iff
Finally,
(R(T),D~(~))
R(T)
T' = Tboi'
Therefore
to
i'
is
is sur
(R(T),UR(T))
is
is welllocated.
The contents of Palamodovfs paper [8] consist in a formalization of the MittagLeffler method.
The main result is theorem 11.2
which is a homological criterion for the applicability of the MittagLeffler principle.
One of the main consequences of this
theorem 11.2 is a new proof of HBrmander's surjectivity theorem for LPDOs
(linear partial differential operators) with constant coef
ficients on the space of distributions
Q'(C2).
However, rather
comprehensive calculations are necessary because he has to verify some approximation properties of the kernels of the differential operators under study. Retach [12] stated a very useful functional analytic characterization of a subspace to be welllocated.
For proof he refers
to Palamodovls homological theorem 11.2 and some of its corollaries. Pt6k and Retach [9]
used this characterization to obtain necessary
and sufficient conditions for the surjectivity of the duals of continuous linear operators in LFspaces.
A s a consequence they also
stated HBrmander's surjectivity theorem for LPDOs
in
B'(i2).
In this paper we intend to prove some characterizations of welllocation which are closely related to those of Retach [ 121, Pt6k and Retach [9].
We give purely functional analytic proofs which do
not make use of any homological argument.
Our functional analytic
approach seems to us simpler and more straightforward.
In a subsequent paper [6] we will use our criteria for welllocation to prove sufficient conditions for the normal solvability of the duals of continuous linear operators in LFspaces.
Some
WELL LOCATED SUBSPACES OF LFSPACES
applications to LPDOs
Q'(n)
in
289
with not necessarily constant
coefficients will also be stated.
2 . RESULTS AND PROOFS
We denote by on
(Y,o)
or
r
rn
or
(Yn,un),
respectively.
K
:=
P
(X
is the closed punitball in K
p E
for
the set of all continuous seminorms
P
E Y : p(X)
Y;
:= (x
For h
p
r
E
l]
analogously we set
E Yn
: p(x)
< 13
rn.
Polars and orthogonal subspaces will always be taken with respect to the dual pair For a subspace
(Y,Y')
N C Y
unless otherwise noted.
we define, as usual,
distp(y,N) := inf{p(y+z)
: z
E N]
(p
E
r, y
E Y).
The set distr(
,N)
L=
{dist ( ,N) : P E
P
is a system of continuous seminorms on
(Y,o)
locally convex (no, separated) topology on p E
r
we have {Rk
: n
E
p E distr( ti}
,N)
iff
is welllocated;
( ~ " 5 ~ ) is complete; m
Obviously, for
pIN = 0.
on
Y'.
We consider the following properties:
(11)
which defines a
is a basis of neighborhoods of
metrizable group topology 0,
(I) R
Y.
r]
0
for a pseudo
R.
290
(VI) S r E T
MENNICKEN and M.
V n E N
3 j 2 n
i d : (%,r+dist
(1) Theorem.
r(
,Yn))
V k z j
+ (%,dist
If
R
(v) 3
,Rj))
i s continuous.
(VI). (Rk,ok)/Rj
i s s e q u e n t i a l l y c l o s e d and i f
k > j
ive f o r a l l
r(
i ) We s t a t e t h e i m p l i c a t i o n s
(I) c) (11) 0 (111) c, (IV) 0 ii)
MOLLER
flexive f o r a l l
o r t h e s e p a r a t e d c o m p l e t i o n of
(Yk,r)
t h e n a l s o (VI) =) (V), i . e .
k € N,
i s reflexi s re
a l l properties
(I), (11), (111), (IV), (v), (VI) a r e e q u i v a l e n t . (11) i s a c o m p l e t e n e s s c o n d i t i o n s .
A d i f f e r e n t completeness
c o n d i t i o n was o b t a i n e d by F l o r e t i n [ 2 ] .
H e c o n s i d e r e d c l o s e d sub
s p a c e s of r e g u l a r i n d u c t i v e l i m i t s of r e f l e x i v e F r e c h e t s p a c e s . R is w e l l  l o c a t e d
iff
d e n o t e s t h e s t r o n g t o p o l o g y on
Y'.
He s t a t e d t h a t
B
Proof
of
(I) =) (11):
i s c o m p l e t e where
(Y' , p ) / R A
(I): Let
be a Cauchy s e q u e n c e i n
(y;):
t h e r e e x i s t s a n i n c r e a s i n g sequence y;

y;
E RA
for all
(kn)z c N n E N
and
(Y'
such t h a t k , l 2 kn.
Thus, (
~
~
2
'
:=) ( Y , Y ~ ) n
d e f i n e s a unique
z'E
(R,oR)'.
( Y € Rn,
,cR).
n E N)
Then
WELL LOCATED SUBSPACES O F LFSPACES
291
By assumption and the HahnBanach theorem we can find such that and
lR
y'
k 2 kn,
for all
which proves the completeness of
(Y' ,5R).
(yn,un),
(Yn,Un).
We conclude
n E N
F o r each
[Kq : q E
i.e.
y'
TL c
let
rh]
n E N.
the second category in
E N)
(Y'
such that for all
q1 E
(Y' , $ R ) .
gory in
u
(1II)i)
find
a'€
for each
Y'
and
j
2
rn E Tn
where
is of
91
rk
'k
by
denotes the convex hull.
cv
n KO = KO is of the second category rn k = l 'k n E N. Thus, for each fixed n E N we can
n
n
such that a'+
Since
KO
n
is fulfilled and
,cR)
such that
By induction we can find qk E n n E N KO is of the second cate
We define
n On Kr = cv( K ) , n k=l 'k
r;
,tr).
k=l
(Y'
be a countable basis
is a basis of neighbourhoods in
By Bairels thoorem, there is a
in
rn
n .E N
Then
holds for all
(k

yk E R,'
(11) * (111): on
.
= z'
Y'
y'E
is a subspace and J convex, we obtain RA
R+ = 1 (a'+R+) J 2 J
R'
j
5 , c :K n
: K 'R
= n

n
+
(a'+R;)
+ $)
(K:
is absolutely
n
MN
 $R
c KO rn
which proves (111).
(IV):
(111) q'
1 yn
q 2 9'2 {
= q n'
F o r each
(n E N)
E Tn
there is a
q'E
Tn+l such that
(cf. e . g . Horveth [ 31 , p.160).
for some (substitute
q
E 'n+l q'
by
If, in addition,
is fulfilled, we may assume that maxtq'
,;I
if necessary).
These pre
liminary remarks enable us to construct inductively a sequence
292
rk
R.
rn
E
MENNICKEN and M .
such t h a t
= rl,
r;
r' n+l
rlyn = r'n
d e f i n e s an
w i t h t h e a i d of
[ 5 ] , (1.4),
Thus
MbLLER
' 'n+l
= ( K n ~
c K':
K:
I'.
r E
n
and
r'
From
r;
I
= r' n '
n + l Yn
= K:
Y , ) O
2 rn
+ Y;
we c o n c l u d e ,
,
n
which p r o v e s (111)s ( I V ) . (IV)
y'E
3
Let
(V):
R;.
+
= y;
y'
The r e q u i r e m e n t s
q
we o b t a i n for a l l
+
y;
E
n
L
r
y;. and
j < k y;
E K:,
Define
¶Iyn
be a s i n ( I V )
= 0
R'
by
1",
E
and
q ( y ) = l(y,yh)l
(kn,kn+l,kn+2)
2""y',
(kn)I C
(n E W).
W
such t h a t ( V )
Substituting
we o b t a i n :
Next we prove f o r an a r b i t r a r y , h u t f i x e d ,
v
"1:
y t ~
n+1
E
Finally,
kn+1
(2)
y;
(y E Y ) .
are fulfilled.
( I ) : Choose an i n c r e a s i n g sequence
=a
y;
and choose
y E Rk
holds f o r each t r i p l e y'E
and
By a s s u m p t i o n , t h e r e a r e
such t h a t
(V)
r
r E
a
qn E
r
n E N
293
WELL LOCATED SUBSPACES OF LFSPACES
Banach theorem t h e r e i s a
for a l l
y
E R.
Since
y h + 2 E Y'
+
y'
yh+2 E
such t h a t
%
,
and by t h e same argument
n+2
t h e r e is a
yh+g E Y'
such t h a t
yh+3 I %
= (Y' + Y h + 2 )
n+3
I %n+ 3
and
I (Y y E R.
for a l l
'
yn+j
E Y'
r(Y)
+ 9 ( n + l )( y )
By i n d u c t i o n , we o b t a i n f o i e a c h q ( n + j  2 )E
and
2n2
I r
j 2 2
such t h a t
and
m
qn =
Define
WE
deduce
i s some
j
C
From
v =n
qn E
r
> 2
such t h a t
and
¶,Iykn
= 0. y E R kn+j
Let
v'E
(R,uR)'.
W e show t h a t
Banach theorem, f o r e a c h
that
YhlRkn
=
.
V'I Rkn
n E N,
Let
y E R
For an a r b i t r a r y
.
y E R
there
Then we o b t a i n
v'E
(R,uR)'.
By t h e Hahn
we can choose some be a r b i t r a r y .
y;
E Y'
such
There is an n 2 2
and M. M d L L E R
R..MENNICKEN
294
y E R
such t h a t
W e conclude
kn'
n 2
(3)
= (Y,Yh)
(y,v')
= (Y,Y>)
 y;+l)
c (y,y;+2 !J=l
+
0)
= (Y,Y;)
'
For

yv+2
y:+1
C
:=

(Y,Y;+2
Y;+l).
according t o ( 2 ) .
qv
kv+l
r
E
q
qv.
v=1
v=l
choose
RL
0)
q
+
h o l d s because of
qIykn
m
s
I(Y,V')I
C
=
qvIykn.
v=1 w e o b t a i n , i n v i e w of ( 3 ) and ( Z ) ,
y E R
F i n a l l y , for e a c h
Define
n1
c
kY,Y;)I
+ y = l

I(Y,Y;+2
%+1)
I
m
s
l(Y,Y$)l
c
+
(Z+r(Y)
+ r(Y) + ¶(Y),
= I(Y,Y&)l
+ qv(Y))
v + l
v'E
w h i c h proves
(R,uR)'. r E
Let us f i x
r a t e d c o m p l e t i o n of
r
and
n s
j s k.
(?,,s)
d e n o t e s t h e sepa
(Yk,r).
The f o l l o w i n g a s s e r t i o n c o m p l e t e s the p r o o f
(4) Proposition.
of the thoeren
(1).
~ s s u m e
W e assert: S = id
(VI*)
:= (Rk,r
+
dist
r(
,Yn))
+ (%,distr(
,Rj))
i s continuous. If
R
i s s e q u e n t i a l l y c l o s e d and i f
are reflexive, Proof.
(V*)
=)
(vI*) (VI*):
we may substitute
(f,,;)
or
(%,uk)/Rj
(v*). Since
distr(
(Rk,r + d i s t r ( , Y n ) )
,Rj)
i s bornological,
by the w e a k t o p o l o g y
295
WELL LOC.4TED SUBSPACES O F LFSPACES
u ( R ~:R,
Fix
(%,%)
R$
y'E
),
i.e.
(i.e.
q E distr(
(VI*)
r'
Choose
,Yn) 3
IRk
Y'
and a n u l l s e q u e n c e
we h a v e t o show t h a t
E R'j in
(y,):
= {q,
:
(%,r
S
Thus, a c c o r d i n g t o
(6)
4 Y V ) + GV(YV
v E IN]c
r
s Y(r(Y)
(51,
k'
distr(
a c c o r d i n g t o (V") ,Yn)).
Since
+ q(Yv)
0.
t
does n o t hold:
such t h a t
qv
(yk,uk).
b
and s u c h t h a t
( ,Yn). 41, such that
qv A := d i s t
Set
y z 0
1 5 %+1
and
vo
E
IN
+
we c a n
= 1 and
Next w e Drove t h a t ( R  R' )
(R,,u(R~,%
+
r
s =(Yv)
i s continuous, there a r e I(Y,Y'>l
q E
We assume t h a t (V*)
i] s a b a s i s on { q v I y k : v E @I Since
),
I 0, the sequence
T: 6 fl V + 2y
Exn] C D
{y,]
is Aclosed w.r.t.
n
V,
for each
n
is bounded in
Y.
r
if
for
(V,Y)
304
P.
s. MILOJEVIC
either one of the following conditions holds: (a)
n
TI
V + C(Y)
is
r
jectionally complete scheme or R
(b) Y
r
and
m = Y"
T:
Q p x = Kx
fl
+
V
is reflexive, = To
with
Q : K x
(c) either with
&,Ax
T:
r
5 n
for
=
ra
is a pro
K: X + Y"
with
continuous,
x E Xn.
for
is demiclosed (and, in particular, u.d.c.),
= Y"
and either
T
is locally bounded and
r
ro = Kx
with
V +' C ( Y )
or
x E Xn
for

is demicontinuous and
Y = Y""
with
=
ra
x E Xn.
5 n V + K(Y)
T = A+N
= Ax
R(K)
=
= To
is locally bounded or Kquasibounded and
= Kx
0 I D c B(xi,r), i=1
u
CONTINUATION THEORY
T: X + Y
A mapping
kX(D)
for each
whenever
x (D) f
5
309
is said to be kcontractive
D C X;
it is condensing
if
x(T(D)) s
x ( T ( D ) ) < x(D)
if
0.
The class of ballcondensing mappings is rather extensive and includes, among others, the classes of compact, kcontractive,
k < 1, and of semicontractive ([S])
type.
Modifying slightly the proof of Proposition 2.1 in [20], have the following extension of it proven in [25] EXAMPLE: 1 . 5 .
Let
ro
with respect to c > 0
and
A: D(A)
and large
An:Dn
+
Yn
A
ping on
X
x,y 6 X
and
I
Y
continuous.
be densily defined, astable X n C D(A),
Suppose that
kb < 1
Y = X
z E J(xy),
A+N
N: X + Y
is Aproper
W.T.
= Kx
and
is either
J
b = c = 1,
ro.
In par
accretive map
(AxAY,~)2 cllxyI12
where
for
to
cstrongly
for each
is the normalized duality map
ping) or cstrongly Kmonotone with I)Kxll s allxll, Q : K x
i.e., for some
o r ballcondensing if
we can take a
(i.e.,
a > 0, and
(cf. also f261).
n,
b = maxl)Qnl). Then
ticular, as
X
(X,Y) with
for
kballcontractive with where
C
we
x € X
and some
x E Xn.
Let us now look at the intertwined representations of astable mappings and their perturbations. ed linear space with the
Y.
We restrict
T: X + Y
there exists a mapping
U: XxX
and
be a norm
D
into the normed linear
ourselves to singlevalued mappings in
DEFINITION 1.9 (f.281).
x E X
Cb(D,Y)
supremum norm of all continuous bounded
functions from the topological space space
Let
+
is semiastable w.r.t.
Y
such that
T(x)
r0
= U(x,x),
if
P
3 10
(i)
The mapping
x + U(x,')lD
for each bounded
D C X.
For each
x E X,
(ii)
To
for
(X,Y)
.s. MILOJEVI~ D
is compact from
U(x,.)
into
Cb(D,Y)
is continuous and astable w.r.t.
i.e., there exists a constant
> 0 and
c
n
1
2
such that
A n important subclass of semiastable mappings are a  c  s t r o n g l y 
Kmonotone mappings with
U(x,)
T
with
satisfying (i) above and
Kmonotone with
c(r) = cr2
DEFINITION 1.10 ( [ Z S ] ) . X,T:
6
T(x)
= U(x,x)
(a)
C(Y)
+
T
for each k < 1
t
C(Y) (c)
U(x, . ) :
for each
Let
D
x E
for
6.
x
T(x)
= U(x,x)
being strongly
and
Q z x = Kx,
x E Xn.
be a subset of a Banach space
U:
6x6 + C(Y)
such that
Then
is of strictly (weakly) &kacontractive x E
6,
fi +
U(x,.):
C(Y)
T
6
into
x
and the mapping
x
+
is ballcondensing for each
x E
U(x,*):
C(Y)
continuous from
D
is 1ballcontractive and into
Y
6
I
6.
is of semi1ballcontractive type if, for each t
U(.,x)
cb(6,y).
is of semiballcondensing type if in (a),
T
type if,
is kball contractive with
(k=l, resp.) independent on
5
i.e.,
U(x,)
and there exist a mapping
is compact from (b)
IIKx/l i allxll,
uniformly for
U( x

,x)
in
x E 5,
is completely
D.
Then, we have the following new important EXAMPLE 1.6 ( [ 2 8 ] ) . w.r.t.
To
and
N:
Let
6c
A: X + BK(Y)
X + C(Y)
U.S.C.
semikballcontractive type with ing type if
6 = c = 1, where
is Aproper and Aclosed w.r.t.
be u.d.c.
and either of strictly
k6 < c,
6 = maxllQnll. TO'
semiastable
or of semiballcondensThen
A+N:
6 + BK(Y)
311
CONTINUATION THEORY
Many o t h e r r e l a t e d examples c o u l d b e f o u n d i n [28]. t h a t monotone l i k e mappings
It t u r n s o u t
c a n b e t r e a t e d v i a t h e Aproper
T
mapping t h e o r y due t o t h e f a c t t h a t t h e y a r e u n i f o r m l i m i t s of
i.e.,
Aproper mappings,
+
T
aG
or a r e s t r o n g l y A  c l o s e d .
G,
i s Aproper
a >
for
and some
0
Our a b s t r a c t t h e o r y i s a p p l i c a b l e
t o them and we r e c a l l some b a s i c needed f a c t s .
1.11.
DEFINITION
x +
K:
Let
is
X
x
T:
from e a c h f i n i t e  d i m e n s i o n a l
U.S.C.
t o t h e weak t o p o l o g y of ( i i ) xn
A mapping
in
is
BK(Y)
subspace
F
of
Y.
i m p l i e s t h a t for e a c h
X
x+
if:
s a i d t o b e quasiKmonotone (i) T
2Y”.
un E T ( x n )
and
l i m sup ( u n , f n ) 2 0 .
f n E K(X,X),
S i n g l e v a l u e d q u a s i  m o n o t o n e mappings from
X
Anto
X* w e r e
i n t r o d u c e d and s t u d i e d b y H e s s [ll] and C a l v e r t and Webb [ 6 ] and t h e n s t u d i e d by many a u t h o r s ( c f . M i l o j e v i 6 and P e t r y s h y n [ 3 0 ] ) . Such mappings a r e u n i f o r m l i m i t s of Aproper mappings.
Namely, w e
have
1.7.
EXAMPLE
Let
a d m i s s i b l e scheme
X
and
Fa
c
Y
b e r e f l e x i v e Banach s p a c e s w i t h a n and
{Xn,Vn;Yn,Qn]
K:
X
+ Y”
a bounded
mapping f o r which c o n d i t i o n s ( i ) , ( i i ) and ( i i i ) of Example 1.1 hold.
Let
Kmonotone
T: X and
Then
(KS+).
+
BK(Y) X
G:
T
+ aG
+
BK(Y)
d e m i c l o s e d and q u a s i 
b e bounded, d e m i c l o s e d and of t y p e
is o f t y p e (KS+) and t h e r e f o r e i s Aproper
ra
and A  c l o s e d w . r . t .
b e Kquasibounded,
a > 0.
f o r each
The n e x t two c l a s s e s of mappings a r e g i v e n by DEFINITION l . l Z . ( a ) Kmonotone (ii) i f
if
xnx
A mapping
T: X
+
BK(Y)
i s s a i d t o be pseudo
( i ) of D e f i n i t i o n 1.11 h o l d s and in
X
and i f
un
E
T(xn)
and
fn
E
K(xnx)
are
P.
312
l i m sup ( u n , f n )
such t h a t
(b)
T:
X
S
t h e n f o r each element
0,
g E K(xv)
u(v) E T(x),
exist
s. MILOJEVIC
and
gn C K(xnv)
v
E
X
there
such t h a t
i s s a i d t o b e g e n e r a l i z e d p s e u d o Kmonotone
IB K ( Y )
( i ) of D e f i n i t i o n 1.11 h o l d s and
if
(ii) i f
x
X
in
and
un E T ( x n ) ,
and l i m s u p ( u n , f n ) s 0
Y
in
xn
f n E K(xnx)
imply t h a t
I n the singlevalued case,
u E T(x)
w i t h unu
and
(un,fn)+O.
t h e f i r s t c l a s s of pseudomonotone
mappings was s t u d i e d by L e r a y and Lions[16]
and e x p l i c i t e l y t h e y
were i n t r o d u c e d ( i n a somewhat d i f f e r e n t way) and s t u d i e d by B r 6 z i s K = I , Y = X").
(here,
[2]
o t h e r a u t h o r s ( c f . Lions [
L a t e r on t h e y were s t u d i e d b y many and Browder [ 31 )
171
.
G e n e r a l i z e d pseudo
monotone mappings w e r e i n t r o d u c e d b y Browder and H e s s
[5].
These
two c l a s s e s o f mappings a r e a l s o u n i f o r m l i m i t s of Aproper mappings More p r e c i s e l y , we h a v e ( c f . [30,24]).
a s w e l l a s s t r o n g l y Aclosed.
EXAMPLE 1.8.
Let
ra
Kquasibounded, and
K
X
and
=
Y
{Xn,Vn;
be r e f l e x i v e ,
T:
X + B K ( Y )
be
a n a d m i s s i b l e scheme for ( X , Y )
Yn,Qn]
a bounded mapping f o r w h i c h c o n d i t i o n s ( i ) ,( i i ) and ( i i i )
of Example 1.1 hold.
Let
G:
X + B K ( Y )
b e bounded d e m i c o n t i n u o u s
and of t y p e (KS+). (a)
If
tinuous,
a >
0,
then
R(K) T
If
= Y"
+ aG
i s d e m i c o n t i n u o u s and Kmonotone w i t h
T
+
T(g(0,r))
Aclosed, (b)
T
UG
i s closed f o r each
r > 0
and
T
w.r.t.
w e a k l y con
ra
i s Aproper and A  c l o s e d w . r . t .
w e a k l y Aproper and w e a k l y Aclosed T
K
f o r each
i s strongly
Ta.
i s p s e u d o Kmonotone and e i t h e r d e m i c l o s e d o r
and i f
i s Aproper
i s c l o s e d for e a c h
K
i s weakly c o n t i n u o u s w i t h and Aclosed
r > 0
and
w.r.t. T
ra
for
K(0) = 0 ,
a > 0,
i s s t r o n g l y Aclosed,
then T(B(0,r))
weakly
CONTINUATION THEORY
Aproper (c)
w.r.t.
and weakly Aclosed If
r
closed f o r each then
ra'
i s g e n e r a l i z e d pseudo Kmonotone,
T
Ta
Aproper and A  c l o s e d w . r . t .
uous,
313
>
0.
Moreover,
Aclosed w . r . t .
and
+
T
aG
K
is
is
T(B(0,r))
i f i n addition,
i s s t r o n g l y Aclosed,
T
a > 0
for
then
i s contin
weakly A  p r o p e r and w e a k l y
'a*
The n e x t g e n e r a l c l a s s of n o n l i n e a r mappings i s t h a t of type
(m).
DEFINITION 1.13.
(KM)
T: X
i s s a i d t o be of
BK(Y)
t
( i ) of D e f i n i t t o n 1.11 h o l d s and i f
if
un E T ( x n ) ,
s o
A mapping
f n E K(xnx) u E
imply
with
un
u
xn
in
in
x
X
and
and l i m s u p ( u n , f n ) s
Y
T(x).
T h i s t y p e of mappings was i n t r o d u c e d b y B r 6 z i s [ Z ] when Y = X",
K = I
and l a t e r s t u d i e d by
They a r e a l s o of s t r o n g l y Aclosed
mny a u t h o r s ( c f . Lions
type.
[17]).
Namely, we h a v e p r o v e n i n
[ 241 t h e f o l l o w i n g
1.9.
EXAMPLE
X
Let
a s i n Example 1.1 and Then
T
and T: X
Y
K:
be r e f l e x i v e ,
BK(Y)
t
i s s t r o n g l y Aclosed,
X
+
Y"
bounded and
Kquasibounded and of t y p e (KM)
w e a k l y , A  p r o p e r and w e a k l y A  c l o s e d
ra.
w.r.t.
W e n o t e a l s o t h a t weakly c l o s e d and
weakly c o n t i n u o u s map
( c f . [ 281 ) .
p i n g s a r e a l s o s t r o n g l y Aclosed
W e c o n c l u d e t h i s s e c t i o n b y l o o k i n g a t some n o n l i n e a r p e r t u r To t h a t end w e n e e d i n t r o d u c e t w o
b a t i o n s of Fredholm mappings. more r e l a t e d schemes. i(A)
that
= 0, B
I
If
A:
X
+
Y
i s c o n t i n u o u s and F r e d h o l m w i t h
t h e n t h e r e e x i s t s a compact l i n e a r mapping A+C:
mensional w i t h
X
t
Y
is a bijection.
dist(x,Xn)
t
0
for
Let x
in
EXn] X
C X
and
C:
X
+ Y such
be f i n i t e d i 
Pn:
X
t
Xn
a
3 14
P. s
linear projection.
Define
linear projection.
Then
complete scheme for
If and
A: D(A)
V f X,
that
. MILOJEVI~
Yn = B(Xn)
TB = (Xn,Pn; Yn,Qn]
t
(X,Y).
c X + Y
= V
Y
is a Fredholm mapping with
is a bijection and then
by the closed graph theorem.
Y.
complete scheme f o r
For
Let
rB = (Xn,Pn; Yn,Qn]
{Yn,Qn}
x E V,
Pn: V + Xn = Bl(Yn)
The schemes
be a
is a projectionally
then we have again a compact linear
B = A+C: V
we define
Qn: Y + Yn
and let
by
C: X
Bl
t
Y
i(A) = 0 such
is continuous
be a projectionally
x = B'y
for some
Pnx = BlQny.
y E Y,
Then the scheme
is a projectionally complete scheme for (V,Y).
rA, TB
FB
and
induced by
A
have a number of addi
tional useful properties needed in discussing the equation Ax + N x = = f
as demonstrated in our works [ 261 and
1271.
Which scheme one
uses depends on a situation. Our first result deals with ballcondensing perturbations (see [ 261
,
[ 271 and [ 281 ) .
= V c X + Y
EXAMPLE 1.10.
(a) Let
A: D(A)
i 2 0, D C X
open and bounded and
kballcontractive with where
6 = max)lQnII
for
kb < c
rA
Aproper and Aclosed w.r.t. (b)
Let
N: 6 + C(Y) II(A+C)xll if
5
V f X.
A:
= V c X
I(Ax1II A.
cI/xlI for
x E X,
Then the mapping
C(Y)
U.S.C.
c/lxllI for
2
Then
6 n
A+N:
V
and either 6 = c = 1,
x1 E X1, 4
C(Y)
the is
TA' t
Y
as in part (a) with
Aclosed w.r.t. (c)
D(A)
6 +
or ballcondensing if
and
complement of the null space of
N:
be Fredholm of index
be Fredholm of index zero, 6 = maxl(QnIl
T = TB
where A+N:
6
fl V
relative to
if
+ C(Y)
V = X
and
and
D
r r
and = fB
is Aproper and
r.
If either
N
ballcondensing with
H = (AIX )l 1 IIQnll, then A+N or
is compact o r
N(A+C)'l
is Aproper and Aclosed
is
CONTINUATION THEORY
3 15
*
w.r.t.
TB' of index z e r o i s given
A l a r g e c l a s s of F r e d h o l m mappings
([lo]).
LEMMA 1 . 2
If
+ X"
X
A:
i s l i n e a r and of t y p e ( S ) ,
then
i t i s Fredholm of i n d e x z e r o . I f a l i n e a r mapping inequality
(1.3), t h e n i t i s o f t y p e
of i n d e x z e r o . pact, F: X
then
W e a l s o know
IC
s a t i s f i e s the Glrding l i k e ( S + ) and c o n s e q u e n t l y Fredholm
t h a t if
C:
+ X
X
i s Fredholm of i n d e x z e r o .
or k  s e t
i s a linear kball
Y
b
X + X"
A:
Fredholm of i n d e x z e r o ( s e e [ 3 2 ]
i s l i n e a r and com
More g e n e r a l l y , i f
c o n t r a c t i o n , then IF
is
).
F o r n o n l i n e a r p e r t u r b a t i o n s of t y p e ( S ) we h a v e , i n v i e w of Example 1.1,
EXAMPLE 1.11.
Let
X
b e a r e f l e x i v e Banach s p a c e ,
t i n u o u s l i n e a r a n d of t y p e ( S ) and q u a s i b o u n d e d and s u c h t h a t
is either
(X,X*).
FA
Then
or A
TB and
N:
X
+
X"
X
+
A
that
r
or a n y p r o j e c t i v e o r i n j e c t i v e scheme f o r A+N
a r e Aproper
i s of t y p e ( S ) (or ( S + ) ) A+N
con
Suppose t h a t
some c o n d i t i o n s on
i s o f t y p e ( S ) ( Q r( S + ) )
r.
and A  c l o s e d w . r . t .
L e t us n o t e t h a t i f A and N a r e o f t y p e (S+), s o i s If
X"
demicontinuous,
i s of t y p e ( S ) .
A+N
A:
N
A+N.
which imply
a r e p r o v i d e d by Examples
1.3
1  4 w h i c h a l s o t r e a t i n t e r t w i n e d monotone and s t r o n g l y monotone nonlinear perturbations.
Explicity,
l e t u s l o o k now a t
semiball
condensing p e r t u r b a t i o n s .
We c a n e a s i l y d e d u c e t h e f o l l o w i n g g e n e 
r a l i z a t i o n o f Example 1 . 1 0
(see [28]).
EXAMPLE 1 . 1 2 . i z 0,
D c X
(a)
Let
A:
D(A)
open and
N:
5 +
ballcontractive w i t h
k8 < c
= V c X
C(Y)
+ Y
U.S.C.
b e Fredholm o f i n d e x and e i t h e r semik
or s e m i  b a l l  c o n d e n s i n g i f
8 = c 1,
316
c
where
and A  c l o s e d
+ C(Y)
kb < c
is
.
MILOJEVI~
a r e a s i n Example 1.10 ( a ) .
Then
5
A+N:
+ ($3
fl V
FA.
w.r.t.
p a r t ( b ) o f Example 1.10 i s v a l i d i f
Analogously,
(b)
6
6
and
is Aproper
N:
s
P.
and e i t h e r s e m i  k  b a l l  c o n t r a c t i v e
U.S.C.
or s e m i  b a l l  c o n d e n s i n g
6 = c = 1,
if
with
b
where
c
and
a r e as i n that p a r t ( b ) . F i n a l l y f o r m o n o t o n e type p e r t u r b a t i o n s o f F r e d h o l m m a p p i n g s w e have t h e f o l l o w i n g r e s u l i s o f t h e a u t h o r [ 26,273
EXAMPLE 1.13. K:
X
mapping
Let
and
X
be r e f l e x i v e B a n a c h s p a c e ,
Y
a l i n e a r homeomorphism and suppose a l i n e a r c o n t i n u o u s
Y*
t
(a)
A:
X
t
satisfies condition (1.3).
Y
Then
is Kquasibounded.
weakly Aclosed
= ( y , ~ x ) for (b)
Let
w.r.t.
x X,
E xn, Y,
A+N
If
Y
weakly Aproper
EXAMPLE 1 . 1 4 .
(a) o r part
ra
w.r.t.
T'
and
X,
Y,
N
X
(a),
Y
be
and
(Qny,Kx)
A:
X
+ Y
demi
pseudo K  m o n o t o n e
Y
t
X + Y
A+N:
is weakly Aproper
ra. i(A)
= 0
K,
ra,
Then
and
+
X
t
is Bquasithen
A+N
is
rB. N
be e i t h e r a s i n p a r t
G:
X
t
tN
+
pG
Y
be bounded, d e m i 
i s strongly A  c l o s e d
i s o f t y p e (KM) and i s A  p r o p e r
i n o t h e r cases f o r each
Y
i s a s above,
w.r.t.
A
A
N:
and
w h e r e B = A+C
(b) o f Example 1.13 and
if
for w h i c h
Yn,Qn}
N:
and w e a k l y A  c l o s e d
Let
t
is weakly Aproper
be a s i n p a r t
Then
w.r.t.
c l o s e d and of t y p e ( K S + ) . w.r.t.
Ta
(KS+)
(BM),
X
E Y.
is H i l b e r t ,
bounded and of t y p e
X + Y
= {Xn,Vn;
Kquasibounded.
and w e a k l y A  c l o s e d (c)
y
and
K
c o n t i n u o u s and o f t y p e with
ra
A+N:
N:
Let
or o f t y p e (KM) and s u c h t h a t
e i t h e r g e n e r a l i z e d pseudo K  m o n o t o n e A+N
.
p > 0
and
and A  c l o s e d
t > 0.
CONTINUATION THEORY
3 17
CONTINUATION THEOREMS FOR APROPER AND STRONGLY ACLOSED MAP
2.
PINGS
In this section we shall prove a number of continuation results for Aproper like and strongly Aclosed like mappings. Most of the results have been first announced in a written form in [28]
(without proofs).
Applications of the results of this section
to equations involving nonlinear perturbations of Fredholm linear Their applications to other
mappings can be found in Section 3 .
classes of nonlinear mappings like, for example, ballcondensing pergurbations of (strongly) accretive,or Kmonotone mappings, etc., were given in [28] and will be published elsewhere. Throughout this section and
r' = {En,Vn,Fn,Wn]
V
X
shall denote a subspace of
an admissible scheme for
(V,Y).
T o facilitate the statements of our results, we separate the following condition on If for some
(2.1)
H(t,x)
tk E (0,l)
defined in
with
tk + 1
(g) I 0 for some yk E H(tk,Vnk(unk)) wnk there are zk E H(l,Vnk(%)) such that k
+
x E 5
n V
if
H(t,x)
and a bounded sequence
and
Y,
in
g
Wnk(zk)
a(H(tn,x),
is acontinuous at H(1,x))
tn + 1 uniformly for
that if, and
(g n V):
 k"'
then as
(g)
a.
We say that
as
[O,l] x
H(t,x)
n
for
x E
We say that a mapping
V.
Exn]
1
uniformly for
X
x
E
n
V
then (2.1) holds.
n V + K(Y)
is bounded in
E H(tn,x))'+O
It is easy to show (see [29])
6 n Vn(En),
T: 6
uniformly for
SUP {d(y,H(l,x));y
is acontinuous at
WnH(t,Z) E C(Fn)
(*) if, whenever
5
x E
5
1
and
satisfies condition d(f,Txn)
+ 0 for
318
P
some
f
in
.s . M I L O J E V I ~ x E
t h e n there i s some
Y,
5 n
V
f E T(x).
such t h a t
Our r e s u l t s a r e b a s e d on t h e d e g r e e t h e o r y f o r u . d . c . pings
T:
a s d e v e l o p e d i n [ 151, [ 1 8 ] .
En + K(Fn)
C
of our f i r s t r e s u l t h a s b e e n o b t a i n e d i n [ 2 9 ] ,
Part
map(a)
and t h e o t h e r p a r t s
i n [ 281.
H:
+
[O,l]x(EflV)
(2.2)
D t X
Let
THEOREM 2 . 1 .
K(Y)
b e open and bounded,
Co,el x
(aDnv)
H(t,x)
for
[o,I.]
t E
f
(2.4)
tf
(2.5)
deg(WnHoVn, Vn1( D n V ) ,
H(0,x)
Suppose t h a t ( 2 . 1 ) (a) f
E H(1,x) (b)
f
If
If
E H(1,x) (c)
If
H1
f
PROOF.
t i e s of (2.6)
t E [O,l]
holds with
0)
f
on
n
x E aD
and
V;
n
V;
for a l l l a r g e
0
g = f.
n.
Then,
r,
w.r.t.
the equation
i s f e e b l y approximationsolvable. H1
r,
i s s t r o n g l y Aclosed w . r . t .
the equation
i s solvable.
H1
t E
f o r each
E H(1,x)
for
x E aD
and
i s A  p r o p e r and A  c l o s e d
s a t i s f i e s c o n d i t i o n (*),
a t 1 u n i f o r m l y for
r
g i v e n and
(o,~);
o E
f o r each
r
homotopy w . r . t .
and Aclosed
(2.3)
d
Y
such t h a t
i s a n Aproper
H
in
f
6 rl
x E
V
and
( t o , l ) and some
Ht
to,
H(t,x)
i s acontinuous
i s s t r o n g l y Aclosed w . r . t . then the equation
i s solvable.
A r g u i n g by c o n t r a c t i o n , i t i s e a s y t o s e e t h a t t h e p r o p e r Ho
tWn(f)
and ( 2 . 4 )
imply t h a t t h e r e e x i s t s a n
WnH(O,Vn(u))
for
u E aDn,
nl z 1
t E [O,l],
such t h a t n 2 nl.
T h e r e f o r e , by t h e homotopy theorem f o r t h e f i n i t e d i m e n s i o n a l
319
CONTINUATION THEORY
degree ([15],
[18] ) , we obtain that for
Now, let
(0,l)
e E
be fixed.
n 2 n
1'
Then, arguing again by con
tradiction, we see that the Aproperness and Aclosedness of
H: [O,e] x (aD n2 = n2(e)
(2.7)
n
V)
such that for
nl
;2
wn(f) a'
and (2.3) imply that there exists an
WnH(t,vn(u))
with
n2(C ) 2 n2(C 1)
topy
Fn: [O,l]xO,
n z n 2'
e > e 1,
whenever
+ K(Yn)
t E C0,cl
E aDn,
for
given by
,
Using ( 2 . 7 ) and the homo
Fn(t,u) = WnH(Ot,Vn(U))Wn(f)t
we obtain that
for each
t
E
there exists
E Dn
un
Next, let
there exists
 w
y,
n z n2.
and
(O,e]
such that
E H(l,Vn
n z n2,
E WnH(e ,Vn(un)).
Wn(f)
be increasing and tending to
ek E (0,l)
k(i)
Therefore, for each
such that
u ) k(i) nk(i)
W %(i)
1
(Y,
and
k(i)
)

(f) + 0.
Therefore, the equation
f E H(1,x)
is feebly approximation
solvable if (a) holds and just solvable, if (b) holds. Suppose that (c) holds. for each
n
2 n2(C)
and
C
is solvable in
5 n
d(f,H(l,%))
a(H(ek,xk),
5
E
Then, since (to,l)
Wn(f)
E WnH(e,Vn(un))
fixed, the equation f E H(e,x)
V. Let e k + 1 increasingly and f E H(ek,xk).
tion ( * ) , there exists
H(l,xk))
x E 5
When a given homotopy
n
V
+ 0
as
such that
H(t,x)
k +
Then
and, by condi
f E H(1,x).
is not Aproper, one often
320
P.
.
s MILOJEVI~
needs the following extension of Theorem 2.1. THEOREM 2.2.
Let
D
such that for a given (2.8) (2.9)
in
f
there exists an
Y
Wn(f) @ WnH(t,Vn(u))
Suppose that (2.5) h o l d s .
ann,
nf
f
in
( + ) if
t E [O,l),
n z nf
for
u E D ~ ,t E C0,1l,
n 2 nf.
Then the conclusions of Theorem 2.1 hold.
yn E H(tn,xn)
H: [O,l]xV
A mapping
[xn] C V
with
f
E H(1,x)
we need
Y,
DEFINITION 2.1.
1
u E
To treat the (approximation) solvability of for each
;r
+ K($
for
w,H(o,v,(u))
twn(f) q'
H: [O,l]X(fhV)
be as in Theorem 2.1 and
is bounded whenever
with
+ K(Y)
satisfies condition
yn + f
f o r some
tn E [O,l].
From Theorem 2.1 one easily obtains the following Let
COROLLARY 2.3.
(2.1),
(2.2) hold on
D = B(0,r)
[O,l]xV
H:
[0,1)
>
for each
r
in
Y.
hold for each
f
0.
x
b
K(Y)
V.
satisfy condition ( + ) and
Suppose that ( 2 . 5 ) holds with
Then the conclusions of Theorem 2.1
The degree condition ( 2 . 5 ) holds if, for example,
5,
odd on a symmetric with respect to zero set Ho(x) fl Ho(y)
= 6
whenever
x
#
y
(cf. [28]
Ho
is
o r injective, i.e.
for other types of Ho).
Moreover, we also have the following result, first proved by the author in [23] in the finite dimensional case. PROPOSITION 2.4 ping large
G: fj fl V n,
([26]).
+' K ( Y )
Q,Gxc
Gx
Suppose that there exists an u.d.c. such that for
deg(QnG, D fl Xn, 0) f 0
x E 6 fl Xn
and
mapfor all
(QnY,KX) = (Y,Kx)
for
CONTINUATION THEORY
y E Y
x E Xn,
K: X
and some
u E H~(x), v E GX, x E
(u,v)+ > /IuIIIlvII for
Ho
Then
Suppose that
Y".
3
321
a(Dnv).
satisfies condition (2.5).
We continue our exposition with a second type of continuation results for Aproper and strongly Aclosed mappings. of the theorem has been announced in [21]
aD
n
V
THEOREM 2.5.
Let
H: [O,l]x(hV)
3
D
0, which turns out to be suitable in
at
various applications (cf.
while the rest appeared
We note that we require that (2.10)
for the first time in [28]. below holds on
C 211 ,
[ 221 ).
be as in Theorem 2.1. and a given
K(Y)
Part (a)
Y
in
f
Suppose that satisfy the following
conditions : (2.10)
H
is an Aproper and Aclosed homotopy at
r
on
[O,e]
x
(aD
(2.12)
tf $ H(1,x)
(2.13)
deg(WnHoVn, Vil(D),
for
n
0)
f
0
aD
n V
If H1
f E H(1,x) (b)
If
0 E H(1,x)
(c) at
1
If
V,
w.r.t.
e E (0,l).
for each
V)
x E aD
Suppose that (2.1) holds on (a)
n
0
t E [0,1]
n.
for all large with
g = 0.
is Aproper and Aclosed w.r.t.
Then,
r,
the equation
is feebly approximationsolvable.
H1
is strongly Aclosed w.r.t.
r,
the equation
is solvable without (2.12).
H1
satisfies condition (*),
uniformly f o r
x
E
5 n
V
and
IIt
H(t,x)
is acontinuous
is strongly Aclosed for
322
MILOJEVI~
P.S.
t E
(to,l),
the equation
0
E H(1,x)
is solvable without assuming
(2.12).
PROOF.
(a)
Arguing by contradiction and using the properties of
we obtain an
H1,
(2.14)
t Wn(f)
nl
#
5:
1
such that
WnH(l,Vn(u))
for
u E aDn,
Moreover, we claim that there exists an
o # w,H(~,v,(u))
(2.15)
for
If not, then there would exist
u
n2
tk E [O,l]
n
condition (2.1) some subsequence
tk
H(t,x)
t
( k , ' i) which leads to a contradiction with (2.11).
n 1'
n
n
2'
u
t
to.
at
0
we would get a contradiction to (2.11).
V),
2
E aD such nk nk In view of (2.14), we and
0 E W H(tk,Vnk(un ) ) f o r each k . nk k have that each tk < 1. We may suppose that
by the Aproperness and Aclosedness of
n
such that
nl
2
E aDn, t E C0,ll
that
(aD
t E [O,l],
with
0
#
If
to
on
[O,e]
1,
x
If
to = 1, by
y
E H(l,V%u\),
"k
Hence, (2.15) holds
and, in view of (2.14) and the homotopy theorem, we obtain for each
Therefore, (a) follows now easily from the properties of (b)
each
Let us first suppose that for some
n
2
n2
1,
2
E aD
"k
If each If
H1.
nk
condition
tk E [O,l],
tk + to
and
un E [O,l], u E k nk 0 E W H(tk,Vnk(unk)) for each k. nk
tk = 1, we are done by the pseudo Aclosedneas of
to f 1, we get a contradiction as in (a).
and by (2.1) some subsequence y
1,
Next, if (2.15) does not hold for any
then there would exist with
2
and consequently the conclusion follows from the
strong Aclosedness of n2
n2
H1.
E H(1,V
un ). n k k
W
%(.i). (Ynk(i)
)
Hence, t
0
H1.
to = 1
with
The conclusion now follows from the strong
CONTINUATION THEORY
Aclosedness of (c)
Let
of (2.10), n
B
H1'
c E (0,l)
be fixed.
(2.11), we obtain an
Then, as in Theorem 2.1, in view such that for each
nl = nl(e )
nl
(2.16) with
323
0
@ WnH(t,Vn(u))
nl(e)
for
E aDn,
t E [O,E1,
being an increasing function of
n
the homotopy theorem we obtain that for each deg(WnHeVn,Dn,O) Thus, the equation n
2
e E
nl(c) (to,l).
0 E H(e ,x)
ek
Next, let
1
+
Then, by the acontinuity of there exists an
x E
n
V
n1
2
= deg(WnHoVn,Dn,O)
0 E FnH(e,Vn(u))
and therefore
Consequently, by
0 .
such that
Dn
for some
n
x E 5
at
1
for each
V
with
0 E H(ck,xk).
and condition ( * ) ,
0 E H(1,x).
When one does not have the Aproperness of following more general result holds.
0.
is solvable in
increasingly and
H(t,x)
f
H(t,x),
the
Its proof is essentially
contained in the proof of Theorem 2.5. THEOREM 2.6.
H: [O,l]x(%lV)
Let
D
+ K ( Y )
be as in Theorem 2.1, and
f
such that for each large
n
Suppose that (2.13) holds.
Then the equation
feebly approximationsolvable (solvable) if Aclosed (strongly Aclosed).
in
Y
f E H(1,x)
H1
and
is
is Aproper and
Moreover, the conclusion
(c)
of
Theorem 2.5 holds without assuming (2.18). Finally, let us consider now a third general continuation
324
MILOJEVI~
P.S.
type result.
Its parts (a) and (b) were essentially proven in [27]
(cf. also [ 2 6 ] ) . THEOREM 2 . 7 . f
Y
in
Let
D
be as in Theorem 2.1.
+
H: [O,l]x(l%V)
and
Suppose that a given
satisfy the following condi
K(Y)
tionr
tf
(2.19)
q'
H(t,x)
x E aD
for
n
t E C0,lI.
V,
Suppose that ( 2 . 2 ) and ( 2 . 5 ) hold and (2.1) holds with
g = f.
Then, (a)
If
H1
f E H(1,x)
If
(b)
f E H(1,x)
If
(c) 1
t E
(to,l),
PROOF.
H1
H1
satisfies condition (*), x E 6
the equation
E (0,l)
n
V
nl(e)
be fixed. nl = n l ( e )

for each
nk
(y,
Then (2.1),
k
) = e
w
knk
/lWnk(ynk)
is acontinuous
is strongly Aclosed for
is solvable.
Then, using conditions ( 2 . 2 )
2
such that for each
1
for
n 2 n.1
Therefore,
e.
#
and
[o,e]
t E
u E aDn,
Wnf,Dn,O) = deg(WnHoVn,Dn,O)
for
0
Wn(f) E WnH(e,Vn(u))
n z nl,
is solvable
n 2 nl.
ek + 1
Next, let
w
Ht
H(t,x)
being an increasing function of
and consequently the equation Dn
and
f E H(1,x)
tWn(f) @ WnH(t,Vn(u))
deg(WnHeVn
in
equation
is solvable.
(2.19), we obtain an
with
r , the
is strongly Aclosed w.r.t.
.Let e
(2.20)
the equation
is feebly approximationsolvable.
uniformly for
at
I?,
is Aproper and Aclosed w.r.t.
increasingly and
(f)
for some

(f)l
wnk
some subsequence
Ynk E H ( e
= (lek)llW W
nk
( 2
%(i)
nk(i)
(f)l
E Dn
unk
ktV%
(u
t 0
1  w
%(i)
nk
such that k ) ) and each k .
as
k +
(f) +'
o

and, by for some
CONTINUATION THEORY
325
E H(1,V (u ) ) . Therefore, the equation f E H(1,x) nk( i) %(i) %(i) is feebly approximationsolvable if (a) holds, and just solvable if Z
(b) holds. Now, suppose that (c) holds. un E Dn
for some
zn
solvable in and
xk E fj
n
Y
n z nl,
and each
V
with
E E
auch that
+ and, by condition (*),
Wn(ef)
ek
Let
ekf E H(ek,xk).
(lek)l)fll
E WnH(e,Vn(un))
E f E H(E ,x)
the equation
(to,l).
d(ckf,H(l,xk))
d(f,H(l,xk))
Since
+ 1
increasingly
Then
+ I)ckffl)5 a(H(ek,Xk),H(l,xk)) +
as
0
there exists an
is
k +
x E
t
m
V
such that
f E H(1,x).
COROLLARY 2.8. for each
>
r
Suppose that condition (2.5) holds with H(t,x)
0,
satisfies condition ( + ) instead of (2.19)
and that all other conditions of Theorem 2.7 hold. clusions remain valid f o r each PROOF.
f
in
tion ( + ) .
D = B(O,rf)
Then its con
Y.
It suffices to note that for each
(2.19) holds with
D = B(0,r)
f
in
rf > 0
for some
Y
condition
in view condi
8
As before, when
H(t,x)
is not Aproper, we have the fol
lowing useful extension of the last result. THEOREM 2.9.
Suppose that f o r a given
H: [O,l]x(cnV)
+
K(Y)
f
in
Y
and
the following condition holds for all large
n (2.21)
tWn(f)
< WnH(t,Vn(u))
Suppose that (2.5) holds.
for
u E Dn, t E
fo,ll.
Then the conclusions of Theorem 2.7
remain valid. The rest of the section will be devoted to establishing
P. s
32 6
. MILOJEVI~
various (approximation) solvability criteria for equations of the form f E Ax
(2.22)
using Theorems 2.7
+
Nx
( x E ~ ~ fey) v ,
and 2.9.
These results will be used in studying
In all
nonlinear perturbations of Fredholm mappings in Section 4 . our results below, we can allow
to be the zero mapping.
A
V
Throughout our discussion we assume that Banach space, and
r
an admissible scheme for
= {En,Vn; Fn,Wn]
(possibly multivalued).
K : X + Y*
is a subspace of a (V,Y)
We need the following
condition
(W~Y,KX)= (Y,KX)
(2.23)
Our first result for E q . THEOREM 2.10. and
N: 5
Let
n v +
(2.22)
Dc X
x E
(a)
If
A+N
A,C: 5
be open and bounded,
o
and
(CX,KX) > o
Aproper and Aclosed w.r.t. (0,l) and ( 2 . 1 )
E Y.
is
Suppose that either the homotopy
c E
v~(E~), Y
n
V
b
Y
such that
K(Y)
(AX,KX) 2
(2.24)
for
holds with
H(t,x) on
for
x E aD
= Ax + tNx
[O,€] x
g = f,
+
(aD I7 V )
or ( 2 . 2 3 )
is Aproper and Aclosed w.r.t.
n
V;
(1t)Cx f o r each
holds.
r0,
is
Then
Eq.
(2.22)
is feebly approximationsolvable; (b)
If
A+N
is strongly Aclosed w.r.t.
A+N
satisfies condition (*),
T,
Eq.
(2.22)
is
solvable ; (c)
If
H(t,x)
is acontinuous
CONTINUATION THEORY
at
1
for
uniformly for
t E (to,l),
PROOF.
x E
and
Eq. (2.22)
Ht
is strongly Aclosed w.r.t.
n
x E aD
Ax + ty + (1t)Cx = tf
But, since
(Ax,Kx)

y E Nx.
such that
We may assume that t < 1
Then
(AX,Kx) + t(yf,Kx)
(2.27)
Supposing not, there
t E [O,l]
and
V
for some
for otherwise we are done.
r
is solvable.
We show first that (2.19) holds.
would exist some
327
= (tl)(Cx,Kx)
O
and by (2.24) it follows that (Ax,lCx) in contradiction to (2.27). Now, if for each
g
H(t,x)
E (0,l)
+
t(yf,Kx) > 0,
Therefore, (2.19) holds.
is Aproper and Aclosed on [O,C] X ( k V )
and (2.1) holds with
g = f,
the conclusions
of the theorem follow from Theorem 2.7. Next, suppose that (2.23) holds and let us assume first that either (a) or (b) holds. large
n.
Then we claim that (2.21) holds for all
If not, then there would exist
such that for each
\
E aD
nk
and
tk E r0,lI
k 2 1
for some
yk E NVn (uk). If tk = 1 for infinitely many k , we k S o , we may assume that tk < 1 for each are done by (a) o r (b).
k
which, by condition (2.24), implies that
(2.28)
(AV, uk, KVn uk) k k
= (tkl)(W
"k
cvn
+ tk(ykf,
kuk'
KV
u ) =
"k
KV uk) = (tk1)(CV nk
nk
uk, KVn uk) < 0 k
328
P.
for each
k
and
s.
MILOJEVI~
tk f 0 by (2.24). Uk$ KVn uk) k k

Now, let
k
be fixed.
*
(Ykmf, KVn uk)
k
'
Since
9
we obtain by adding the last two inequalitites that (tkl)(ykf, KVn uk) < 0. k (2.24) it follows that
Hence,
(Avnks, KVn Uk) k in contradiction to (2.28).
+
tk(ykf,
KV
t(ykf, Kv
u
nk
) > 0 and by
Uk) >
"k
Therefore, (2.21) holds for each
n
large and the conclusions in cases (a) and (b) now follow from Theorem 2.9. Finally, still under condition (2.23), assumptions of (c) hold. (0,l)
e E
(2.29)
Then, as above, we obtain that for each
there exists an
tWn(f)
with
nl(e)
each
n 2 nl
@
n1 = nl($)
WnH(t,Vnu)
for

t E
(to,l),
for each
whenever
u E aDn,
t E [0,6], 6 .
n
2
n1
Therefore, for
$Wnf,Dn,O) = det(WnHoVn,Dn,O) f 0
and consequently, the equation
Dn
such that
being an increasing functions of
deg(WnHeVn
in
suppose that the
n z nl.
the equation
c E (to,l).
Wn(cf)
Since
Ht
E WnH(e,Vnu)
is solvable
is strongly Aclosed for each
e f E H ( $ ,x)
is solvable in
The solvability of
f E H(1,x)
5 n V now follows
as in Theorem 2.1 (c). REMARK 2.1.
If we had that
Kx = 0
only if
x = 0,
then the
second part of the proof of Theorem 2.10 can be significantly simplified.
Namely, now (2.21) follows easily from (2.19) and (2.23).
Moreover, when (2.23) holds in Theorem 2.10, it is sufficient to assume that (2.24) and (2.25) hold on
aD
n
Vn(En)
for each
Often in applications one is in a situation of
X
n.
embedded
CONTINUATION THEORY
in some Banach space
l ~  ~be~ the o
Let
Z.
329
norm of
The fol
Z.
lowing result is an application of Theorems 2.7 and 2.9. THEOmM 2.11.
Let
A,C: V
(2.90)
(Ax,Kx) 2 0
and
(2.31)
For a given
f
R
IlxlIo
Y
+
(Cx,Kx) > 0
Y
in
+
K(Y)
for
0
f
be such that x E V;
there are positive numbers
such that if either
I(xI1 = R
and
IIxIIo < r,
r
and
or
then
2 r9
(AX,KX) + (Y,KX) Let
N: X
and
D = {x E V
1
(~,Kx) for
r
(2.1) holds with
= Ax
H(t,x) on
+
tNx
(1t)Cx
x (aD n V )
[O,o]
g = f,
+
y E NX.
and (2.5) hold.
]IxIl < R, ]Ix]Io< r]
either the homotopy Aclosed w.r.t.
2
or (2.23) holds.
Suppose that
is Aproper and
for each
o E
(0,l)
and
Then the conclusions of
Theorem 2.1 hold.
PROOF. [O,s]
Suppose first that
x (aD n V )
for
H(t,x) (0,l)
e E
is Aproper and Aclosed on and (2.1) holds with
g =
f.
Then, in view of Theorem 2.7, it suffices to show that (2.19) holds with
D
tf E H(t,x) Then
IIxIIo
t E [O,l]
for some
< r,
y E Nx
x E aD.
and
We may assume t < 1.
for otherwise we would have by (2.25) that

(AX,=)
(2.32) and, for
If (2.19) did not hold, then
as defined above,
(yf, KX) i
with
+
Ax
ty
+
o
for
y E NX
(1t)Cx = tf,
we would have as
before that (2.33)
(AX,KX) + t(yf, Kx) < 0.
Adding the last two inequalities we get that and consequently
t(yf,KX)
(Ax,Kx) + t(yf,Kx)
>
0,
(tl)(yf,=)
< 0
> 0. By (2.24) it follows that
in contradiction to (2.33).
Thus,
330
P.S.
IIXI/~< of
r
l(xl/ = R
and
(2.3),
MILOJEVIC
and ( 2 . 3 1 )
holds f o r t h i s
Now, i n view
X.
we g e t a c o n t r a d i c t i o n a g a i n a s a b o v e .
Hence,
(2.19)
holds. Suppose now t h a t ( 2 . 2 3 ) h o l d s and c o n s i d e r t h e p a r t s ( a ) and Then i t : s u f f i c e s t o show t h a t
(b) simultaneously. view o f Theorem 2.9.
tk
E
[0,1]
\ E
tk
f
0.
Let
k
a s above and c o n s e q u e n t l y
be f i x e d . IlV
k Hence,
(AVn
k
Uk, KVn Uk) k

nk
(Ykf,
Then
uk//= R
by ( 2 . 3 1 ) .
KVn Uk) 2 0
k
a s i n Theorem 2.10.
which l e a d s t o a c o n t r a d i c t i o n t o (2.28) fore,
t h e n t h e r e would e x i s t
aDn
a s i n Theorem 2.10 and (\Tin uk/(< r
(2.21) d i d not hold,
holds i n
such t h a t t W ( f ) E WnkH(tk,Vn uk) k "k k One may assume t h a t e a c h t k < 1 and o b t a i n s ( 2 . 2 8 )
and
k.
f o r each
If
(2.21)
There
(2.21) holds. Finally,
s t i l l under (2.23),
s u p p o s e t h a t t h e h y p o t h e s e s of
( c ) hold.
Then we o b t a i n a s above t h a t f o r e a c h
exists
= nl(e)
nl
theorem and as i n part
such t h a t
nl(f)
(2.29)
increasing w i t h
e E (0,l)
holds w i t h E.
D
there
as i n the
The c o n c l u s i o n now f o l l o w s
( c ) o f Theorem 2.10.
Now we d e d u c e t h e f o l l o w i n g c o u p l e o f c o r o l l a i r e s u s e f u l i n applications. COROLLARY 2.12.
(2.34)
Suppose t h a t ( 2 . 3 0 )
F o r a given (Ax,&)
+
f
in
(y,Kx) 2
Y
holds a s well a s
t h e r e i s an
(f,Kx)
for y
E
r > 0 Nx,
X
E
such t h a t
V , ) / x / /= ~r;
331
CONTINUATION THEORY
m =
(2.36)
inf (Y,KX) ~vV,lIxllO
m
+
tNx
Y€ Nx
H ( t , x ) = Ax
Suppose t h a t t h e homotopy
r
w.r.t.
and A  c l o s e d
6n
and ( 2 . 1 ) h o l d s on IIx(Io < r}
on
x (aD
[O,o]
g = f
for
V
IIxIlo < r .
x E V,
R.
Let
+
(yf,Kx)
Ex
D =
E
E
v I
E (0,l) I/XII
< R,
R's.
It s u f f i c e s t h a t ( 2 . 3 1 ) h o l d s f o r
a n d some
i s Aproper
f o r each
Il V )
with
hold providedalso (2.5) holds f o r such PROOF.
(1t)Cx
Then t h e conc u s i o n s o f Theorem 2 . 1 1
R.
f o r each l a r g e
+
/[XI/
= R
Then ( 2 . 3 5 )
and
/Ixllo
< r
implies t h a t f o r
Y E Nx, (Ax,&)
I)xII L R ,
whenever

(allfll + c ) k

B
al/x1I2

d
+
h o l d s w i t h such a chosen COROLLARY 2.13.
R
where
br2
2 blIxllo

clIx/I

d + m
 ullfl~~lx~~ 2 0
c ( k ) = ak2
i s such t h a t t h e f u n c t i o n
m 2 0
k B R.
for
Therefore,
(2.31)
R.
Let (2.23),
Suppose t h a t ( 2 . 5 )

(2.3)
holds with
D
(2.34)s
(2.35)
and ( 2 . 3 6 ) h o l d .
a s i n C o r o l l a r y 2.12
f o r each
R
large.
Then t h e c o n c l u s i o n s of Theorem 2 . 1 1 a r e v a l i d .
PROOF.
It s u f f i c e s t o o b s e r v e t h a t ( 2 . 3 1 ) h o l d s , w h i c h was shown
in C o r o l l a r y 2 . 1 2 . F o r q u a s i b o u n d e d mappings of r e s u l t s
Suppose t h a t ( 2 . 3 0 ) h o l d s and
For a g i v e n bx,Kx)
IlxIIo2 r;
we h a v e t h e f o l l o w i n g c o u p l e
.
THEOREM 2.14.
(2.38)
N
f
+ (y,Kx)
in 2
Y
there i s an
(f,Kx)
for
r > 0
y E Nx,
x
such t h a t
E
V
with
P.
332
(2.39)
I)Y~/
aI(xl1
x
[O,c]
(aD
n
as well a s (2.1) with
= R(cl)
for
5n
on
r
V
such t h a t
with
g = f,
< r,
I)xl/
\lxll z R1.
i s Aproper and Aclosed e E
f o r each
for
(2.5) holds
(O,l),
I
D = Ex E V
where
)Ixl/
To t h a t e n d , tWnk(f)
IIVnkukl/ s M
n2.
vn
2
= deg(WnHoVn+pWnGVn,Dn,O)
fact and ( 3 . 4 ) imply that Vk E H(l,Vn
n
n 2 n
yk + vkzk = f
pk
t
0
decreasingly and
for some
yk E H(l,xk),
xk E
and by condition (*) there exists a
x E
6 n
V
zk E Gxk.
6 n V
be such
Then yk
t
such that fEH(1,x).
Often one needs the following extension of Theorem 3.1 when
Ht
+
pG
is not an Aproper and Aclosed homotopy.
Its proof is
essentially contained in the above proof. Let
THEOFEM 3.2.
bounded, exists an
Dc X
H: [O,l]x(%V) no = n(f,p)
be open and bounded, t
K(Y)
f E H(1,x)
f
in
Y
6 n
V
t
BK(Y)
such that there
with
Suppose that (3.5) holds and the equation
and
G:
H1
satisfies condition (*).
is solvable.
Then
f,
P.S.
338
MILOJEVI~
An easy consequence of Theorem 3.1 is the following surjectivity result. THEOREM 3.3.
We omit its proof. Let
G: V + BK(Y)
be bounded,
H: [O,l]XV
satisfy condition ( + ) and (3.1), (3.4) hold on
p E ( 0 , ~ ~ ) Suppose . that
r,
Aclosed w.r.t. with
D = B(0,r)
H1
kG,
x
V
K(Y) for each
is strongly
p E (O,lo),
satisfies condition ( * ) and (3.5) holds
H1
for each
is solvable for each
t
[O,l)
+
f
in
>
r
Then the equation
0.
f E H(1,x)
Y.
Theorems 3.13.3 extend the corresponding results of Milojevi6Petryshyn [30],
where one can find a number of applications to par
ticular classes of uniform limits of Aproper mappings and references to the relevant works of other authors. tions that imply (3.2)(3.4)
Some specific condi
have been discussed in [SO].
Using
Theorem 3.1, one can study the perturbed equation of the form f
E H(1,x)
+
in a manner similar to that in [30].
R(x)
We continue o u r exposition by looking at an analogue of Theorem 2.5 for uniform limits of Aproper mappings. result was first announced in [22]
bounded,
Let
X
C
be open and bounded,
H: [O,l]x(%lV) + K(Y), t pG
on
(3.11)
D
Suppose that
f
in
Y
and
G:
5 n
V + BK(Y)
po > 0
such that
is an Aproper and Aclosed homotopy at 0
x
[O,c]
0 !$ H(t,x)
t
Then,
(aD
bGx
(3.4) with
condition ( * ) .
and its special cases have been
f 221.
discussed also in [Zl], THEOREM 3.4.
The following
n
V)
for
g = 0
for each
x E aD
n
V,
E E
(0,l)
t E [O,I],
and (3.5) hold and
H1
w.r.t.
and
p E
( 0 , ~ ~ ) ;
satisfies
CONTINUATION THEORY
If
(a)
the equation
f E H(1.x)
be fixed.
n1 2 1
+
pWnGVn(u)
As in Theorem 2.5 (a) we obtain an 0 a' WnH(t,Vn(u))
(3.14)
n
Therefore, for each
+ 2
n2
pWw,GVn(u)
for
u E aDn,
u E aDn,
for
f E H(1,x)
+
+
f E H(1,x)
decreasingly and
2
n2
t E t0,lJ.
#
xk E
5 n
pGx V
0
and
is solvable.
be a solution of
This and condition ( * ) imply that
pkGx.
n
n2,
consequently, the equation
pk + 0
1041.
t E
such that for
nl
2
n 2 n 1
deg(WnHIVn+bWnGVnWnf ,Dn,O) = deg(WnHoVn+BWnGVm,Dn,o)
Let
(0,~~)'
c~ E
Then it follows from (3.12)
such that for each
tWi(f) ! $ WnH(l,Vn(u))
(3.13)
for
is solvable without assuming (3.12).
p E (O,po)
that there exists an
for
is solvable.
is strongly Aclosed w.r.t.
0 E H(1,x)
(a) Let
r
is Aproper and Aclosed w.r.t.
H1 + pG
If
PROOF.
pG
the equation
p E (O,po),
(b)
+
H1
339
i ' E H(1,x)
is solvable. (b)
Let
M E ( 0 , ~ ~be) fixed.
Theorem 2.5 (b), we obtain that The solvability of
0 E H(1,x)
Then, using the arguments of 0 E H(1,x)
+
pGx
is solvable.
now follows as in (a).
Analyzing the proof of Theorem 3 . 4 , we see that the following its extension is valid when (3.10) does not hold. THEOREM 3.5. H: [O,l]x(%lV)
for each H1
Let
D C X
be open and bounded,
r
n z no(f,u)
for each
in
Y
and
such that conditions (3.13) and (3.14) hold
+ K(Y) 2
1
each
E ( 0 , ~ ~ fixed. )
satisfies condition (*) and that
w.r.t.
f
H1
+
VG
Suppose that
is strongly Aclosed
p E ( 0 , ~ ~ )Then . the equation
f E H(1,x)
is
solvable. An easy consequence of Theorem 3.4 is the following surject
340
MILOJEVIC?
P.S.
ivity result. THEOREM 3.6.
We omit the proof. G: X
Let
BK(Y)
b
such that (3.10) holds on
[O,E]
Suppose that (3.4) holds with and Aclosed
r
w.r.t.
f
x (V\B(O,R))
g = 0
for each
conditions ( + ) and ( * ) . for each
be bounded and
V,
on
H: [O,l]xV for some
H1 + 1 G
E ( 0 , ~ ~and )
Then the equation
H1
H(1,x)
f E
R
K(Y)
+
>
0
and
is Aproper satisfies is solvable
Y.
in
In the rest of the section we shall extend Theorem 2.7 and the subsequent results of Section 2 to the uniform limits of Aproper mappings. THEOREM 3.7. bounded,
These results have been announced in [27],[28].
Let
Y
in
f
D
C
X
and
be open and bounded, H: [O,l]x(@pJ)
G:
n
V + BK(Y)
such that (3.1),
K(Y)
+
6
(3.4) and ( 3 . 5 ) hold and
Suppose that
u
E
(O,~,)
f E H(1,x) PROOF.
H1
+
C(G
H1
and
is strongly Aclosed w.r.t.
satisfies condition (*).
Then the equation
is solvable.
(Sketch)
Let
E (O,po)
(3.1) and (3.17) one can find
and
E E
nl = nl(&)
(0,l)
Choosing c k
H1 + vG,
+
1
be fixed.
such that for
It follows that the equation c f E H(c,x) + pGx
of
T' for each
Using
n 2 nl
is solvable.
and using also (3.4) and the strong Aclosedness
we obtain that
f E H(1,x)
+
vGx
is solvable.
CONTINUATION THEORY
341
Finally, the conclusion follows using condition ( * ) .
m
As usual, when (3.1) d o e s not hold, one has the following extension of the last result. THEOREM 3.8.
H: [O,l]x(6flV) is an
Let
6 n
G:
K(Y)
b
no = no(f,D)
V
BK(Y)
b
1
2
so that for
f E H(1,x)
and there
E (O,po)
n z no
is strongly Aclosed f o r
satisfies conditions ( * ) .
H1
Y
in
f
u
such that for each fixed
Suppose that (3.5) holds, H1 + p G
E ( 0 , ~ ~ and )
be bounded,
Then the equation
is solvable.
We continue our exposition by looking at various solvability criteria for equations of the form
(3.20)
( x E ~ ~ vf e,y )
f E Ax + Nx
based on Theorems 3 . 7 and 3 . 8 .
These results will be also used in
studying nonlinear perturbations of Fredholrn mappings in Section As usual,
V
X,
denotes a subspace of a Banach space
admissible scheme for
(V,Y)
and
K:
X + Y*
4.
an
a (possibly multi
valued) given mapping. THEOREM
3.9.
bounded, (3.21)
A,C:
D c X
Let
fi n
(Ax,Kx)
V
b
be open and bounded,
Y
and
N:
fi n
V
b
K(Y)
0, (Cx,Kx) > 0 and (y,Kx)
B
G: d
;r
n
V
b
BK(Y)
such that for
0
x E aD
n V,
Y E Gx. (3.22)
F o r a given
(AX,KX)
(3.23)
f
+ (Y,KX) B
For each
p E
Y,
in
(f,Kx)
for
x E aD
n
v,
y
E NX.
(O,po),
deg(Wn(A+C)Vn+CLWnGVn,Vn
1 (D),O) f 0
for all large
n.
342
P.
Suppose t h a t
+
A
p E ( 0 , ~ ~and ) H
+
1G
+
tNx
+
PROOF.
Suppose f i r s t t h a t
Theorem
3.7,
that
= f,
or ( 2 . 2 3 ) h o l d s ,
Then Eq. ( 3 . 2 0 )
(1t)Cx.
( 3 . 1 ) and ( 3 . 4 ) h o l d .
+
UGX
n
x E aD
with
I.(
E
(O,po)
(3.17) d o e s n o t h o l d f o r some
x
E
a D
Then
Ax
+
i s solvable.
Then, i n view of
+
ty
(1t)Cx
+ pz =
tf
n
We may assume
p + 0
and
V
Let
we a r e done b y c o n d i t i o n ( * ) . that
g
for
Suppose t h a t e i t h e r
i t s u f f i c e s t o show t h a t (3.17) h o l d s .
H(1,x)
f !$
I?
(*).
s a t i s f i e s condition
(3.1) and ( 3 . 4 ) w i t h
= Ax
H(t,x)
MILOJEVIC?
i s s t r o n g l y Aclosed w . r . t .
pG
A+N
satisfies
where
+
N
s.
b e f i x e d and s u p p o s e and
V
for some
f o r otherwise
y
E
Nx
t E [O,l). and
z
E
Gx.
Hence ,
(3.24) t
and
(Ax,Kx)
#
+
by ( 3 . 2 1 ) .
0
+
t(yf,Kx)
u(z,Kx)
= (tl)(Cx,Kx) < 0
B u t , s i n c e by ( 3 . 2 1 ) ,

(Ax,Kx)
(yf,Kx)
(3.22)
 ~ ( z , K x )
(Ax,=)
that
pi +
+
t(yf,Kx)
(f) E "'k k, f
E
hen
H(1,x)
holds f o r
uk E a D
"k
+
E
H(1,x)
piGx
by c o n d i t i o n
ui I7 V
uiWnkGVn,(uk)
+
0,
and
0.
h(z,Kx) > 0
holds.
Let
I f f o r each
( 0 , ~ ~b e) s u c h
pi
fixed,
uk E a D n
for
and e a c h
k i s solvable a s i s t h e equation
(*).
Now, a s s u m i n g t h a t
t h e n f o r e a c h such
tk E [ O , l )
f o r each
pi E
i t s u f f i c e s t o show t h a t
(s)) + biWn GVn (u,) k k +
i
(3.17) h o l d s .
c o n d i t i o n (3.19) h o l d s .
W H(l,Vn k nk f
+
I n view of Theorem 3.8,
pi
f o r each
Thus,
suppose t h a t (2.23)
0.
(tl)(yf,Kx)
0 and by ( 3 . 2 1 )
i n c o n t r a d i c t i o n t o (3.24). Now,
0,
k.
ui,
t h e r e would e x i s t
t W
( f ) E Wn H(tk,V%&)) "k k t k < 1 bY t h e above o b s e r v a 
such t h a t Each
(3.19) d o e s n o t
CONTINUATION THEORY
tion.
k
Let
be f i x e d .
Then
+
zk E GV
and
nk
(3.25)
(u,)
pIwn
and,
k
(w A X ~ , K X ~+) "k
setting
(tkl)(Wn
(w
t
tk f 0
contradiction.
k
we g e t
= + pi(wn
k
Z ~ , K X) =
k
= (tk1)(Cxk,KXk) < 0
Cxk,KXk)
by ( 3 . 2 1 ) .
+ Mi(Zk,Kxk)
 wn kf , K x k )
yk nk
k and
(f)
W
"k xk = v n k ( u k ) ,
+ tk(Ykf,Kxk)
(Axk,Kxk)
w
= t
(2,)
(y,) + "k f o r some yk E N V ~(u,)
k
nk
+ (ltk)WnkCVnk(uk)
+ t
(u,)
AVn
W
343
Now, a s i n t h e f i r s t p a r t , we o b t a i n a
(3.19) h o l d s and Theorem 3 . 8 i s a p p l i c a 
Therefore,
ble. I n t h e n e x t few r e s u l t s we assume t h a t embedded in a Banach s p a c e THEOREM 3 . 1 0 . Suppose t h a t with
A,C:
Let (3.21)
D = {x E
V
1
Theorem
3.7,
+
+
K(Y)
V.
.
and
and ( 3 . 2 3 ) h o l d
(3.20)
i s solvable.
I n view of
+
(1t)Cx
have by ( 3 . 2 1 )
(3.17) d o e s n o t h o l d .
bz = tf.
and ( 2 . 3 1 )
(Ax,KX)
tf E H(t,x)
y E Nx Then
and
bounded.
Suppose t h a t t h e o t h e r con
Then E q .
x E a D fl V ,
G
with
(2.31)
and t h a t
( 3 . 1 ) and (3.4) h o l d .
Suppose n o t and l e t
+
X
We h a v e
i t s u f f i c e s t o show t h a t ( 3 . 1 7 ) h o l d s f o r
t E [O,l]
ty
G,N:
V\[O]
be f i x e d and s u p p o s e t h a t
t = 1. Ax
and
3.9 h o l d on
Suppose f i r s t t h a t
some
Y
< R , )IxlI0 < r]
IIxI/
PROOF.
1
+
V
.
1/*110
w i t h t h e norm
h o l d s on
d i t i o n s o f Theorem
Let
Z
i s continuously
X
+
z E Gx
IIxIlo < r
pGx.
p E
( 0 , ~ ~ ) .
Then,
for
We c l a i m t h a t
be s u c h t h a t
f o r o t h e r w i s e w e would
that

(yf,Kx)
 p(z,Kx)
s 0
and (3.26)
(Ax,Kx)
with
t f 0
that
(yf,Kx)
+
t(yf,Kx)
by ( 3 . 2 1 ) .
> 0
+
p(z,Kx) = (tl)(Cx,Kx) < 0
Adding t h e l a s t two i n e q u a l i t i e s w e g e t
and by ( 3 . 2 1 )
P.S.
344
MIL~JEVI~
(AX,KX) + t ( y  f , ~ ~+ ) ~ ( z , K x )> 0 , i n contradiction t o ( 3 . 2 6 ) .
(2.31) holds f o r this
X.
Now,
t r a d i c t i o n a g a i n a s above.
n v,
x E aD
some
f E
+
f E H(1,x)
+ pGx
H(1,x)
Therefore,
pGx
(*), t h e (2.23)
We c l a i m t h a t e i t h e r f o r e a c h
@
CO,ll,
t E
+
WnAVn(u)
x E
5
n
V
ij
i s solvable i n
by c o n d i t i o n
tWn(f)
aD
n
V
+
LGx
t h e equation
p E
f o r each
(O,po).
i t s proof)
(cf.
Thus,
for
c~ E
f o r each
fl V.
( 0 , ~ ~ ) .
f E H(1,x)
equation
i s solvable.
E ( 0 , ~ ~b e) f i x e d .
h o l d s and l e t n
and
( 3 . 2 1 ) , we g e t a conf € H(1,x)
Thus, e i t h e r
f o r some
//x/l = R
and
i t f o l l o w s from Theorem 3.7
Now s u p p o s e t h a t
(3.27)
r
i n view of
or ( 3 . 1 7 ) h o l d s on
I n the l a t t e r case, that
I)X//~
O.
with
then
t o t h i s e q u a l i t y , we g e t t h a t
i n contradiction t o (5.20)or and
i tQoNx
t E [O,l],
and
and
n
for a l l large
0
H p ( t , x ) = tAx
b e f i x e d and
0
#
0)
M o r e o v e r , v a r i o u s non
i s s t r o n g l y Aclosed
f o r some
366
G
P.
s.
MILOJEVI~
h a v e b e e n d i s c u s s e d i n S e c t i o n 1 and l a t e r on i n t h e s e c t i o n .
A s we h a v e remarked i n S e c t i o n 4,
2
(Ax,Kx) 2 aI/x/l
b e weaken i n c e r t a i n c a s e s t o
a
0.
and
Since
N
and A  c l o s e d i s bounded,
the
fi
s a t i s f y (3.1) and ( 3 . 5 ) r e s p e c t i v e l y , a s P S i n c e a l s o A+N s a t i s f i e s c o n d i t i o n ( * ) , t h e con
and
(3.4).
well as
X
G:
i s a s i n Theorem 5.3
fi
Suppose
Then
(3.5)
H(t,x)
(1.3).
i s bounded and g e n e r a l i z e d p s e u d o Kmonotone
Kquasibounded
(KS+).
(b)
b e r e f l e x i v e and s e p a r a b l e Banach
Y
open bounded,
type
where
and
X
a c o n t i n u o u s l i n e a r mapping t h a t s a t i s f i e s
that
of
Let
elusions f o l l o w from Theorem
5.3.
I n t h e above r e s u l t one c o u l d have u s e d t h e homotopy of t h e H(t,x)
+
UG
= Ax
and s u c h t h a t
+
+
tNx
A+C:
V
+
+ 1Gx
(1t)Cx
Y
with
is a bijection
C
compact and l i n e a r
( s e e Theorem 5 . 4 ) .
More
g e n e r a l l y , we h a v e COROLLARY 5.10.'
Let
X,
Y,
K,
A
and
G
be a s i n Corollary 5.9
3 67
CONTINUATION THEORY
G
with
odd.
Suppose t h a t
5.5
+
Ta
and
5.7 i s a p p l i c a b l e . If
N
ra
w.r.t.
satisfies
C = fJlPo
Then E q .
s a t i s f i e s condition
A+N
W G is s t r o n g l y A  c l o s e d
REMARK 5 . 2 .
Kquasibounded,
A+N
has Property (P).
I n e i t h e r case,
Corollary
i s e i t h e r g e n e r a l i z e d pseudo
(5.11)(5.13) w i t h
e i t h e r one of c o n d i t i o n s
PROOF.
X + Y
o r of t y p e (KM) w i t h
Kmonotone
position
N:
a s i n Pro
(5.1) i s solvable.
( * ) and
b t Example 1.13.
N
+
m
i s g e n e r a l i z e d p s e u d o Kmonotone, C o r o l l a r y 5.10
d e m i c o n t i n u o u s mapping of t y p e (KS+) and
(BM) mapping, where
B
=
and o f t y p e
1.13).
( c f . Example
A+C
general
A
a Bquasibounded
N
is a
A
i s p s e u d o Kmonotone.
N
i s a H i l b e r t s p a c e , C o r o l l a r y 5.10 i s v a l i d f o r
Fredholm mapping of i n d e x z e r o and
I n view
N.
5.9 and 5 . 1 0 a r e v a l i d when
of Example 1.13, C o r o l l a r i e s
Y
+
Hence,
h o l d s w i t h P r o p e r t y ( P ) r e p l a c e d by t h e b o u n d e d n e s s of
If
A
W e c o n c l u d e o u r d i s c u s s i o n by l o o k i n g a t p s e u d o monotone A = A +A2 w i t h
l i k e p e r t u r b a t i o n s of Fredholm mappings of t h e f o r m A1
p.d.
and symmetric from
compact, where
X
i n a H i l b e r t space an
A
5.11.
H
and E Xn.
Iquasibounded
H
1
A2:
and
X + H
The a b o v e c o r o l l a r i e s a r e v a l i d w i t h s u c h
H.
= X
C
H
+
H
be closed p o s i t i v e
and c o n t i n u o u s and b i j e c t i v e from
H
To =
{Xn,Pn; A1(Xn),
Let
A2:
X + H
Pn]
a scheme f o r
b e l i n e a r and c o m p a c t ,
(X,H) N:
and e i t h e r d e m i c l o s e d p s e u d o Imonotone
and o f t y p e ( I S + ) .
(5.11)(5.13)
bounded.
D(A)
A1:
( a ) Let
ed pseudo Imonotone.
tions
+
i s a Banach s p a c e c o n t i n u o u s and d e n s i l y embedded
d e f i n i t e and symmetric i n
A1(X,)
= X c H
and we e x p l i c i t y s t a t e t h e f o l l o w i n g
COROLLARY
onto
D(A)
Then E q .
Let
G:
X + H
Suppose t h a t and e i t h e r
(5.1)
with
X + H
o r generaliz
odd, d e m i c l o s e d
s a t i s f i e s e i t h e r one of c o n d i 
N
ro
b e bounded,
X
h a s P r o p e r t y ( P ) or
i s solvable.
N
is
(b)
Suppose t h a t
type ( I M )
A2
A1,
and
G are as i n (a),
I
w i t h t h e embedding
(5.11)(5.13).
of
.s . MILOJEVI~
P
368
To
Suppose
X
N:
+
H
i s of
compact and s a t i s f i e s e i t h e r one has Property ( P ) .
Then E q .
(5.1)
i s solvable.
[24],
+
A1
+
A2
t N + WG
c a s e ( a ) and j u s t
>
i n view of P r o p o s i t i o n 4 i n
It s u f f i c e s t o observe t h a t ,
PROOF.
0.
i s Aproper
s t r o n g l y Aclosed
To
and A  c l o s e d w . r . t .
in
t E [O,l],
i n case ( b ) f o r
H L e t u s now i l l u s t r a t e how one c a n a p p l y some of Theorems
3.93.13
t o monotone l i k e p e r t u r b a t i o n s of F r e d h o l m mappings.
Y = X*,
e x a m p l e , when COROLLARY 5 . 1 2 .
Let
we have D
b e a bounded
X,
i n a r e f l e x i v e Banach s p a c e mapping of t y p e
(5.1)
By Example 1.13,
w.r.t.
ra ra
A
= {Xn,Vn; ] :V,:X s a t i s f i e s (2.23),
t i s f y (3.21),
part A.
+
for
a continuous l i n e a r x E a D
and
Suppose t h a t f o r g i v e n
+
N
and J
+ pJ A+N
i s Aproper
N:
X
f
in
IB
Km
X*,
and Aclosed
s a t i s f i e s condition C = J
i s odd and
N
and
(*).
K = I
sa
H
c o u l d be a g e n e r a l i z e d pseudo
o r of t y p e (KM) mapping w i t h a p p r o p r i a t e l y c h o s e n l i n e a r
I n e a c h c a s e one n e e d s c h o o s e
Theorems 3.93.13, S e c t i o n s 2 and involving,
X*
t h e c o n c l u s i o n follows f r o m Theorem 3.9.
A s i n t h e above c o r o l l a r i e s , Kmonotone
X
i s solvable.
PROOF.
Since
A:
0
symmetric n e i g h b o r h o o d o f
(Ax,x) 2 0
( S + ) with
d e m i c l o s e d and pseudo monotone.
Then E q .
For
3.
K,
C
and
G
i n using
and some o f t h e i r c h o i c e s h a v e b e e n d i s c u s s e d i n T h u s , numerous s p e c i a l c a s e s of t h e s e r e s u l t s
s a y , monotone l i k e n o n l i n e a r p e r t u r b a t i o n s c o u l d b e
e a s i l y deduced.
We do n o t c a r r y t h i s o u t e x p l i c i t e l y .
369
CONTINUATION THEORY
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J. LERAY and J.L. LIONS, Quelques resultats de Vieik sur les
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(1972), 143. 19*
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20.
P.S. MILOJEVId, A generalization of LeraySchauder theorem and surjectivity results for multivalued Aproper and pseudo Aproper mappings, J. Nonlinear Analysis, TMA, 1 (1977), 263276.
21.
P.S. MILOJEVId, On the solvability and continuation type results for nonlinear equations with applications, I, Proc. Third Internat. Symp. on Topology and its Applic., Belgrade,
1977, 468485. 1977 77T3327* 22.
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CONTINUATION THEORY
24.
P.S.
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MILOJEVI6, Approximationsolvability of some nonlinear
operator equations with applications, Proc. Intern. Sym. Funct. Diff. Equat. and Bifurcation, July 1976, S g o Carlos, Brasil, Lecture Notes in Math., vol. 799, 1980, pp.289316, Springer Verlag, (Ed. A.F. 25.
126).
P.S. MILOJEVI6, Fredholm alternatives and surjectivity results for multivalued Aproper and condensing like mappings with applications to nonlinear integral and differential equations, Czechoslovak Math. J. 30 (105) 1980, 387417.
26.
P.S.
MILOJEVId, Approximationsolvability results for equations
involving nonlinear perturbations of Fredholm mappings with applications to differential equations, Proc. Intern. Seminar Funct. Anal., Holomorphy and Approxim. Theory, August, Rio de Janeiro, Brasil, Lecture Notes in Pure and Appl. Math.
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P.S.
MILOJEVI6, Continuation theorems and solvability of equa
tions involving nonlinear noncompact perturbations of Fredholm mappings, Atas do 129 Semindrio Brasileiro de Andlise, ITA, S g o Jose dos Campos, 1980, 163189. 28.
P.S.
MILOJEVI6, Theory of Aproper and pseudo Aclosed mappings,
Habilitation Memoir, UFMG, Belo Horizonte, Brasil, 1980,
PP. 1190. 29.
P.S. MILOJEVI6 and W.V. PETRYSHYN, Continuation theorems and the approximationsolvability of equations involving Aproper mappings, J. Math. Anal. Appl. 3 ( 6 ) (1977), 658692.
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PETRYSHYN, Continuation and surjectivity
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31.
L. NIRENBERG, Topics in nonlinear functional analysis, Courant Institute Lecture Notes, 1974.
32.
R.D. NUSSBAUM, The fixed point index and fixed point theorems for kset contractions, Ph.D. Dissertaion, Univ. of Chicago, Chicago, Ill. 1969.
33.
W.V. PETRYSHYN, Direct and iterative methods for the solution of linear operator equations in Hilbert spaces, Trans. h e r . Math. SOC. 105 (1962), 136175.
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.s. MILOJEVIC
34. W.V. PETRYSHYN, On the approximationsolvability of equations involving Aproper and pseudo Aproper mappings, Bull. Amer. Math. SOC. 81 (1975), 223312.
35.
S.I. POHOZAEV, The solvability of nonlinear equations with odd operators, Funct. Ana i Priloienia, 1 (1967), 6673.
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Departamento de Matemdtica Instituto de CiGncias Exatas Universidade Federal de Minas Gerais Caixa Postal, 702
30.000

Belo Horizonte

MG, Brasil
Functional Analysis, Holomorphy and Approximation Theory, JA. B ~ R O S(ed.J O 0 NorthHollandhblishing Company, 1982
NEW
EXAMPLES
OF NUCLEAR F R ~ C H E T SPACES
WITHOUT BASES(*)
V.B.
Moscatelli
We p r e s e n t a c o n s t r u c t i o n of a n u c l e a r F r 6 c h e t s p a c e w i t h o u t b a s i s which i s d i f f e r e n t from a l l o t h e r s also [4]). different,
T h e r e i s , of c o u r s e ,
([7],
[2]
and [ 11 ; s e e
a r e a s o n why t h i s c o n s t r u c t i o n i s
and i t i s t h a t i t was o r i g i n a l l y d e v i s e d t o s o l v e a com
p l e t e l y d i f f e r e n t p r o b l e m , n a m e l y , t h e f o l l o w i n g o n e , r a i s e d by Dubinsky i n a p r i v a t e c o n v e r s a t i o n (P)
Let
E
Must
(1978):
b e a F r e c h e t s p a c e h a v i n g no c o n t i n u o u s norm. be i s o m o r p h i c t o t h e p r o d u c t of a s e q u e n c e
E
of F r b c h e t s p a c e s e a c h h a v i n g a c o n t i n u o u s norm? Spaces such a s
s,
t h e power s e r i e s s p a c e s of f i n i t e or i n 
f i n i t e t y p e and t h e s p a c e s o f a n a l y t i c f u n c t i o n s h a v e c o n t i n u o u s norms.
On t h e o t h e r h a n d , c l a s s i c a l s p a c e s w i t h o u t c o n t i n u o u s
norms s u c h a s
UI
and t h e s p a c e s of c o n t i n u o u s or i n f i n i t e l y d i f 
f e r e n t i a b l e f u n c t i o n s on a n open s u b s e t of
Rn
c a n b e shown t o b e
i s o m o r p h i c t o p r o d u c t s of F r b c h e t s p a c e s w i t h c o n t i n u o u s norms. Problem ( P ) seems t o have b e e n a r o u n d for some t i m e and a p a r t i a l s o l u t i o n t o i t was g i v e n by Dubinsky h i m s e l f i n 1967 ( s e e
[ 3 ] ) by showing t h a t the a n s w e r i s p o s i t i v e i f quence s p a c e ( i ) cp C A C
("IT h i s
x. w,
By t h i s one means a s p a c e (ii)
=''1
and (iii)
t a l k i s a m o d i f i e d v e r s i o n of [
81.
x
E
i s a p e r f e c t se
such t h a t :
(x,T(x,1'))
i s Fre'chet;
374
here,
of c o u r s e
xX (cf.
MOSCATELLI
V.B.
[6]),
=
r(rln)
:
cp
and
while
c
ltnqnl
1,
D = Rn,
n > 1 with
For the inq € L2,
has eigenvalues) from the
behaviour of
S
(see [8]).
U
niqueness fails in the presence of eigenvalues in the onedimensional case (see [2]). all
k, k'
T o what extent
q
can be recovered by using
information has recently been investigated by Faddeev
[ 91 and Newton [lo].
382
MICHAEL O'CARROLL
For obstacle scattering, i.e.
L = A,
D = Rn/C,
Ccompact
and convex, with Dirichlet, Newman o r impedance boundary conditions the local curvature of large
C
is obtained from the asymptotically,
behavior of the scattering operator.
Ikl, Ik'I
From the
local curvature the obstacle is determined uniquely up to Euclidean motions (see [ 111 , [ 121 and for the direct problem [ 181 ) . Now we turn to some new results and methods for solving the onedimensional inverse spectral and scattering problem.
These
methods are also used to solve nonlinear evolution equations, the Kortewegde Vries equation being the best known example.
For sim
plicity of presentation we focus our attention on the SturmLiouville operator on the real line and the Dirichlet operator
LD
.
Similar
methods and results also hold for other 1dim operators and evolution equations as well as their discrete versions (see DTLund [ 3 ] ) . The, by now, classical method f o r solving the inverse spectral and scattering problems is by the GelfandLevitan [l3] Faddev [9] eqs. [l4]
o r Marchenko
in the SturmLiouville case, the ZacharovShabat
or AKNS [15] method for more general operators.
For these in
verse scattering methods applied to the solution of nonlinear evolution equations see Kruskal, et.al. [16] and the review of Faddeev
[ 9 1 , AKNS [ 151 and Flaschka and Newel1 [ 171. We will discuss specifically four topics: 1.
Socalled local trace formulas.
2.
Solution of inverse scattering using 1.
3.
The SturmLiouville o r Schroedinger eq. as a completely integrable Hamiltonian system of constrained classical harmonic oscilators and solution of nonlinear evolution equations.
4.
Geometry of isospectral sets of the Dirichlet problem.
ON THE IMrERSE SPECTRAL AND SCATTERING PROBLEMS
383
First let us discuss the spectral properties of the operator 0
L =  
d2 2 dx
each
k E R,
[
Iq(x)l(l+x
s(x),
+
Lfi = k2fi
2 )dxc:
,
x
k f 0, there are two solutions
R
(see [ 2 ] ) .
fi, i = 1,2
For
of
defined by

f s f (x,k) 1
x+m
e ikx
f2 =
2 fl + 1 f (x,k) T1 T1
fl =
R2 1 q f2 +  f2(x,k) T2
which implies
fi
By considering the Volterra integral equations obeyed by
we
have

, e m e+m 1
X+ m
f1(x,k)
eikx fl,
e
ikxd
and
L
Im k 2 0, when The
S
ikx
T
R2(k)
ikx
T I T1
R I R 1,
k
admit analytic continuation in
to
doesn’t have eigenvalues.
operator can be completely specified by the ”scat
tering” matrix
L
may have a finite number of negative eigenvalues, denoted by
pi
0: 3 U
=:
U(x)
such that
TI
pologically onto the ball U(n(x),r) with radius
r].
Then, as above for domains in
Cn,
3
of on
Bbn(G),
H(G).
separate the points of
We define a new boundary distance on
here
flG
H(G)
maps
U
to
around n ( ~ )
we regard the following family
holomorphic functions
394
P. PFLUG
:= (f : f
3
holomorphic on
H(G),
sup
If(x)l*AH(,4n )(x)
ho
.
G,
LE(G)
o u t s i d e of
with
with
WELL GROWING HOLOMORPHIC FUNCTIONS
399
These inequality in mind one deduces from a ) :
1 .s c J z V  w v Jdist(zV,aG)
8 21 
10
which gives the expected contradiction. As an easy corollary the following existence theorem follows from the basic properties of the kernel function. COROLLARY.
Assume
sequence in
G
G
with
zv
t
zo
E aG
2
there are functions
m
[zw)w=l be a
as in the theorem and let
\lfvll= 1,
fw E Lh(G), If,(zV)
then, for any
X,
0
< 1 < 1,
such that
I ' dist(zw1 ,aG)'
It should be mentioned that the condition
G
has a C2smooth
boundary can be weakened into the following direction [ 141 THEOREM.
Assume that
G
.
is a bounded pseudoconvex domain in
with the following cone condition along
Cn
aG:
E aG
wv + z o
there exist reals r < 1, a 2 1 and a sequence a such that U(ww, r1zOw" I ) n = @ . (r, a , ww may depend
on
U(z,p)
for any
zo;
zo
:= the ball around
with radius
z
p).
Then the following statements are true
1)
for any boundary sequence
there exists a Lh(G)function 2)
for any boundary point
c G
{zV)
2
z*
converges to infinite i.e.
f
zw
with
E aG, lim

t
z*
E aG
suplf(zv)l
=
m ,
the kernel function
K(Z,Z) =
m,
z+z*
zEG
3)
there exists a function values along
aG.
f E LE(G)
with infinite boundary
400
P.
The p r o o f
PFLUG
o f t h i s theorem i s o b t a i n e d u s i n g t h e t h e o r e m of BanachThe d e t a i l s w i l l b e o m i t t e d
S t e i n h a u s and a g a i n S k o d a ' s t h e o r e m . h e r e , compare [ REMARK.
.
141
E s p e c i a l l y t h i s r e s u l t can be understood
way, n a m e l y , any bounded pseudoconvex domain i n c o n d i t i o n i s a l r e a d y convex w . r . t .
i n the following Cn
w i t h t h e cone
t h e H i l b e r t space
Lt(G).
This
o b s e r v a t i o n i s , f o r e x a m p l e , u s e f u l when i n v e s t i g a t i n g t h e S e r r e problem A t
[la].
t h e end of t h i s s e c t i o n we g i v e a s l i g h t s h a r p e r v e r s i o n of t h e Catlin [4].
above c o r o l l a r y which was g i v e n by D. THEOREM.
Assume
G = [r
which i s g i v e n a s zo F aG
with
t o b e a bounded pseudoconvex domain i n
G
01
1,
a constant
C such
t h e r e e x i s t s a func
= 1 and
c o n s i s t s i n modifying t h e proof
OF THE
111) HYPOELLIPTICITY
and l e t
,...,0,l).
Then t h e r e i s a n open n e i g h b o r h o o d that,
r
Cn
g i v e n above.
ZPROB~M
I t ' s w e l l known t h a t i n t e r i o r r e g u l a r i t y of t h e a  p r o b l e m h o l d s , i.e.

a&
an equation
= 0,
be
=
Q
w i t h a smooth ( 0 , q )  f o r m
a l l o w s a smooth s o l u t i o n
Q
(q 2
l),
f o r pseudoconvex d o m a i n s .
More d i f f i c u l t i e s a r i s e d i s c u s s i n g t h e r e g u l a r i t y u p t o t h e bounda r y because i f boundary

then
bf = fig
Q
can
Q
a (0,l)form,
%a = 0 ,
Cm
up t o t h e
behave b a d l y n e a r t h e boundary.
The
problem t h e n c o n s i s t s i n f i n d i n g a s p e c i a l s o l u t i o n w i t h t h e c o r r e c t
4 01
WELL GROWING HOLOMORPHIC FUNCTIONS
boundary b e h a v i o u r .
in t h a t d i r e c t i o n h a s b e e n f o u n d by J . J . Kohn [ l l ] .
The f i r s t r e s u l t THEOREM.
Let
a E
and assume
be a s  c l o s e d 
af =
which
b e a Cmsmooth, bounded pseudoconvex domain i n 6"
G
:= { ( O , l )  f o r m s
C:l(c)
: Cm up t o t h e boundary]
form t h e n t h e r e e x i s t s a f u n c t i o n
a
f
E
C"(8)
G.
We want t o e m p h a s i z e t h a t t h i s t h e o r e m h a s b e e n u s e d q u i t e
o f t e n i n t h e l a s t time s o l v i n g problems of p e a k  p o i n t s etc.,
for
holds.
T h i s r e s u l t s g i v e s g l o b a l r e g u l a r i t y a l o n g t h e b o u n d a r y of REMARK.
to
compare, f o r example, [ 5 , 8 , 1 3 ]
and p e a k  s e t s
.
Whereas t h e l a s t theorem a n s w e r s t h e q u e s t i o n of g l o b a l b o u n d a r y r e g u l a r i t y t h e l o c a l b o u n d a r y r e g u l a r i t y of t h e 3  p r o b l e m seems more d i f f i c u l t .
But,
f i r s t , l e t u s g i v e a p r e c i s e d e f i n i t i o n what
l o c a l boundary r e g u l a r i t y means: g i v e n a n open s u b s e t
on
G,
Cn
in
U
and a :closed
p s e u d o c o n v e x , which i s smooth on
B
find s o solution
on
of
G
ZB
= a
U
(p,q)form
n 6
a (q r l )
is it possible t o
B
such t h a t
is
Cm
on
un 6 ? The a n s w e r t o t h i s q u e s t i o n i s , i n g e n e r a l , no a s t h e n e x t example due t o J.J.
EXAMPLE.
Kohn w i l l show:
Let
G c C2
b e a bounded pseudoconvex domain w i t h
Cmsmooth b o u n d a r y w i t h t h e f o l l o w i n g p r o p e r t y : G
Take a c u t  o f f
n u(0.1) =
function
X
E
{z E
u(o,i)
C;(U(O,l))
: x2
with
a 2 >...> on
K,
with
n z 2.
Let
assumes the values
2 E
Ki = {t E K; It1 = ail,
Let us assume that the result has been proved for
i = l,Z,...,n. n1.
Let
K’ = K2 U.. .U Kn. B y hypothesis, there is a polynomial It1
then the constant function
agrees with the Fcharacteristic function of
U
1.4, the
on
0
and
p(x)
I
K.
K C X 1, x E x,
Therefore
420
J.B. PROLLA
we may assume that t E K
n
there is
U,
is nonempty.
K\U
ft E lft(u)I
By continuity of that
Ift(y)l
there are
Consider
y E V(t).
for all
. .,tn E
K
fx = f
.f
tl,.
for all
.s 1,
tl
n u
and
u E K.
Since
such that
.... ftn .
t2
F o r each
ft(t) = 0, ft(x) = 1
there exists a neighborhood
ft
< e
such that
fj
x E K\U.
Fix
K
n u
K
V(t)
il U
is compact,
u.. .u
c V(tl)
fx E 0
Then
t such
of
V(tn).
fx(x) = 1,
and
while
NOW
hx = 1

f o r all
u
Ifx(y)I
for all
Y E K
< E,
belongs to
fx
Ihx(u)l
By continuity of
hx
such that
thx(y)I
there are
xl,x 2,...,xm f E 8
and
K;
n
hx(x)
U.
y E K
n
there is a neighborhood
in
y E W(x). such that
K\U
Since
U.
W(x)
x
of
is compact,
K\U
K \ U c W(x,)
W(x,).
U...U
.... hxm .
f = h
defined by
while
= 0,
u E K;
for all
< e,
for all
< e
oa
1, f o r all
h
lhx(u)ll
Consider
E
Ifx(u)I s 1,
We claim
that
(i)
If(y)l
(ii)
< e,
If(y)ll
Ihx.(y)
I
y E K\U,
< e,
Put
y E K.
while
for all
Iv1v2
n
U.
y E W(x,)
Ihx.(y)I
i
1
1< i
for some for all
15 j
Hence (i) is true. f o r all
..... vkll
k = 1,2,...,m.
To prove (ii), let
k = 1,2,...,m.
rn.
L
h
J
vk = h ( y ) Xk
(iii)
y E K
then
1
since
Y E K\U
for all
< e,
Indeed, if Hence
for all
y E K
m,
n
U.
We claim that
< E Clearly, (ii) follows from (iii), by taking
421
STONEWEIERSTRASS THEOF2EM
k = m.
We prove (iii) by induction.
For
The induction step is verified as follows. Then
lvnl
L
lvnll < o
and
1
Iwn


+
n
maxClvll
Ivnll)
Let us denote by as follows:
if, and only if,
a(x)
X
for all
= a(y)
onto
X
n
Y;
= g(rr(x)).
subset
Let
G C F,
the set
fl(G) fl(G)
[x]
of
x
is open in
Let us define
i.e.
Y,
D
C
g
g E @(Y;F). is open in
63
is a subalgebra,
D
is a separating subset of
rr
be the x E X, Let
X/G.
ir(x).
g: Y + F
(Any
such that
Indeed, for every open
X,
and
= T'(g'(G)). Y,
this means that
is continuous.
f = gon
belongs to
is a subalgebra of Hence
C(Y;F).
is compact.
Moreover, in this case
Indeed, let
y E Y
and let
X/G)
as
C(Y;F)
03 = ( g E C ( Y ; F ) ;
If G
modulo
X
be the
is continuous and f o r each
B y the definition of the quotient topology of
gl(~)
Y
and let
X/a
There is a unique
We claim that
(modulo
y
I
be constant on each equivalence class
has this property.)
f E G
x
a E 0.
modulo
y = ~ ( x ) is the equivalence class f E @(X;F)
if we denote
0
U.
then we say that
quotient topological space of quotient map of
iE.
the equivalence relation defined on
X/G
x,y E X,
if
1vn11)3
fg E N ( K , c ) ,
the Fcharacteristic function of
g
....* vn1'
v = v1v2
11
bnl,
Clearly, (i) and (ii) show that
f(x)

vn
*
s max(1v11,
by
Let
imply
11 = I wn v
*
(iii) is clear.
k = 1,
V
Y
Y
GI.
C(Y;F).
Moreover,
X
is compact, if
is a 0dimensional space.
be an open neighborhood of
y
in Y.
J.B. PROLLA
422
K = Y\V
Then
ft(y) = 1,
such that
t E K,
For each
is compact.
ft(t) = 0.
there is f t E @ ( Y ; F )
Then
and
t E Ut
are both open and closed, there are
tl,t2,
...,tn E
and
Ut = A . n
U...U
1 B = W
and
A
t1
.
fl... n Wt
n
0 B = 6.
Then
Let u s consider the map
of all X
are both open and closed,
Y from
f E @(X;F)
Then
h E 03
be g i v e n .
0
while
h
dule,
F o r each
in
K
l/f(t)g(t)l) < E, By Theorem 2 . 2 , l!f(x)h(x)I/ nclosure
of
x E K,
K/(G
< E, h
(K).
then
[x]
b e a compact
i s a subalgebra of
K
@(K;F),
i s i t s equivalence c l a s s
By h y p o t h e s i s , t h e r e i s
for a l l
t E [x]
d(f1K;hlK)
= 0.
for all
x E K.
in
K C X
i s a v e c t o r s u b s p a c e w h i c h i s a n (G \ K )  m o 
K c C(K E )
module
G IK
Then
Let
Conversely, l e t
@(X;E).
0
n
K,
becauso
Hence, t h e r e i s Therefore,
f
g E h [x] h
such that i s closed.
E h
such t h a t
belongs t o t h e
STONEWEIERSTRASS THEOREM
COROLLARY 2.6.
h c @(X;E)
Let
G c @(X;F)
be a separating subalgebra and let
be a vector subspace which is an Gmodule.
f C @(X;E),
f
then
if, for each
h
belongs to the nclosure of
x E X
and
E
>
there is
0,
THEOREM 2.7 (StoneWeierstrass). and let
429
f E @(X;F).
Then
f
Let
G
g E h
F o r each
if, and only
such that
be a subalgebra
C @(X;F)
belongs to the %closure of
G
if,
and only if the following conditions hold: (1)
given
(2)
given
x, y
such that
4x1 PROOF.
let f
[XI on
g(y); f(x)
f
g E G
0, there is
such that
[x],
choose
Then
,
If
Then
c(x;F).
that, for each
x E X,
COROLLARY 2 . 9 valued field.
and
f
K c [x]
0
h E G
G,
If
X/G.
f
agree on any such that and agrees with
compact.
By Theorem
belongs to the nclosure of
G
0
Let
is ?tdense in
= h,
modulo
belongs to
and a fortiori on any and
f
f(x) f 0, there is
g = f(x)(h(x))lh
E = F,
c(F;F).
0 E G.
[x]
Conversely,
By condition (l),
satisfy conditions (1) and ( 2 ) .
compact.
[x]
g E G
there is
0.
COROLLARY 2.8.
in
f
with
is constant on each equivalence class
2 . 5 with
in
f
f E C(X;F)
h(x) f 0. f
g(x)
x E X
f(x) f f(y),
with
X
The conditions are easily seen to be necessary.
is zero on
KC
in
(i
c C(X;F)
be a separating subalgebra such
there is
a E G
with
a(x) f 0.
Then
G
@(X;F).
(Weierstrass).
Let
(F,I.I)
be a nonarchimedean
Then the algebra of all polynomials on
F
is ndense
430
J.B.
DEFINITION 2.10.
Let
(E,I)
(F,
medean v a l u e d f i e l d
PROLLA
11)
b e a normed s p a c e o v e r a n o n  a r c h i 
1 * 1 ).
We s a y t h a t
of for s h o r t ,
polynomial approximation p r o p e r t y ,
h
if t h e v e c t o r s p a c e
has t h e Weierstrass
E
the property ( W ) ,
of a l l p o l y n o m i a l s
n
C
t E F +
ait
k
k= 0
a ,al,. ..,an
where
THEOREM 2 . 1 1 .
F.
on
m
h
Clearly,
Let
h
i s ndense
in
f o r any
h = @(X;F) 8 E
i s ndense
in
Ivhen
i s a nonempty
2.2
G c @(X;F)
C
@(X;F)
LI
@(F;E).
X.
i s a @(X;F)module.
x E X.
By C o r o l l a r y
On
2.6,
0 T1
0dimensional
N o t i c e a l s o t h a t when
space, then X
@(X;F)
i s 0dimensional
i s f u l l , t h e r e i s no need i n t h e proof
of Theorem 2 . 2
THEOREM 2 . 1 3 .
in
contains the
i s s e p a r a t i n g over
h
@(X;E).
of Theorem
In f a c t , a n o b v i o u s m o d i f i c a t i o n of
t o pass t o t h e quotient.
t h e proof
G
X.
b.
because
Now
C(X;E).
b(x) = E ,
i s s e p a r a t i n g over
i s separating.
@(X;F)
Clearly,
E.
@
G
and
i s ndense
t h e o t h e r hand
and
(W).
x E F,
f o r any
u s assume t h a t
h = @(x;F)
X
@(F;E).
o v e r a nonarchimedean
(E,!l*!l)
i s a n Gmodule;
h] = E
@(X;F) 0 E
Let
in
i s ndense
c @ ( F ; F ) b e t h e a l g e b r a of a l l polynomials
By C o r o l l a r y 2 . 6 ,
THEOREM 2 . 1 2 .
,...,
have p r o p e r t y
G
and l e t
constants.
PROOF.
0,1,2
@(F;E) be t h e v e c t o r space introduced i n Defini
C
kb(x) = { g ( x ) ; g E
Then
1)
(F,I
Let
t i o n 2.10,
=
n
A l l normed s p a c e s
valued f i e l d PROOF.
E E,
shows t h a t t h e f o l l o w i n g i s t r u e .
Assume t h a t
X
i s compact and 0  d i m e n s i o n a l .
h c @(X;E)
b e a f u l l s u b a l g e b r a , and l e t
s u b s p a c e which i s a n Gmodule.
F o r each
f
E C(X;E)
Let
be a v e c t o r we have
431
STONEWEIERSTRASS THE0RF;M
d(f;h) = sup{d(f(x); where
h ( x ) = (g(x);
G c @(X;F)
If GlK
g E h]
for
x E XI,
h(x));
x E X. K C X,
is a full subalgebra, then for each
is a full subalgebra of
On the other hand, a sub
@(K;F).
space of a 0dimensional space is 0dimensional.
Hence the fol
lowing result follows from Theorem 2.13 in the same way that Theorem 2.5 follows from Theorem 2.2. THEOREM 2.14.
G c C(X;F)
X
Let
be a nonempty 0dimensional space, and let
be a full subalgebra.
space which is an Gmodule. the n closure of there is
g
E h
COROLLARY 2.15.
G
C C(X;F)
Let
h
@(X;E)
C
f E C(X;E),
F o r each
if, and only if, for any
h
such that Let
X
Ilg(x)f(x)l(
0,
0 .
T1
be a 0dimensional
be a unitary subalgebra.
be a vector
space, and let
Then the following are equi
valent. (a)
G
is separating.
(b)
G
is full.
(c)
G
is Itdense.
PROOF.
(a)
G.
(b)
follows from Theorem
0dimensional.
(b)
E = F,
and noticing that
G
G = h,
even when
X
is not
(c) follows f r o m Theorem 2.14, by taking G(x)
for all
= F
x E X,
since
is unitary. T o prove (c)
*
(a), just notice that any ndense subset of a
separating set is separating. a
1.7,
T~
space,
C(X;F)
Now, since
is separating.
X
is 0dimensional and
J.B.
432
PROLLA
REFERENCES
1.
CHERNOFF, P.R., R.A.
RASALA, and W.C. WATERHOUSE,
Weierstrass theorem for valuable fields.
The Stone
Pacific J. Math.
27 (1968), 233240. 2.
KAPLANSKY, I., The Weierstrass theorem in fields with valuations.
3.
Proc. Amer. Math. SOC. 1 (1950), 356357.
MACHADD, S., and J.B. PROLLA,
An introduction to Nachbin spaces.
Rend. Circ. Mat. Palermo, Serie 11. 21 (1972), 119139.
4.
PROLL.4, J.B., Nonarchimedean function spaces.
In Linear
Spaces and Approximation (Edited by P.L. Butzer and B,Sz.Nagy), Birkhluser Verlag Base1 (1978), 101117.
5.
S'WIERSCZKOWSKI, S., The pathfunctor on Banach Lie Algebras. Indag. Math. 33 (1971), 234239.
Universidade Estadual de Campinas Instituto de Matemgtica, Estatistica e CiGncia da Computaqzo Campinas

BRASIL
Functional Analysis,Holomorphy and Approximation Theory, JA, Batroso (ed.) 0 NorthHolland hblishing Company, 1982
SEMIMARTINGALES AND MEASURE THEORY
Laurent Schwartz
Pellaumail [l] was the first to introduce semimartingales as defining measures on the previsible sigma field with values in the space
Lo
of measurable functions (Theorem ( 3 . 2 ) ) .
These de
velopments went o n and arrived to some definite result with the converse theorem by Dellacherie (theorem
(4.2)).
There are numerous
articles of Pellaumail and Metivier on the subject.
$1. VECTOR MEASURES.
Let
il
pological vector space, metrizable and complete.
(il,5)
with values in
values in
E,
{y(B), B E S]
B
a to
A measure
on
E,
for
E
Banach, is a function on
6 with
countably additive.
It is automatically bounded:
is a bounded set in
E.
I t is n o longer the case if
is not locally convex, and it is necessary even to add the sup
plementary condition: ed.
E
be a set equipped with a sigma field, and
the convex hull of the previous set is bound
One gets in these conditions a good theory of integration; in
particular, the "little" dominated convergence theorem of Lebesgue or theorem of bounded pointwise convergence is true: is a sequence of Bore1 functions,
to
0,
p(rpn)
converges to
0.
IcpnI
L
1,
if
(cp,)
n€N converging pointwise
Then, if it is true that many
measures are given as set functions more as functionals (even the Lebesgue measure on
R
is initially given only as a function of
434
SCHWARTZ
L.
intervals) it is more interesting to consider that just as a way of finding a measure, but to give a definition of a measure as a functional: (1.1)
A measure
E,
p
on
(IR,O)
with values in
of the space
BO
of bounded Borel functions
is a linear map into
E , se
quentially continuous for bounded pointwise convergence, i.e. verifying the little bounded convergence theorem of Lebesgue. wrote very interesting articles on these
Erik Thomas [ Z ]
[st].
vector integrals and also more recently Bichteler [ S ] , KUssmaul
I n this case, there is no more convexity condition since the
BO
unit disc of
E
For for
E
is convex, and
p E X(BO;E).
Banach, one introduces
positive function, and
p*,
arbitrary another a little more complicated functional, and
one makes a theory of integration. gible if
I.((B’) = 0
for
A bore1 set
B’C B ;
B’E O,
B
is pnegli
18
an arbitrary set
is
A
pnegligible if it is contained in a pnegligible Borel set; a real function function.
f
is pmeasurable if it is
pa.e.
equal to a Borel
For integrability, it is more interesting to introduce
the topology of
X1(p)
on the space
fines a product measure in
X1(p)
if and only if
verges to
0
Xb(BO;E),
i.e. if and only if
E BO,
uniformly for
$
completion of
B(Q
161
L
a function
cp E BO
6 + (cpp)($) = p(ep$) and
by
cpp
BO:
p(cpiQ)
1.
X1
cpi
converges to
converges to
Then
for this topology:
pip
0
de‘On
in
0
in
E,
is more o r less the
we shall say that
f
is
of functions of (fn)nEN and which is Cauchy in X’(U).
pintegrable if there exists a sequence BQ,
converging
The space
S1(u)
is integrable if quotient bas
L1(p)
a general
pa.e.
to
f,
of pintegrable functions is a vector space f
is integrable,
X1(u)
is complete, and
is metrizable and complete (FischerRiesz)
.
If1 ts One
dominated convergence theorem of Lebesgue: if (fn n€N
435
SEMIMARTINGALES AND MEASURE THEORY
converges f
wa.e.
to
i s vintegrable,
cular
f n dy
Ifn]
f,
fn
converges t o f dy.
converges t o
a p a r t i c u l a r l y important r o l e , i n t e g r a b l e i f and o n l y i f
= y(hf),
92.
while,
f n and
g,
5
S1(p),
in
f
f(w) =
hU
f
play hu
is
(hu)(f) =
and t h e n
(fh)p.
MARTINGALES, LOCAL MARTINGALES, SEMIMARTINGALES
A s always,
p r o b a b i l i t y on
8,
terminal time,
+m)
R
8
w i l l be a s e t ,
(6+ =
(Zt)tcG+
[O,+m]
a sigma f i e l d on
,
a f a m i l y of Ameasurable
i s a r e a l f u n c t i o n on
t++X(t,w)
Xt:
variable
n

and r i g h t c o n t i n u o u s
8+
X
zt,
sigmafields
('Gt
R,
t
s)
A r e a l process
t h e random
The p r o c e s s i s s a i d continu &
c a d l a g ( c a d l a g i s a n a b b r e v i a t i o n of a F r e n c h e x p r e s s i o n : d r o i t e avec l i m i t e s
i t means i n E n g l i h s r i g h t c o n t i n u o u s
with l e f t l i m i  t s ; I preserve the French a b b r e v i a t i o n ) i f , f o r
w,
X(w):
the trajectory
6,
t +X(t,w),
IR
is
Aalmost
every
cadlag.
W e do n o t d i s t i n g u i s h b e t w e e n two i n d i s t i n g u i s h i b l e p r o 
cesses, A c
R+
i.e.
x R
whose t r a j e c t o r i e s a r e t h e same f o r 1  a l m o s t w i l l be s a i d Xnegligible
P
every
if i t s p r o j e c t i o n on
R
w; is
1 n e g l i g i b l e . A r e a l martingale
Mt
i s a r e a l cadlag process,
M
i s 1integrable f o r every
t,
(2.1 bis)
for
A
E Gs,
and
integrable:
verifying:
1
[ E ( M ~ / z ~=) M~
(2.1)
t
a.e. 2
s,
a
contain
adapted:
IR,
Ztmeasurable.
is
1
R,
we want t h e r e i s a
ing a l l the 1negligible subsets, increasing
X
then
and i n p a r t i 
The p r o d u c t m e a s u r e s
i s pintegrable,
f o r products,
uintegrable,
r e a l p i n t e g r a b l e ;
h
hf
g
for Ms
t > dX =
s,
or M t d1 *
The most u s u a l of c o n t i n u o u s m a r t i n g a l e s ( w i t h c o n t i n u o u s
436
L.
SCHWARTZ
t r a j e c t o r i e s and n o t o n l y c a d l a g ) i s t h e r e a l Brownian m o t i o n
for which
Bo = 0
Xa.e.,
of t h e sigma f i e l d
,
meter
and
t
for
BtBs,
2
i s independent
s,
and f o l l o w s a Gauss law on
Gs,
B,
R,
or p a r a 
i.e.
2
1 ex / 2 ( t  s ) ___
dx
J2r;
fi " t h e f i r s t time a p h y s i c a l
A s t o p p i n g time i s , i n t u i t i v e l y ,
e v e n t happens".
For instance, i f
B o r e 1 s u b s e t of
R,
X
i s a cadlag process,
t h e e n t e r i n g time
T
of
X
in
a
A
defined
A,
by
[t E
T(w) = Inf (T(w)
i s taken equal t o
if
+m
R+;
E A]
X(t,W)
i s never i n
X(t,w)
A)
is a
stopping time.
M a t h e m a t i c a l l y a s t o p p i n g t i m e i s a f u n c t i o n on
with values i n
6+,
such t h a t
It i s a random v a r i a b l e ( i . e .
xT
i s
t E
Y
hT+,
{w C n; T(W)
),measurable);
if
t ] E Zt.
i s a process,
WX(T(W),W). A s t o p p i n g time a l l o w s t o s t o p a p r o c e s s :
process,
T
a stopping time,
(t,w)*X(t,tu)
XI,
t z T(UJ),
if
T
a stopping time,
tingale.
if
stopped process,
x(T(w)), i f
A f u n d a m e n t a l t h e o r e m b y Doob s t a t e s t h a t ,
and
X
h
n,
is:
t
5.
T(w).
M
i s a martingale
MT
i s a g a i n a mar
if
t h e stopped p r o c e s s
is a
X
From where t h e n o t i o n of a l o c a l m a r t i n g a l e , where t h e
word l o c a l h a s no t o p o l o g i c a l s e n s e : (2.2)
M
i s a l o c a l m a r t i n g a l e i f t h e r e e x i s t s an i n c r e a s i n g se
quence
( T ~ )
of s t o p p i n g t i m e s c o n v e r g i n g s t a t i o n a r i l y
nc@J to
+m
( f o r every
stopped process
MTn
w,

A c c o r d i n g Doob's
Tn = Mo
+m
for large
n),
such t h a t each
i s a martingale.
theorem, a m a r t i n g a l e i s a l o c a l m a r t i n g a l e
b u t of c o u r s e t h e c o n v e r s e need n o t b e t r u e .
In particular,
if
M
SEMIMARTINGALES AND MEASURE THEORY
is a local martingale, A real process every
w,
Mt V
need not be integrable!
has a finite variation if, for Xalmost
has a finite variation,
V(w)
IdVs(d
Then a real semimartingale is a real process written at least in one way as a sum
to
is not unique. and
V
However, if
X
to be continuous, with
M
X
X = V+M
finite variakion and a local martingale. &ion
437
+.
T(wr)
of the stopping times, a fortiori by the stochastic intervals ]S,T]
=
{(t,w);
< t s T(w)],
S(w)
S
H
Then it occurs that if
T
and
stopping times, S
5
T. X
is real bounded previsible and
is a semimartingale, it is possible to define: f
(3.1) not separately for every
w,
but globally:
kX(H)
is a Xclass of
real Xmeasurable functions (?,class,for the equality 1a.e.:
M~(H) a definite value at a given point
cannot assign to only
Xa.e.);
One puts pM(H); gale,
pX(H)
X = V+M;
E Lo(n,@,?,).
Mt E
that, if quence
M
2 6: ( X )
for every
t.
M
w,
but
The method is relatively long.
it is trivial for
it is relatively easy if
one
it remains to define
V,
is a square integrable martin
Afterwards, a deep theorem says
is a local martingale, there exists an increasing se
(T,)
of stopping times, converging stationarily to
such that every stopped process
MTn
+a,
be (in a non unique way) sum
of a martingale with finite variation and a square integrable martingale.
Therefore one can define
stationary limit for
n +
+m,
c~ Tn(H);
M one gets
and passing to the
pM(H).
Probably too much
time has been devoted in the past to find more o r less simple constructions of this stochastic integral (they are never simple). It remains, in the idea of non probabilists, some confusion; they have a tendency to believe that the stochastic integral is something "which can be built" (painfully) (and even they have the idea that the result could depend on the process of building).
But
there exists a theorem of existence and uniqueness analogous to the theorem which says that there exists a positive (non finite) unique measure on
R,
Lebesgue measure, whose value on an interval
[a,b] ,
SEMIMARTINGALES AND MEASURE THEORY
a
S
b,
is
439
ba:
THEOREM (3.2) (stated by Pellaumail [l]).
If
X
martingale, there exists a unique measure
pX
on
is a real semi
Jl = B+ x
n
,
equipped with the previsible sigma field
B = pr6,
E = L0(n,Q,X),
are stopping times, S
such that, if
S
T
and
with values in i
T,
The existence results of the construction given above; the uniqueness from the fact that the stochastic intervals nerate the sigma field.
If
pX(Hn)
in
converges to
0
(Hn)ncN Lo, i.e.
converges to
0,
]S,T]
l~,]
gei 1,
in probability, but not
X
a.e.; however it is possible to extract a subsequence which converges
1a.e.
I t is even possible to show that, if we put
f
the
(H*X)t define a semimartingale
H  X vanishing at the time 0,
and generally it is this semimartingale which is called stochastic integral of pointwise,
for every
with respect to
]H,(
t,
LO.
$4.
H
L
but
I, (H;X)*
not only
X.
If
(H,)
converges to n@J ( H ~ . x ) converges ~ to o in
= sue I(Hn.X)tl tER+
converges to
0
0 LO
in
CHARACTERIZATION OF THE SEMIMARTINGALES AS MEASURES ON
THE PREVISIBLE SIGMA FIELD If
X
is a semimartingale, the measure
the fundamental following properties:
ux
on
(ill,&)
has
440
L. SCHWARTZ
0)
it vanishes on
1)
it is timeadapted:
(4.1)
]O,t] 2)
fO}x0
x n, bx(H)
if
uX(H)
H
is carried by is Ztmeasurable (trivial);
E Lo(n,B,h)
it is 0localizable:
n'c 0 ,
(trivial);
H
if
g+ x n ' ,
is carried by
n'
is carried by
.
("I
What is interesting is that there is a converse, so that it is a characterization of semimartingales: PROPOSITION 4.2 (Dellacherie,
( a , @ ) = (g+xn, pr6)
If
[5]).
with values in
is a measure on
p
E = Lo(R,B,X),
vanishing on
timeadapted and nlocalizable, there is a semimartingale
(O]xn,
X,
see
unique but to indistinguishibility, vanishing at the time
such that
p = px
.
The principle of the proof is the following.
1x1
stopping time allows first to prove that random variable
0,
A technique of
is majorized by a
M: n + R+ ; therefore there exists a probability
with respect to which
M
is integrable.
Then, the space
X'
?,
BO
of bounded previsible functions being isomorphic (it is a com
mutative C*Banach algebra ! ) to a space (see
[ 6 ] ) says that B5
where
p:
BB
b
Lo(f2,Q,?,)

L2(n,B,h)
a theorem by Maurey
C(K),
factorizes by Lo(n,B,X),
(a) is the multiplication by a measurable function a.
if we put
A"
= const.
1
,
l+a linear continuous from B5 the unit disc of
BO
we see that into
M"
L2(n,B,x");
is bounded in
L 2 ( n ,&,A''
: BIB
4
Lo(n,B,h")
the image by
);
and
p
1" 1'.
Then, is of
This
easily allows to prove that whatever be the subdivision
(*)
This result was found many years ago; it has been proved step by step. One can find a complete demonstration in Schwartz [ 4 ] , prop. ( 9 . 2 1 , page 17.
441
SEMIMARTINGALES AND MEASURE THEORY
bounded, i.e. to
X
is what we call a quasi martingale with respect
And it is known that a quasimartingale, majorized by an
X".
integrable random variable, and right continuous in probability, is
It remains to apply Girsanov
defined by a unique semimartingale. if
theorem (see [ 7 ] ) : also for
X
is a semimartingale for
1" A ,
it is
A.
$5. VARIOUS APPLICATIONS Of course it is necessary not to confuse the semimartingale process dX:
X
X
and the measure
is a primitive of
X
there is between
and
px
defined by it, which can be called
dX,
dXs = clx(lo,tl x n ) ; Xt = the same relationship as between a
dX
F
function with finite variation fined by it,
F
on
R
and the measure
is a primitive of the measure
dF.
dF
de
The above cha
racterization has various applications:
(5.1) Refined theorems proved in the last years on semimartingales become trivial with this conception. F o r instance Girsanov theorem is trivial since
Lo(n,Q,X) = L 0 ( n , Q , X ' )
if
1'
 1.
Unfortunately, we needed
Girsanov theorem to prove the proposition ( 4 . 2 ) . the generalized Girsanov theorem: not equivalent to to
X,
if
X
martingale.
X = V+M
X'
has
base
X,
but is
is a semimartingale with respect
it is with respect to
the decomposition
near map
),,
if
But let us take
1'.
I t is not at all trivial with
since a Xmartingale need not be a 1'
But it is here trivial since there is a continuous li
Lo(fl,Q,X)
+ L o ( f l , Q , ) , ' ) .
On the other hand, if
X
is a
442
SCHWARTZ
L.
for a family of subsigma fields
n,@,),,(st)
martingale for
 .
t€Rl usual decomposition, since if it does not mean that vial since
V
8 = (8,)
, i t remains a semi
I t is not at all trivial with the X = V+M
M
and

t€R,
is adapted for
t €EL 7
are adapted!
But it is here tri
is 8adapted and nlocalizable.
px
,
(St)
(In fact, it was
just proved more or less in this way in the past).
(5.2)
I t is only recently (see Jacod [ 8 ] ) that instead of integrating with respect to
dX
bounded previsible processes,
integrable previsible processes have been integrated. I t gave many surprises.
H
tingale and but if
X
For instance, if
bounded previsible,
is only dMintegrable,
local martingale.
HM
HM
X = M
is a local martingale;
is no longer necessarily a
Even the definition of the
processes introduced some difficulties.
is a local mar
H
dXintegrable
Now, with the new defini
tion, it is sufficient to go back to the general theory of integration.
I n particular the general dominated convergence theorem
of Lebesgue holds.
(5.3)
Emery [9]
recently introduced a topology on the space
8h
of semimartingales,which makes it metrizable and complete. We gave remarkable properties (in particular for the continuous dependence of the solution of a stochastic differential equation with respect to the data). well as it should have been. by the space
This topology has not been "accepted" as But it is just the topology induced
Mes(R,B,E) c Sb(BB;E),
able and complete; the subspace
Mes
it is well known,
metriz
of the continuous maps on
B8
for the pointwise bounded convergence of sequences is closed and the subspace of the adapted localizable measures is closed in therefore
Sh
is complete.
Mes,
SEMIMARTINGALES AND M E A S U W THEORY
443
(5.4) The infinite positive measures (that is with no necessarily finite values) are very familiar (Lebesgue measure on
R!).
The space of the signed non everywhere defined measures in much less known.
However it exists, as well as the space of non everywhere
defined measures with values in a metrizable complete topological vector space if and
p p
E.
is such a measure,
P = U
Pk,
kFN is defined everywhere on Pk.
For instance if f(x)dx
We consider only ufinite measures; grosso modo,
f
Wk E 0 , Wk
The theory is very simple.
is a an arbitrary real Bore1 function on
is a such a measure with
!Rk = [k,+k] n
{If1
measures have no primitives, since an interval of of being integrable).
has no reason
Therefore we shall call a
formal semimartingale a formal measure on
Lo(n,B,X),
formal semimartingale,
(0 ,@) =
adapted and localizable.
p (H)
X
has a meaning for
not necessarily for every bounded previsible previsible
(Such
I: k].
R
H, pXintegrable or not,
H'X
(R+xn, If
prB),
X
The possibility of writing
is a
H pXintegrable,
H; but for every
always makes sense as a
formal semimartingale; and it is a true semimartingale iff dXintegrable.
R,
More than not everywhere defined measures,
I shall say better formal measures.
with values in
is kintegrable
H
is
H  X without any res
triction "liberates" completely of the usual integrability conditions and makes easier a lot of operations; it is only necessary, at the end of the computations, to see if the result is a formal or a good true semimartingale (exactly as if we want to solve the partial differential equations using derivatives in the sense of distributions, we first find a solution as distribution, and we see at the end of the computation whether it is a function or not and what is its regularity).
I just wrote an article on the formal semimargingales.
L. SCHWARTZ
444
BIBLIOGRAPHY
1.
J. PELLAUMAIL:
Sur l'intbgrale stochastique et la d6composition
de DoobMeyer, Ast6risque no
9, Soci6t6 Mathematique de France
(1973). 2.
E. THOMAS: L'intggration par rapport a une mesure de Radon vectorielle, Annales Inst. Fourier XX, fasc. 2 (1970), 55191; and: On Radon maps with values in arbitrary topological vector spaces, and their integral extensions, Preprint (Department of Mathematics, Yale University).
3.
K. BICHTELER:
Stochastic integration and LPtheory of semi
martingales, Preprint (University of Austin (Texas), Sept.
1979). 3'. KUSSMAUL:
Stochastic integration and generalized martingales.
Research Notes in Math., Coll. n , Pitman Pub. London,
4. L. SCHWARTZ:
1977.
Semimartingales sur des vari6t6s, et martingales
conformes sur des vari6tes analytiques complexes, Lecture Notes in Math. no 780, Springer, 1980.
5.
C. DELLACHERIE and P.A. MEYER:
Probabilites et potentiels,
chap. V 5 VIII, chap. VIII, $4, p. 400, Hermann no 1385, Paris 1980.
6. B . MAUREY:
SBminaire MaureySchwartz 197273, expos6 no XII,
Ecole Polytechnique Paris; and: Th6orAmes de factorisation pour des op6rateurs linhaires
a
valeurs dans les espaces Lp,
Ast6risque nP 11, Sociht6 Mathematique de France (1974). Shinaire de Probabiliths X , Strasbourg 197475,
7. P.A. MEYER:
Lectures Notes in Math. no 511, Springer 1976, p. 376. 8.
J. Jacod:
Calcul stochastique et problhmes de martingales,
Lecture Notes no 71, Springer 1979.
9. C. DELLACHERIE, P.A. MEYER, M. WEIL: Shminaire de Probabilitbs XIII, Strasbourg 197778, Lecture Notes in Math. no 721, Springer 1979. 10. L. SCHWARTZ:
Semimartingales formelles, Lecture Notes in Mathematics, nP 850, Springer 1981.
gcole Polytechnique Plateau de Palaiseau


Centre de Mathgmatiques 91128 Palaiseau Cedex

France
Functional Analysis, Holomorphy and Approximation Theoty, JA. Barroso led.) 0NorthHollandF’ublishing Company, I982
ON SEMISUSLIN SPACES AND DUAL METRIC SPACES
Manuel V a l d i v i a
I n t h i s p a p e r we s t u d y some p r o p e r t i e s of a c l a s s of t o p o l o g i c a l s p a c e s i n c l u d i n g t h e KSuslin
s p a c e s and h e n c e f o r t h we o b t a i n
some new r e s u l t s i n t h e t o p o l o g i c a l v e c t o r s p a c e s t h e o r y .
A i s a s t a r s h a p e d m e t r i z a b l e subset of a
l a r l y , we p r o v e t h a t i f Hausdorff
E,
topological v e c t o r space
E
bounded s u b s e t of
Particu
intersects
A
s u c h t h a t e v e r y c l o s e d and
i n a compact s e t , t h e n
A
is
separable.
We u s e t h e r e H a u s d o r f f t o p o l o g i c a l s p a c e s . v e c t o r s p a c e s u s e d h e r e a r e d e f i n e d on t h e f i e l d complex numbers. B(E,F)
and
respectively
3
If
(E,F)
If
W(E,F)
The t o p o l o g i c a l of t h e r e a l or
K
i s a d u a l p a i r we d e n o t e by
t h e weak,
u(E,F),
s t r o n g and Mackey t o p o l o g i e s on
E,
. i s t h e t o p o l o g y of a t o p o l o g i c a l s p a c e
ACT]
T
and
a s u b s e t of
T,
we d e n o t e by
t o p o l o g y by
3.
The t o p o l o g i c a l d u a l of a l o c a l l y convex s p a c e
is
E‘
.
X(E’,E)
E”
the s e t
i s t h e t o p o l o g i c a l d u a l of
t h e t o p o l o g y on
compact s u b s e t s of
E.
E’ On
with the
A
E’[B (E‘ , E ) ]
E
induced E
We d e n o t e by
of t h e u n i f o r m c o n v e r g e n c e on t h e p r e 
E”
,
X(E” ,E’
)
i s t h e t o p o l o & T of t h e
u n i f o r m c o n v e r g e n c e on t h e p r e c o m p a c t s u b s e t s of u s u a l , we i d e n t i f y
.
is
A
w i t h a s u b s p a c e of
EN
E‘[B (E’ , E ) ]
.
As
by t h e c a n o n i c a l i n 
j e c t i on. By a “ w e a k l y ucompact
g e n e r a t e d l o c a l l y convex s p a c e ”
G
446
M.
VALDIVIA
we mean a l o c a l l y convex s p a c e
which h a s a s e q u e n c e of weakly
G
compact s u b s e t s whose u n i o n i s t o t a l i n of bounded
H
s o that i f
I.
of
B z
H
E B,
i s contained i n
i s starshaped i f t h e r e i s a vector then
{txo
+
(1t)z
: 0 i t
DEFINITION.
A t o p o l o g i c a l space
a P o l i s h space s e t s of
2.
in
E
space
{x,]
and a mapping
i
x
c
I.
in
B
l] C B .
P of
cp
and a mapping
u
{cp(x) : x
c
cp
from
P
there exists
i n t o t h e c l o s e d sub
in
P
PROPOSITION 1.
so that
t h e r e i s a P o l i s h space E,
then
from
V
P
(zn)
h a s an adherent p o i n t
i f there is a Polish
i n t o t h e compact s u b s e t s of F ,
and g i v e n a n a r b i t r a r y p o i n t of
z
t h e r e i s a neighbourhood
cp(z),
[7].
cp(U) C V ,
P
converging towards an element
i s KSuslin
F
A topological
compact s u b s e t s of
P
cp(x).
P] = F
and a n e i g h b o u r h o o d z
i s semiSuslin i f
n = 1,2,...,
which i s c o n t a i n e d i n
P
[lo]:
E
i s a seqaence i n
zn E cp(xn),
such t h a t
U
I]
such t h a t t h e f o l l o w i n g c o n d i t i o n s a r e s a t i s f i e d :
If
and E
P
A topological space
in
2
c
SEMISUSLIN SPACES
The f o l l o w i n g d e f i n i t i o n was g i v e n i n
x
for some
A.
: i
{Ai
is f u n d a m e n t a l
H
s e t s i n t h e topological vector space
i f e a c h bounded s e t i n A subset
A family
G.
space
E
i s KSuslin
and a mapping
cp
from
i f and o n l y i f P
into the
s u c h t h a t t h e two f o l l o w i n g c o n d i t i o n s a r e
satisfied:
2.
x
and
If zn
(x,)
i s a sequence i n
E cp(x,),
n = 1,2,...,
P
c o n v e r g i n g t o w a r d s an e l e m e n t
then
(zn)
h a s an a d h e r e n t p o i n t
447
ON SEMISUSLIN SPACES AND DUAL METRIC SPACES
in
which i s c o n t a i n e d i n
E
PROOF.
Let u s suppose t h a t E
p r o p o s i t i o n and x
E
P,
U
of
(Un)
L e t u s t a k e now a p o i n t
z n E cp(xn),
g
zn
t i o n says t h a t cp(x).
E
v e r i f i e s t h e two c o n d i t i o n s of t h e
i s n o t a KSuslin
a neighbourhood
neighbourhoods
to
cp(x).
xn
in
E
(zn)
P
Un
n = 1,2,
U,
~ ( x ) and a f u n d a m e n t a l s y s t e m of
of x
Then t h e r e i s a p o i n t
space.
n=1,2,... and a
The c o n d i t i o n 2 of t h e p r o p o s i 
z
h a s an a d h e r e n t p o i n t z
U,
cp(xn) q! U,
such t h a t
...
On t h e o t h e r h a n d ,
~ ( u , )$
such t h a t
6
in
~ ( x )because
E zn
which b e l o n g s
E
U,
n=1,2
,...
T h e r e f o r e we a r r i v e t o a c o n t r a d i c t i o n . E
C o n v e r s e l y , l e t us suppose t h a t a mapping from a P o l i s h s p a c e
u
that
and a n e i g h b o u r h o o d U
of
= E
(cp(x) : x f P]
z
in
P
of
V
such t h a t
i s KSuslin.
i n t h e compact s u b s e t s o f
P
~ ( z )i n
E,
cp(U) c V .
x
and a s e q u e n c e
which h a s n o t a d h e r e n t p o i n t i n
compact, i f
M
in
cp(x).
,...,zn ,...3 ,
M
t e and t h e r e f o r e t h e r e e x i s t s a p o s i t i v e i n t e g e r
... ] n cp(x)
( ~ ~ ~ , z ~ ~ + ~ E, v.i d. e. n]t l.y , bourhood
X
of
x
positive integer znl
E
B fl cp(x
such t h a t n1
n
B
Let
~ ( x = ) $,
B
) c B fl q ( X ) = $ ,
in
Since
n
no
cp(x)
On t h e o t h e r h a n d ,
which a r e n o t KSuslin
[lo].
P
cp(x) i s is fini
such t h a t
hence t h e r e i s a neighWe c a n c h o o s e now a
so that
x
Therefore
P r o p o s i t i o n 1 allow u s t o o b t a i n t h a t e v e r y KSuslin i s semiSuslin.
does
zn 6 ~ ( x , ) ,
E,
E X. "1 hence a c o n t r a d i c t i o n .
n
P
b e t h e c l o s u r e of
cp(X) r7 B = $ .
l a r g e r than
"1
= Q.
(x,)
(zn)
n = 1,2,...
no+P'
in
cp
Let u s suppose t h a t
which converges towards
rzno,zno+l,...,z
such
t h e r e i s a neighbourhood
Then t h e r e i s a s e q u e n c e
{z1,z2
be
E
z
and g i v e n a n a r b i t r a r y p o i n t
n o t v e r i f y c o n d i t i o n 2.
i s the s e t
cp
Let
q.e.d. space
t h e r e a r e semiSuslin spaces
448
M.
LEMMA 1.
Let
F
VALDIVIA
be a s e m i  S u s l i n t o p o l o g i c a l s p a c e .
F
m e t r i z a b l e c l o s e d s u b s p a c e of
then
E
If
is a
E
i s KSuslin.
PROOF.
S i n c e e v e r y c l o s e d s u b s p a c e of a s e m i  S u s l i n s p a c e i s semi
Suslin,
[lo],
w e t a k e a mapping
L e t us t a k e i n
compact.
n = 1,2,.
..
cp(x)
Let
(U,)
t h e sequence
P,
i n the
(zn)
~ ( x )i s
then
(xn)
xn = x ,
siich t h a t
i s a n a r b i t r a r y sequence i n
cp (x)
,
zn F cp(xn),
has an a d h e r e n t p o i n t i n
i s c o u n t a b l y compact.
results that
Let
P
and t h e r e f o r e
rp(x)
LEMMA 2 .
(zn)
If
n = 1,2,..., Hence
i s an a r b i t r a r y point i n
x
P
T a k i n g a c c o u n t of P r o p o s i t i o n 1, i t s u f f i c e s t o
tion are verified. show t h a t i f
from a P o l i s h space
s o t h a t t h e c o n d i t i o n s 1 and 2 of t h e d e f i n i 
E
c l o s e d s u b s e t s of
rp
Since
i s metrizable i t
E
i s compact.
q.e.d.
b e a s u b s e t of a t o p o l o g i c a l v e c t o r s p a c e
A
rp(x).
E
b e a s e q u e n c e o f c l o s e d c i r c l e d s u b s e t s of
E[Z].
which v e 
r i f i e s t h e following conditions:
1.
If
z
an a r b i t r a r y p o i n t
of
a r e given t h e r e i s a positive i n t e g e r 2.
If
(mp)
and a p o s i t i v e i n t e g e r
A
n
P
such t h a t
z
E
p
n U P P'
i s a n a r b i t r a r y s e q u e n c e of p o s i t i v e i n t e g e r s t h e
set
[n i s nonvoid
Then PROOF.
Let
Empup
= i,2, ...I]
n
A
and c o u n t a b l y compact. A[3] N
i s a semiSuslin space.
b e t h e s e t of p o s i t i v e i n t e g e r numbers w i t h t h e
d i s c r e t e topology.
By
NN
of c o u n t a b l e many c o p i e s of
u s c o n s i d e r t h e mapping A[J]
: P
rp
we r e p r e s e n t t h e t o p o l o g i c a l p r o d u c t N.
Then
from
NN
NN
i s a P o l i s h space.
i n t o t h e c l o s e d s u b s e t s of
such t h a t i f
x = (x1,x2
Let
,...,xP' ...) E
NN
44 9
ON SEMISUSLIN S P A C E S AND DUAL METRIC SPACES
then
CP(~)rn Expup :
...I] n
A.
P = i,2,
Using condition 1 of this lemma it follows that A = [pp(x)
x(")
Let
converges towards
If
cp(x(")). teger
n(p)
p
2
x
in
NN
NN.
such that the sequence
Let us take a point
zn
such that
x(~) = x
P (x("))
for every P' towards x in
n 2 n(~),
NN.
because
Then, if
n(p), zn E cp(x ( n ) ) c xPuP
{.An)
...I
then
: n = 1,2,
zn
E
[n CYpup
: P = i,2,
and since this set is countably compact, point
in
zo
by (1).
has an adherent
(zn)
...!I
: P = i,2,
n
A = rp(x)
q.e.d.
THEOREM 1.
Let
be a starshaped metrizable subset of a topolo
A
E[3].
gical vector space intersects
E
A
Therefore conditions 1 and 2 of the definition are sa
tisfied.
PROOF.
...11 n
which belongs to
A[3]
3? Expup
E[3]
(1)
is the maximum of the finite set of natural numbers
If yp
of
in
is a positive integer there exists a positive in
of the convergence of n
,...,xP( " ) ,...) E
= (xin),x$n)
(x'"))
N
: x E N ).
If every closed and bounded subset of
in a compact set,
A
A[3]
is a KSuslin space.
Obviously, it sufficos  t o prove the theorem when the origin belongs to
point of
A
A
and every segment which joints an arbitrary
with the origin lies in
there is a sequence
(U,)
A.
Since
A
is metrizable
of closed circled neighbourhoods of the
M. VALDIVIA
450
origin i n
such t h a t
E[3]
(un n is
R
f u n d a m e n t a l s y s t e m of
A : n=1,2,
...I
n e i g h b o u r h o o d s of t h e o r i g i n i n
O b v i o u s l y c o n d i t i o n 1 o f Lemma 2 i s v e r i f i e d on
(Un).
If
A[3]. (mp)
i s a n a r b i t r a r y s e q u e n c e of p o s i t i v e i n t e g e r s t h e s e t
i s nonvoid,
because t h e o r i g i n has i n i t .
To p r o v e t h i s l e t
t h i s s e t i s compact.
bourhood of t h e o r i g i n i n
q
Let u s s e e now t h a t
V
b e an a r b i t r a r y neigh
Then t h e r e i s a p o s i t i v e i n t e g e r
E[3].
such t h a t
u 9 n ~ c v An. If
z
E
n
(mqUq)
A,
n
E Uq
then
A
and i t f o l l o w s t h a t
¶
[n
Empup
: p=1,2,
and t h e r e f o r e ,
...I] n
A C
( m9 u9 ) n
A C m ( U nn) q 9
t h e s e t ( 2 ) i s bounded i n
E[3].
i t i s evident t h a t t h i s s e t i s closed i n i s compact. verified.
I t allows u s t o conclude t h a t
Let
PROOF.
v
9
O n t h e o t h e r hand,
and, t h e r e f o r e ,
it
intersects
A[%]
i s a semiSuslin
i s KSuslin.
A[3]
b e a s t a r s h a p e d and m e t r i z a b l e s u b s e t o f a t o 
A
pological vector space E[J]
9
We h a v e t h u s p r o v e d t h a t c o n d i t i o n 2 of Lemma 2 i s
s p a c e , and by Lemma 1, THEOREM 2 .
A[3]
c rn ( V n A ) c m
E[3].
If
e a c h bounded and c l o s e d s u b s e t of
i n a compact s e t ,
A
By Theorem 1,
A[3]
i s KSuslin
A[3]
i s separabla.
and t h e r e f o r e L i n d e l 8 f , [ 7 ] .
Since every metrizable Lindelbf space i s separable, complet e , COROLLABY 1 . 2 .
t h e proof
is
q.e.d. Let
g i c a l v e c t o r space
A
E[3].
b e a m e t r i z a b l e convex s u b s e t o f a t o p o l o I f e v e r y bounded and c l o s e d s u b s e t of
ON SEMISUSLIN SPACES AND DUAL METRIb SPACES
E[3]
intersects
NOTE 1.
A
in a compact set,
A. Grothendieck asks in
A[s]
451
is separable.
[4] if every FrBchetMonte1 space
is separable.
J. Dieudonne gave an affirmative answer to this
question in [3]
.
C. Bessaga and S. Rolewicz proved in [ 2 ]
that
every metrizable Monte1 topological vector space is separable. This result can be obtained from our Corollary 1.2 taking THEOREM
3
3.
Let
A = E.
be a metrizable topological vector space.
E
be a topological vector t o p o l o g on
E
Let
coarser than the ori
ginal topology such that the following conditions are satisfied: 1.
There is a fundamental system of neighbourhoods of the
origin of 2.
which are closed in
E
E[3].
Every bounded subset of E is relatively countably compact in
EC3l. E[3]
Then PROOF.
is a semiSuslin topological space.
(Un) be a fundamental sequence of circled neighbour
Let
hoods of the origin in A = E
E,
which are closed in
that lemma is satisfied.
On the other hand, let
quence of positive integer numbers.
n compact.
E[3]
(mp)
be a se
The set
Empup : p=i,2, ...I
and closed in
E[3]
and, therefore, 3countably
Consequently, condition 2 is satisfied.
It follows that
is a semiSuslin space.
THEOREM 4. 3
E
Let us take
I t follows straightforward that condition 1 of
in Lemma 2 .
is bounded in
E[3].
Let
E
be a metrizable topological vector space.
be a topological vector topology on
E
Let
coarser than the original
topology such that the following conditions are satisfied:
1.
There exists a fundamental system of neighbourhoods of the
origin in
E
which are closed in
E[3].
452
M.
E v e r y bounded s e t i n
2.
Then
E[3]
i s a KSuslin
i s r e l a t i v e l y compact i n
E[3].
topological space.
of Lemma 2 i s s u c h t h a t
b u i l t i n t h e proof
x E N
f o r every
x = (x1,x2,
E
EC31.
9.e.d.
THEOREM
5.
If
.
rp(x)
i s compact i n
It follows s t r a i g h t f o r w a r d , s i n c e i f
the s e t
and c l o s e d i n
E
and t h e r e f o r e compact i n
E[3]
i s a Fr6chet space,
E'[b(E'
i f and o n l y i f PROOF.
N
...,xn, ...)
i s bounded i n
A
E
By P r o p o s i t i o n 1, i t s u f f i c e s t o p r o v e t h a t t h e mapping
PROOF.
E[3]
VALDIVIA
,EN)]
E"[X(E",E')]
i s KSuslin
i s barrelled.
L e t us s u p p o s e f i r s t t h a t
i s KSuslin.
E"[X(E" , E ' ) ]
Let
b e a n a b s o l u t e l y corivex c l o s e d and bounded s u b s e t o f E " [ u ( E " , E ' ) ] .
G r o t h e n d i e c k proved t h a t e v e r y c o u n t a b l y s u b s e t of tinuous i n E"[u(E"
Since
,E')].
E"[a (E" ,E' )]
[4],
[ 71
E'[w (E' ,E"
hence
.
)]
and t h e r e f o r e Hence
E"[p(E" , E ' ) ] ,
S i n c e each sequence i n
i t follows t h a t
).(E",E')
A
i s a Lindel8f
i s u(E" ,E' )compact and c o n s e 
A
E'[u(E'
,E")]
is barrelled.
A
and
u(E",E') A
If
E ' [ p (E' , E ) ]
E
is
[S]
,
c o i n c i d e s on t h i s sequence,
i s r e l a t i v e l y compact i n
By u s i n g Theorem 4 w e o b t a i n t h e c o n c l u s i o n . Let
A
i s r e l a t i v e l y u(EN, E ' )  c o m p a c t .
i s equicontinuous i n
hence i t i s e a s y t o o b t a i n t h a t
COROLLARY 1.5.
A[a (E" , E ) ]
of t h e o r i g i n which a r e h ( E " , E ' )  c l o s e d .
a bounded s e t i n
,E')].
it follows t h a t
i s a F r 6 c h e t s p a c e which h a s a f u n d a m e n t a l s y s t e m of
neighbourhoods
E"[X(E"
i s c o u n t a b l y compact i n
i s barrelled.
L e t u s s u p p o s e now t h a t
E"[p(E",E')]
A
i s KSuslin
E"[h(E" , E ' ) ]
i s KSuslin
t o p o l o g i c a l space quently
,
E'[@(E' ,E)]
i s equicon
A
be a Fr6chet space.
If
E
q.e.d.
i s distinguished
453
ON SEMISUSLIN SPACES AND DUAL &QCTRICSPACES
then
NOTE 2.
G
(EN ,E' )]
E"[
I n [4]
i s KSuslin. G r o t h e n d i e c k g i v e s a n e x a m p l e s of a F r 6 c h e t s p a c e
A.
G'[p (G' ,G" )]
such t h a t
i s n o t KSuslin
i s not barrelled.
H
example o f a F r 6 c h e t s p a c e $(HI ,H)
and
f
,*). H
w(H'
g ' [ X ( ~, H ' ')]
THEOREM 6.
E
Let
PROOF. that
If
such t h a t
Komura g i v e s i n
E"[X(E",E')]
i s KSuslin
i s Lindelbf. A
and t h e r e f o r e
E"[X(E"
Since
,E')]
is
E"[
i s an
A
(E" ,E' ) ]
(E" ,E' )  c o u n t a b l y
,
compact
Hence E'[p (E' ,E" )]
q.e.d.
DUAL METRIC SPACES
A l i n e a r topological
if
A
is
E"[X(E",E')]
i s L i n d e l b f and
i s X (EN ,E' )  c o m p a c t .
is barrellled.
11.
5
i s b a r r e l l e d i t f o l l o w s f r o m Theorem
E'[p(E',E")]
[ 41 i t f o l l o w s t h a t
i s Lindelbf
is barrelled.
E'[p(E',E")]
Conversely, i f
)]
an
i s barrelled
H'[M(H',H")]
a b s o l u t e l y convex c l o s e d and boJnded s u b s e t o f
A[A (E" ,E'
[6]
i s KSuslin.
be a Fr6chet space.
E"[X(E",E')]
Lindelbf.
Y.
i s a n example of a n o n  d i s t i n g u i s h e d
space such t h a t
i f and o n l y i f
5.
b e c a u s e o f Theorem
T h e r e f o r e G"[X (G" ,G')
l o c a l l y convex s p a c e
E
i s dual metric
i t h a s a c o u n t a b l e f u n d a m e n t a l s y s t e m of bounded s e t s and i n E'
each
3 (E' ,E)bounded s e q u e n c e i s e q u i c o n t i n u o u s [ 91 , p . 11. A l i n e a r t o p o l o g i c a l l o c a l l y convex s p a c e
E
i s (DF)
h a s a c o u n t a b l e f u n d a m e n t a l s y s t e m of bounded s e t s a n d i n $(E',E)bounded
E'
i f it each
s e t which i s c o u n t a b l e u n i o n of e q u i c o n t i n u o u s s e t s
i s i t s e l f equicontinuous [4]. Obviously,
e v e r y (DF)space i s d u a l m e t r i c .
The f o l l o w i n g
two t h e o r e m s g i v e some c l a s s e s o f d u a l m e t r i c s p a c e s which a r e n o t (DF) *
454
M.
THEOREM
7.
pology
3
If
compatible with the dual p a i r
E',
(E,E')
such t h a t
i s a d u a l m e t r i c s p a c e which i s n o t (DF).
E'[3]
Let
PROOF.
Since
i c
I]
b e a maximal o r t h o n o r m a l s y s t e m i n
i s not separable t h e r e i s a p a r t i t i o n of
E
...,I n ,...
11,12,
n = 1,2,... : i
:
(xi
many s u b s e t s
(xi
i s a nonseparable H i l b e r t space t h e r e i s a t o 
E
on
VALDIVIA
Let
6 In]
In
i n countable
i s not countable,
b e t h e c l o s e d a b s o l u t e l y convex h u l l of
An
in
such t h a t
I
E.
E.
We d e n o t e by
@
a l l t h e s u b s e t s of
E
of
t h e form m
with
a b s o l u t e l y convex bounded and s e p a r a b l e and
A
n i t e l y many n o n  z e r o p o l o g y on
E'[3]
3
If
3
i s t h e to
i s compatible w i t h t h e d u a l p a i r
i s a dual metric space.
(DF)space because
3
{A,
: n=1,2,
Moreover,
...I
i t i s not i t s e l f equicontinuous.
Mackey s p a c e s .
(E,E')
i s not a
E'[3]
E'[3]
and
q.e.d.
The d u a l m e t r i c s p a c e s o b t a i n e d u s i n g Theorem
7 a r e not
Theorem 8 g i v e s Mackey d u a l m e t r i c s p a c e s which a r e
(DF).
F o r t h e n e x t theorem l e t u s t a k e a Banach s p a c e t h e r e i s a n o n  s e p a r a b l e a b s o l u t e l y convex and wekly set
8,
i s a s t r o n g l y bounded s e t
w h i c h i s a c o u n t a b l e u n i o n of e q u i c o n t i n u o u s s e t s i n
not
a fi
o f t h e u n i f o r m c o n v e r g e n c e on t h e e l e m e n t s of
E'
i t i s evident t h a t and
s e q u e n c e of r e a l numbers.
(X,)
X
in
F.
Let
E
t i n u o u s f u n c t i o n s from
b e t h e Banach s p a c e X[o(F,F')]
into
F
compact s u b 
C(X[O(F,F')])
K,
so that
of con
w i t h t h e u n i f o r m con
vergence topology. THEOREM 8. such t h a t
In
E"
t h e r e i s a v e c t o r subspace
E'[u(E',L)]
L
containing
i s a non(DF) d u a l m e t r i c s p a c e .
E
455
ON SEMISUSLIN SPACES AND DUAL METRIC SPACES
PROOF.
Let
M
b e t h e l i n e a r h u l l of
is
M
t h a t t h e o r i g i n of
in
Gb
X
in
F.
X[a(F,F')].
Let
b e a se
(Un)
of t h e o r i g i n i n
q u e n c e of a b s o l u t e l y convex n e i g h b o u r h o o d s m
Let u s suppose
.
n
Un = { O } For each p o s i t i v e i n t e g e r p n= 1 { p Un : n = 1 , 2 , ...I i n t h e f a m i l y of n e i g h b o u r h o o d s of t h e o r i g i n
X[u(F,F' )]
such t h a t
( p X)[u(F,F')]
h a s t h e same p r o p e r t y .
Given t h e p a i r
positive integers there exists a f i n i t e set
A
in
P"
(p,n) F'
of
such t h a t
Ao n p X c p U n pn
being
t h e p o l a r s e t of
Ao P"
If
P
A
in
Pn
i s t h e l i n e a r h u l l of
(M,P)
i s a d u a l p a i r and
Since
P
u(M,P)
F.
: p , 1 , 2 ,...)
(Apn
coincides with
u(F',F)
h a s a c o u n t a h l e Hamel b a s i s i t f o l l o w s t h a t
i s m e t r i z a b l e and, t h e r e f o r e , s e p a r a b l e .
in
F', in
X.
X[u(F,F')]
But t h i s a c o n t r a d i c t i o n
w i t h the h y p o t h e s i s .
It permits t o a s s e r t t h a t t h e r e i s a point which i s n o t
X[u(F,F')]
s y s t e m of n e i g h b o u r h o o d s of f u n c t i o n from fi(xo) we t a k e
= 1,
X[o(F,F')]
= 0,
fi(x)
a bounded n e t i n
E.
x E X
(g,ux)
= g(x),
(f,ux)
= 0, Let
H E 51,
let
51 H*
If
in
be a f u n d a m e n t a l fi
[O,l]
be a continuous such t h a t
il,iz E I (fi
and vil : i E I, S
t h e r e i s a element
f
in
c v
iz is
)
E"
t o this net.
let
f o r every
I}
and l e t
X
vi.
Consequently,
x 6 X,
x E X
in
E
i s a d i r e c t s e t and
w h i c h i s a(E" ,E' )  a d h e r e n t For each
xo
: i
(Vi
i n t o the i n t e r v a l
(I, s)
iz i il.
Let
Gb.
x
ux
b e t h e e l e m e n t of
g E C(X[u(F,F')]).
such t h a t
E'
Obviously,
(f,uXO)=l,
(xo3.
b e the f a m i l y of a l l t h e c o u n t a b l e s e t s i n b e t h e c l o s u r e of
H
in
E"[u(E" ,E')]
.
E. Let
If
M. VALDIVIA
456
L
is a subspace of
E"
containing
Then there is a sequence
,E' ) ]
E"[u(E"
point in
wn The set r) {W,
(g,)
.
which has
E
f
as adherent
Let
x
= cx E
: n=1,2,.
..]
x1 E X,
there is a point
in
Let us suppose that f E L.
E.
: ign(x)gn(xo)i
{x,]
is different from
x1 f xo
Ign(xl)gn(xo)
1 r ~ .
0 such that letting
rm
is finite, there exists a constant
k
be finite and
Now let
T'C
that
m
is finite.
into
I.
Since
I .
Let
f(Vi)
E
I)
it holds
> 0 be given.
We can assume
denote the second projection of
for every i € Io,
R
is a bounded subset of
r'
from Weierstrass theorem for differentiable functions, there exists a polynomial
Further, given that
on
q
R,
without constant term, such that
r', there exists I ) . Hence, letting
y E
Y = (m,i,rl
i E I .
JI =
and
r > 0
such
from (1) and ( 2 )
cpq',
it follows
Since
C!
1 tm
is a constant and
r(m+l)
r'
is finite, taking
E
small
enough it follows Py(cpOf

qaf) L
Now it is enough to remark that
1, q o f
hence from Lemma 4 we conclude that A vector space A c C(X;R)
if
M c C(X;E)
v Y E i'.
E af]
Suppqof c K ,
and

cpof E R[f]
.
is a module over an algebra
AM c M.
THEOREM 6 (reduction to the compact open case).
be a module over a strongly separating algebra
Let
M c C:(X;E)
A c C:(X).
Then
G. ZAPATA
468
A=R C
From Lemma 5, there exists Fix one such Since
such that Suppeg f E
8g E AM c
since
M
8 = 1
such that
on
l"c Tm, we can assume
Suppf.
from Remark 3 it follows that there exists
s 1,
V
y E
Supp8,
fi
r'.
f f 0.
and assume
,
py(8(fg))
f = Of
Since
m