NORTH-HOLLAND
MATHEMATICS STUDIES Editor: Leopoldo NACHBIN
~unZtionolAnalysis
Klaus D. BIERSTEDT
Josh BONET John HORVATH Manuel MAESTRE Editors
NORTH-HOLLAND
PROGRESS IN FUNCTIONAL ANALYSIS
NORTH-HOLLAND MATHEMATICS STUDIES 170 (Continuation of the Notas de Matematica)
Editor: Leopoldo NACHBIN Centro Brasileiro de Pesquisas Fisicas Rio de Janeiro, Brazil and University o f Rochester New York, U.S.A.
NORTH-HOLLAND -AMSTERDAM
' LONDON
NEW YORK
.
TOKYO
PROGRESS IN FUNCTIONAL ANALYSIS Proceedings o f the International Functional Analysis Meeting on the Occasion of the 60th Birthday of Professor M. Valdivia Periiscola, Spain, 22-27 October, 7990
Edited by Klaus D. BIERSTEDT University of Paderborn Paderborn, Germany
Jose BONET Technical University of Valencia Valencia, Spain
John HORVATH University of Maryland College Park, MA, USA
Manuel MAESTRE University of Valencia Valencia, Spain
1992 NORTH-HOLLAND - AMSTERDAM
LONDON
NEW YORK
TOKYO
ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 211. 1000 AE Amsterdam, The Netherlands Distributors for the U.S.A. and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC, 655 Avenue of the Americas N e w York, N.Y. 10010, U.S.A.
L i b r a r y o f Congress C a t a l o g i n g - i n - P u b l i c a t i o n
Data
I n t e r n a t ~ o n a lF u n c t l o n a l A n a l y s ~ sM e e t l n g ( 1 9 9 0 Peiiscola, Spain) proceedings o f t h e I n t e r n a t i o n a l Progress I n f u n c t ~ o n a la n a l y s l s F u n c t l o n a l A n a l y s l s M e e t ~ n go n t h e o c c a s l o n o f t h e 6 0 t h b ~ r t h d a y o f p r o f e s s o r M. V a l d i v i a , P e n i s c o l a . S p a l n . 2 2 - 2 7 O c t o b e r 1 9 9 0 / e d ~ t e d b y K l a u s D. B l e r s t e d t [et al.1. p. cm. -- ( N o r t h - H o l l a n d mathematics s t u d i e s 170) ISBN 0-444-89378-4 1 . F u n c t l o n a l analysis--Congresses. I . V a l d i v l a . M a n u e l . 1928. 11. B l e r s t e d t . K l a u s - D l e t e r . 111. T i t l e . I V . S e r l e s . QA319.158 1990 515'.7--dc20 91-42139
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CIP
ISBN: 0 444 89378 4
O 1992 ELSEVIER SCIENCE PUBLISHERS B.V. All rights reserved N o part of this publication m a y be reproduced, stored in a retrieval system or transmitted. in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior wrltten permission of the publisher, Elsevier Science Publishers B.V., Permis sions Department, PO. Box 521, 1000 A M Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication m a y be made i n the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred t o the copyright owner, Elsevier Science Publishers B.V., unless otherwise specified. N o responsibility is assumed b y the publisher for any injury andlor damage t o persons or property as a matter of products liability, negligence or otherwise, or f r o m any use or operation of any methods, products, instructions or ideas contained i n the material herein. pp. 191-200, 367-382: Copyright not transferred. Printed in The Netherlands
PREFACE Whcn t l ~ 60th c birthday of Prolcssor Man11c.I Valtlivia w;ls apl)roaclling, some of his fornier stutlcnts from tlie two universities of Valc~lriatlccitletl to liold an illtcrnational functiottal analysis ~tieetingin his Itonour. T l ~ Orgat~izi~tg c Contmittcc (corlsisting of K.D. Bicrstedt, J. Bonet. J. Horvitl~and hl. Maestro) was formetl (luring the sunlnlcr of 1989, morc than onc year ahcad of the scllcduled tlatc of tllo congrcw. Profes.wr Valdivia is an intcrnatiollitlly rccogtlizt:d f111ictiorla1 analyst and one of tllc hcst known S1)anish mathcmaticiarls. Ilc: has ~)ul)lisIictlniore tltatl 125 artklcs, rnost ol them in renowned international journals ( i l l E~iglisli),ant1 a mosograph or1 locally convcx spaccs Stutlic3, Ilc lras I)c-rllinvitrcl to Inany uriivcrsities in the North-Hollancl serics hlatllc!~l~ntics atitl Ilia given invited talks at many nicrrtirlgs all ovclr th(: \\*orlcl.Valtlivia usctl to Iccturc ahroatl I)y writing in E~igliulion tI~c.I)lac.l;l)oartl, I ~ l talkitlg t i n Spitnish. In 1988, to the surprisc of the auclict~ce,Iic finally started sl)caki~~g ICnglisll iri his Ic.ctures at itltcrtiatio~ial con fel-cncw. Professor Valtlivia tnadc irnl)orta~itco~itril)utio~ls to scvcral arcas of functiotial a~~alysis. Arllolrg tllc topics wl~icl~ Ilc: l)c.c.atl~c*b ~ i ~ o for, ~ r s ivc. ol~lyI I I ~ I I ~ ~ Otlic I I closctl g r a ~ ) I ant1 ~ ope11 mapping I.hcorcms, I)arrellc-(I,I)orl~ologici~l iuitl cll~.ral~or~lological spaces, Frbclict ant1 (DF)-spaccs, srqucncc space ~.c.l)rc.sc~~tirtio~ls of sl)iiccs of fllnctio~lsant1 distril)~~tiot~s, and Ior~iological artd of f3,-cornplcto spacc*s wl~icl~ fir(*r~ot11-co~~~plctt'. IIe solved some prol)le~nsof Grotllcntlieck, i t t particuliir o t ~ c0 1 1 1)rot111e1~ of t ~ ~ i ~rcflcxivc lly SI)BCCS. Ijy giving it I,c;u~tiful cllarac.torization of thc totally rcflc~xivcI:ri.cl~et sl)i~ccsi t ~ ~l.hcir tl (IIIAIS.111 r ~ e t ycars, ~t 11(! turnctl his intcrcst to B R I I ~ Csl);t(.cs I I ~llc*oryi ~ ~ i ( 111orc l, s~)(~.ilicilIly, to \ ~ ~ i (.onlpact~iess i k atltl rcsollltions of tile itlcritity. '~'II(. tvork of Viildiviil has llatl a profoll~itlitlil~ircton CUIICtiatla1 ar~alysisworldwitlc. It was used alltl tllc*~ltiot~e~d i l l r~litllyarl.i(:Ies i\~ldwas rcfcrrcd to r.g. i l l tllc I,ooks o l Iiiitllc, Jarellow ant1 1'brc.z (:;tl.rc.ras/I3o1lct. Valdivia camc to Valcncia as a Flrll I'roft-ssor i n t l ~ (Fitct~lt~ * of SC~CIIC(: of tlrc I.l~iivcrsity of Valcnc.ia in 19G5. 111 tlic ticxt ycbi\l.,IIC \\-as one of tllc fot~riclcrsof tlrc Section of Mathclnatics (later 011, F'il~~Ity 01 hlaI.l~('l~ii~tit.~). 'I'II(*~ I I I ~ I I Y S ~ Sg1.0111)of Valclncia origitbatetl will1 I ~ i t t ~Ilowe\rcr, . Iris i~lfluencc.i t 1 f u ~ l c t i o ~alli~lysis ~ ; ~ l i l l Sl)aitl w c ~ far ~ t I)cyontl Valc~~cia, atitl olie 111aysay that. nowatlays f ~ ~ ~ ~ ( . t iit~~iilysis o t ~ i l l is ollcSof I.11c. ~nosl.l)ror~~inctll lopics in triatlicmatics it1 Spaill, tluc. to hlii11uc.1Villtlivii~. III turrr, tllc. work of Spanis11 nlatllctnaticiaris i l l furlctiol~alatialysis is c l l l i l c * I ) ~ ~ I I I ~ I I ~ illt.c*rr~ntiol~HIly. ,IIL
A thoro11g11a1)preciatioll of Valclirii~'~ ~ l ~ i ~ t . l ~ c ' t ~ ~\\'ark ; ~ I . i ca110 . i ~ l pr(-cisc: I.C~I!I~~:IICCS 10 his n~aititl~coremscall IJe fo1111(l i l l .I. I lorviit.l~'si~l.ticlt% at tllc I)cgint~i~~g of tl~cseProccvdit~gs (wlricl~corrcspontls to his Icct~lrc,cluri~~g 1.11(. olw~li~lg ccrc~ilot~y of 1.I1e. I'ciliscola rrlc,cting). Recognizing Valtlivi;r's i~llportancofor mathematics both in and out of Valcncia, the two
universities of Valer~ciawere quite 11appy t,o s p o ~ ~ s tolr~ cint,crnat,ional functional analysis ~ n c c t i ~ on l g the occasior~of his G01,11birt,l~day,and o t l ~ c rs p o ~ ~ s ojoincd rs them. \Vc woultl likc to acknowletlge their support and, in particular, we would also likc t o t,hanli t h e membcrs of the IIonour Committ,ee i ~ this i rcspcct. T h e "Centro tlc Estudios dc la Funci6n Pi~blica" a t Periiscola, Castell6n, ahout 130 k m from Valencia was chosen as the site of t h e congress. This was a perfect clloice in scveral rcspccts, and we are very grateful t o tllc people i r ~charge of the Celitro cle Est,udios for creating a good (worl.;ing) a t ~ n o s p t ~ e r e for the n~ect~ing. T h e topics of t h e corlgress were centeretl arour~tlt h c ( n ~ a n y )i~ltcrcstsof I'rofessor Valtiivia in f r ~ ~ ~ c t i oanalysis. ~lal I11 S r p t e ~ n l ~ cIgS!), r the organizing committ,ec contacted 30 specialists (from several countries) ant1 i~ivitcdtlicm to attentl t h c Pciiiscola 111eeti11ga11t1 t o present l h e main talks. Practically cveryl~odyreplied immrtliat,cly and agreed to come. Wlicn t,he annor~ncer~lrnt of t h e 111c(>t,i11g apl~carctl(c.g. in t l ~ cE u r o p e a ~ ihlathematical Newsletter and in tlic Notices of t l ~ c.A~ncricanh/Iatlle~naticalSocicty), togcl.l~erwit11 ;t list of the main spcakcrs, the o r g a ~ ~ i z ci o~ ~~ng~ l l i t t crcccivetl e many letters with requcsts for participa.tion in t h e congress and Inany abstracts for consideratio~ii l l t h e section of short, communications. A carefr~ls e l e c t i o ~was ~ ~natlc,allti we t l ~ a n kscvcral colleagues who helped us ill the cvaluation. T h e last weeks bcfore tllc r n c e t i ~ ~wcrc g t l ~ tiliie c for Inally prcl~arat,ions,ant1 wllc~ltlic two g callicxt o Vale~~cia, s l ~ o r t l ybefore tzhecongress, foreign members of the o r g a n i z i ~ ~committcr thcy rcalizctl how m u c l ~work liatl alrca.tly bccn tlo~reby t.lieir Spanish collcagues. During t h e congress, t h e amount of worlc con~~rct.etl wit,ll tlie organizat,ian was sornet,i~nesovcrwhelming. T h o orga.r~izingcornmittcc gratefully a ~ l i ~ ~ o ~ lthat ~ c lmuch g ~ s of all this was done by t h e members of thc S u p p o r t i ~ ~Committ,cc, g R I I wc ~ t.ha111ttlicm for t h e i ~ i v a l u a l ~ l c help. During t h c wcck a t Pciliscoln ( O c t o l ~ c 22--27, r 1990), 2S invit,ctl lcctures of 50 ~ ~ l i r ~ u t7c s , invited lectures of 25 minut,es a ~ l t :3S l short communica.t,ions of 15 minutes wcrc prcscntcd. (l'hc Sclictlule of 1,ectures can b e f o u ~ ~i ltl l t h e editorial part of t l ~ i sLook.) Probably none of tlie 96 participants will cvc:r I'orget t l ~ vsl)c:ctac:~~lilr I ~ r g i ~ ~ nofi ~tllg~ cmeeting, wit11 a severe thur~dcrst,ormduring t l ~ efirst Iccturc; we all adlr~iredRicl~artlAron for goir~go n with some pertinent remarks w l ~ c ntllc \rorltl l ~ c c a tla.rl.; ~ ~ ~ant1 e t l ~ elights welit on and off scveral times. Fortunately enough, t l ~ cwcatlicr improvetl, and t h e meeting turncd out t o be t,he success we had hopcd alici worltctl for. From t h e list of speakers and lcctures a~rtlfrom tlic abstracts it had alrcady 11ec.11o1)vious t l ~ a ttlic mathematics woultl b e interesting and very good. It renlains to thanlt the spca.licrs for t,heir e x c e l l c ~ ~inspiring t, and well prepared talks, the ct~airperso~ls and all thc participar~tsiri thc mccting for their help, their interest and for many st,irrir~lati~~g tliscussions. B u t also t h e social part of t.hc congress turned out fine: T h e old city of I'eiiiscola is quite pictl~rcsquc,a.nd the visit t o its ca.st.le provided 11s with b c a ~ ~ t i l ' sights ul a.nd nice pl~otos.T l ~ excursioli e on Wcdncsclay afternoon sl~owedanother intcrcsting pl;tco, Morclla, ~ ~ too o tfar f r o ~ nPciiiscola. All spealicrs of invited talks at, the rnecti~lgl ~ a dbee11 invited to submit a n article to t h e Proceedings volume. T h e papers were not t o cxcec~tl20 pages, u~llessautl~orizedby t h e
Preface
vii
organizi~rgco~rriniltcc,wllicl~also f ~ ~ ~ ~ c t i o;LS l l cc~tli1,ors d of tllc I'roceedir~gs. T h e original tlcadli~rewas March 1, 1991. Ilitlec~l,111ostof tllc sl~caltcrscontril,uted, Lut t h e last papers only arrived in May. T h c articles were refereed, and we tlla~llcrof papers, final corrections had t o b e made a t Padcrborn (where tllc originals werc s c ~ ~ ttluo ) , t o lack of tinie. (We wo~ildlike t o tliank Mrs. Dudtlcck-Buijs for I ~ e rco~nljc.tericc.) At t l ~ i sp o i ~ ~we t , also thank tllc editor of "Notas tle MatemAtica", Professor 1,col)oltlo Naclll~in,tlrc pul~lisher,Drs. Sevcnster, and the Elsevier/North-IIolland company for i~icludi~lg: this book in their series "Mat,hematics Studies". As a gla~rcea t the t,al~lrof cont,er~t,s slloivs, t,lie p r c s e ~ Proceetli~igs ~t volume contains 27 articles on various interesting areas of prosc,~lt-(l;tyfu~lctio~lal il~~alysis: Banacll spaces and their geometry, weal; conrpact~lcss;operator itlcals, tc~lsorproducts a.nd tensor norms; Frkchct, (DF)- ant1 (12)-spaces, I)ilrr(~IIotlSl)ilCcS; seilue1lc(:, fu~lctioriand distribution spaces, as well as infinite dinrc~lsio~lal Iiolonlorljhy (t.oget,ller wit11 a "touch" of automatic continuity ant1 a p l ) r o s i ~ n a t i ot l~~~c o r y ) .S ~ I I Iof~ t,lic : rctports ivc receivetl frorn referees corlfirm our i m p r e s s i o ~tliat. ~ t11e i~ut,l~ol.s I I R V ( ~ ti1lx~11rr~uclicare with their articlcs and that Inany of the pal>ersprcscnt i ~ ~ l l j o l . t rcsl~lts ; ~ ~ ~ t a ~ ~ ~nc:tllods cl in active fields of research. l t , t l ~ i scollection of papers, dedicated 1.0 Professor We Iiope, and are ( ~ u i t ec o ~ l f i d ~ ~tllat hlanucl Valdivia on t h e o c c a s i o ~of~ l ~ i s60111 I > i r t h d i ~\vitlr ~ , best wishes for t h e future, will prove hcll,ful, valual~leand i ~ l s p i r i ~for ~ g a ~ ~ y l ) o t linterestetl y in f~~nctiorial analysis and related fiolcls.
Collcgc I'arlt/l'adcrl~or~~/Vitlc~~cia, July 1991
K.1). Ijierstetlt, J . l3011(>t,.J. IIorviitll, M. Maestrc
INTERNATIONAL FUNCTIONAL ANALYSIS MEETING on the Occasioil of the 60th Birthday of Professor M. Valdivia Pefiiscola, Castcll6r1, 22
-
27 October 1990
SPONSORS Presidcncia de la Gerieralitat Ministcrio d c Educacibri y Ciencia Conselleria dc Cultura Educaci6 i C i e ~ ~ c itle a , la Generalitat Valenciana Univcrsitlad Polit,&cnic:a de Valr11t:ia Universitat de Valcncia Facultad d e h4atemiticas, Universitat tlc Valencia E.T.S.I. Arquitectura, Universidad Polit6c11ica dc Valcncia E.T.S.I. Telecomunicaciones, 'LTnivcrsitlad Politbcnica tlc V a l e ~ ~ c i a Departalllent0 d e Anrilisis Matemit,ico, Universit,at, tle Valencia Departa.mento d e Matemitica Aplicacla., Universidatl Polit4cnica d e Valencia A y u ~ ~ t a m i e n tdoe Pefiiscola
HONOUR COMMITTEE Molt IIonorahle President de la Cencralitat D. Joan Lern-la Ilustrisi~noSefior Conscllcr clc Eclucaci6 y Cultura D. A ~ i t o l ~ Escarrd io Ilustrisimo Scfior Director Gclieral tle lJ~~ivcl.sitlades I).Antonio Clcmcnte Magnifico Scfior Rector d e la Universitat d c Valcncia D. IZam6n Lapiedra M a g ~ ~ i f i cSefior o Rector d e la Univcrsitlatl I'olit,kcnica tle Valencia I). J u s t o Nicto
ORGANIZING COMMITTEE Prof. Prof. Prof. Prof.
Klaus D. Bierstcdt, Univ. P a t l c r l ~ o ~ nl~eclcral , Republic of Gerlnany Josd Bonet., Univ. Politbrnica tlr Valencia, Spain John IIorvrith, Univ. h4aryla11tl,(:ollrge Park, Maryland, USA Manuel hlaestre, Univ. Valcncia, Spain
SUPPORTING COMMITTEE Prof. Prof. Prof. Prof. Prof.
Pablo Galinclo, Univ. Valencia Donlingo Garcia, Univ. Vnle~icia Manuel Lbpez, Univ. PolitCcnica dc Valrntia Vicente Montesinos, Univ. Politircnica de Vale~icia Jose Luis Santos, Univ. Polit6cnic.a tle Valc~~cin
INTERNATIONAL FUNCTIONAL ANALYSIS MEETING on the Occasion of the 60th Birthday of Professor M. Valdivia Pefiiscola, C i ~ ~ t ~ l l 622 l 1 , 27 October 1990 -
SCHEDULE OF LECTURES Monday, October 22 Morning Session Chair: 1i.D. Bierstcdt
~ o r ~ s f1111ctions 9.30 R.M. Aron, Weak-star c o r ~ t i ~ ~ r al~nlytic 10.30 J. Schmcts, O n t l ~ cextcrit of thc (non-) quasianillytic. classes 12.00 Opening Ceremony, i ~ l c l ~ ~ t l i ~ ~ g : J. IIorvith, T h e mathematical \vorlts of h l a ~ l r ~ Valtlivia cl Reply of Professor M. Valtlivia Afternoon Session Chair: J . Mujica
16.00 Ilcfu~lcliol~s 12.30 A. I'clzyliski, 'I'raaslatioi~ i ~ l ~ i ~ ~)roj(!ctioi~s r i i ~ ~ ~ t ill s1)accs of diffcrcntiirblc functions ancl Paley's cflcct 15.00 Visit t o Morella
Thursday, October 25 Morning Session Chair: A. Pclczyriski 9.00 G. I'isicr, Colnplctc b~untl(*tl~lc~s~ for lii~11i1('11 sl)irccs
Cl~airs:A. Pictscl~(S), M. hliu.slrc. (Ii) 11.30 W. Scl~acllcrmaycr,Slicil~gs,sc~lc*ctio~~s i111tl al)l)lici~tio~ls (S) 12.30 J. Oril~llcla,Coilll)actncss i l l f1111ctio11 s1)ilcW (S) 13.30 W.11. Dcrrick, Irllcrior pro~)~rtic!s ancl fixccl points of ccrtniil disconti~luouaopcralors
(S) D.N. Zarnatlze, .lames sup-t llcorcm [or I?ri.chc:t spilc(3 irrlcl gcncralizatioll of Crotl~c~ldicck's I ~ o r n o ~ r ~ ~ r t11i~orc111 p l r i s ~ ~( I~) ) Afternoon Session Chair: J . Dicstc!l
17.00 IONSORS A N D CORI~II'YI 1:I:S SCI~EDUI,L< O F LECTUILES LISTOF PAH'I'ICIPANTS P R O T O C ODEL ~ , ~ACTO D E A I ' I : I ~ I ' I J I ~ A REPLYOF PROFESSOR M. VAI,I)I\'I,~ LISTO F C O N ' I ' R I ~ ~ U ~ ' O I ~ S T h e mathc,matical works o f hl;~~rucsl V'tltlr~i'r
J. H o r v i t h Regularity properties of (LF)-sl),rcc's
D. Vogt Some applications of a t l e c o ~ ~ ~ l )io11 o \ i l1 1 1 ( ~ l 1 1 0 ~ l
A. Aytuna, T. Terzioglu 011t h e ra.nge of t h e norcl m;rp for cli\sschs ol' ILOIL-cl[lasia~lirlytic functions J. Bonet, R. Meise, B.A. Taylor Biduality in Frdchct and (1,B)-sl)ncc\
K.D. Bierstedt, J. Bonet Holomorplric ~ ~ i i ~ p p ior l ~I,ou~ltl(~l gs t~.1)('011 ( I > l ~ ) - s ~ j a r c ~
P. Galindo, D. Garcia, M. Maestre Linearization of Iiolomorpllic r r ~ ~ l l ) l ) i01~ ~I IgO \I I I I ( I C ~ (y11c'
J. Mujica Spaces of 1iolornorl)hic f u n c t i o ~ ~a~rtl s gc~111s ~ I cI ~ u o t i o ~ ~ t s J.M. A n s e ~ ~ l iR.M. l, Aron, S. P o n t e Automatic co~itiliuityof i ~ ~ t c r t w i ~i ~r r ctol)ologici~l ~rs vcct,or s l ~ r c c s
K.B. Laursen Barrelled furlction spaces
L. Drewnowski, M. Florencio, P.J. Paill On d i ~ t i n ~ u i s l ~Frbchet ed J. Bonet, S. Dierolf
S~)RCCS
Prcquojedions arid their dual5
G. Metafune, V.B. Moscatelli I'rol,lcms from the I'6rcz Carrcra\/l3o11c't Ijoolc
S.A. Saxon Int,crior properties and fixcxl ~)oirl(.s ol' c.o~.ti~i~r ( ~ ~ S ( . O I I ~ . ~ I IOI ]I )~~I' ITSR ~ , ~ I ' S
W.R. Derrick, L. Nova G.
...
Vlll
ix xv xviii xix xxiii
xxviii
Table ot Coflteflls
Functioaal analytic aspects of gcomctry. Linear extending of m c t r i a and relatcd problems C. Bessaga 1,ototsky-Schnabl operators on the unit interval and degenerate diffusion equations F. A l t o m a r e
On
weakly Lindelof Ranacll spaces
J. Orihuela
Distinguished suhscts in vector seclucncc spacm F. Bombal Wcak topologies on bo~tncleclsets of a Datlaclt space. Associated function spaces J .G. Llavona Factorization of mltltilirlcar operators
J. Taskinen Cotnl)lcx geodesics on convcx domains S. Dineen, R.M.T i m o n e y Continuity of tcnsor procll~ctopcriitors bcl.wrcn spaces of Rocl~nerint.egral,le ft~~lrtions A. Defant, K. F l o r e t Compact convcx scts in tltc two-climcnsional complcx lincar spacc wit11 the Yost properly E. Bellrends
S~)IIIP relnarks "11 a Ii111itc1a.s~of a.pl,roxitt~atio~~ i(It-als F. Cobos, T.Kiihn Solnc factorization ~>ropcrtics of co~nposilion01,crators H. Jarchow Eigcnvalucs of nuclear opcrntors on 'I'sirclsot~apace
A. Pietsch Absolutely sutntni~~g surjections fro~nSol,olcv spaces in tllc utlifor~nnorm A. Pekzyriski, M. Wojciecllowski
Progress in Functional Analysis K.D.Bierstedt,J. Bonet,J. Ho~ath& M. Maestre (Eds.) O 1992 Elsevier Science Publishers B.V. All rights reserved.
The Mathematical Works of Manuel Valdivia John Horvath Department of Mathematics, University of Maryland, College Park, MD 20742, USA Introduction On January 12, 1968 Ricardo San Juan Llosa, professor of mathematics at the University of Madrid, wrote me the following: " I read cursorily the magistral work of Prof. Kothe on topological vector spaces. A professor ("catedratico") from Valencia, Prof. Valdivia, whose doctoral dissertation
I directed, works in this and I think that he will obtain results; he seems to me to be a good researcher."
I hardly know more prophetic words, since the same year appeared the first articles of Valdivia on the closed graph and open mapping theorems, and with these a torrent was released.
In the course of the last twenty-
two years he published about hundred and twenty papers on topological vector spaces and related subjects, in which he solved difficult open problems, created new concepts, and inspired an enormous quantity of research in Spain as well as in a large number of other countries. Valdivia's papers contain over a thousand theorems.
It would be to-
tally impossible to do justice to all of them in a limited amount of space.
I have therefore selected those which are the simplest to state. This has the disadvantage that precisely the most difficult, most general and deepest results will not be mentioned. On the other hand, I will not be able to give an idea of the proofs, which are often constructed with the stupendous ingenuity that characterizes the work of Valdivia.
I shall use mostly the terminology and notation of Bourbaki's treatise { 5 } , which differ sometimes from those used by Valdivia.
Numbers in square
brackets refer to the works of Valdivia, those in curly brackets to the works of other authors, listed at the end. To lighten the exposition, I shall occasionally use a loose language. Thus I will speak about a "space" instead of a "locally convex Hausdorff topological vector space"; I shall
2
J. Horvafh
say "inductive limit" when I mean "the finest locally convex topology for which certain linear maps (which most often will not be specified) are continuous". I will also omit hypotheses which are obvious from the context. 1.
SEQUENCES AND FAMILIES OF CONTINUOUS, INTEGRABLE AND HOLOMORPHIC FUNCTIONS Valdivia's research career did, however, not start with the explosion
of 1968. In his doctoral dissertation [ I ] he considers concepts which are usually defined for maps from a topological space into a metric space or a uniform space. He characterizes uniformly convergent, equicontinuous, Cauchy, and quasi-uniformly convergent sequences of maps from a metric space X
into a metric space Y, and then uses the characterizations to
define these concepts when X let
and
be a topological space, Y
X
Y
a regular topological space, and (f,) a
sequence of continuous maps from X a map
f : X+Y.
are both topological spaces. Thus into Y which converges pointwise to
Given a neighborhood V
of
y = f(x), let
n0eN
be
such that f (x)EV whenever ntno. Valdivia says that (fn) converges n uniformly to f if for every X E X and every neighborhood V of f(x)
n f-'(~) is a neighborhood of x; if X is compact and Y a n n2no separated uniform space, then the definition coincides with the usual one. the set
The limit function f
is then continuous.
Another example: a family of maps f. at
the point
the subfamily
XEX
:
X+Y
(ieI) is equicontinuous
if given a closed set F c Y
and an open set A > F ,
(fjIjEJ, consisting of those f. for which
verifies the condition that
n f:l(~)
J
f.(x) E F,
is a neighborhood of
jeJ shows that under mild assumptions concerning X
and
J
x.
Valdivia
Y the Arzela-Ascoli
compactness theorem holds. In the same vein he introduces quasi-uniform convergence, called quasi-approximative in [ 7 1 , which leads to necessary and to sufficient conditions for the limit of a net of continuous maps to be continuous. In the last section of [ I 1 Valdivia introduces conditions which ensure that the limit of a sequence of maps will be absolutely continuous. same section and in [ 2 ] he obtains conditions for
In the
The mathematical works of Manuel Valdivia
to be satisfied. The basic result he proves is the following: Let p(X)
<m
and
f (x)+f(x) almost everywhere on X. Then (1) holds if and only if n the set function p(E) = lirn sup11 f dpl is absolutely continuous. If E n m
X = U X with X . c X and p(Xj) < m , then the additional j=1 j J j+l condition that lim 1 f dp = 0 uniformly in n will imply (1). One of n j-xoX\XJ his theorems is the useful generalization of the Lebesgue dominated converp(X) =
m,
gence theorem called sometimes "Pratt's lemma" 128, Chap.4, Th.17, p.92). Article [31 is related to a paper of San Juan 129).
Let
D
be a
connected open subset of C
having 0 on its boundary, and ("'n'nt~ a sequence of positive numbers. The problem consists in finding necessary and sufficient conditions that given a sequence
(an) of complex numbers
there exists a function f, holomorphic in D, such that
for
ZED
and
n > 1.
Valdivia gives various criteria, distinct from
that of San Juan. For the proofs, he equips the set
A(D)
of holomorphic
functions satisfying (2) with different equivalent metrics for which A(D) is compact. 2. BARRELLED SPACES AND THEIR ANALOGUES
Bourbaki introduced barrelled spaces in order to characterize those spaces for which the uniform boundedness principle is valid.
Let me recall
that a barrel is an absorbing, balanced, convex, closed set, and that a space E
is barrelled if every barrel is a neighborhood of
valently, if every subset of the dual
0
or, equi-
E', which is bounded for the weak
topology o(E1,E), is equicontinuous. The class of barrelled spaces is trivially stable with respect to inductive limits but a closed subspace of a barrelled space is not necessarily barrelled. Dieudonne proved that every finite-codimensional subspace F of a barrelled space E
is bar-
relled, and Valdivia [17; A, Chapter I, 93, 2(9), p.471 proves the same
4
J. Horvath
for subspaces F
of countable codimension.
The result was proved inde-
pendently with a different proof by S. Saxon and M. Levin (30) and a simple proof was given by J.H. Webb (36).
To prove the theorem, one considers an
o F c F 1 cF2c-..cE of subspaces, each of codimension one in the next, and starting with a barrel T = T0 in F one increasing sequence F
constructs a barrel
=
such that Tn+lnFn = Tn.
Tn in Fn
U Tn is a barrel in E, the battle is won.
that
If one can show
Therefore, Valdivia's
n>O proof is based on a theorem about increasing sequences of balanced, convex sets, which explains the title of [171, and which is presented with a minor simplification in (35, pp. 54-55), where on pp. 211-217 a generalization due to M. De Wilde and C. Houet can be found. Valdivia returns to sequences of convex sets in [501, where he makes use of the topology T
of uniform convergence on sequences converging to co in the Mackey sense (19, §28,5}. Let me recall that a sequence (x in n the space E convernes to X E E in the Mackev sense if there exists a
0
bounded subset B
of
E
with the following property: for E > O
there is
x - X E EB whenever n t N; it is a Cauchv sequence n in the Mackey sense if for c > O there is an N E N such that x - x E C B n m whenever n,mtN. The space E is locally complete if every Cauchy an N E H
such that
sequence in the Mackey sense converges in the Mackey sense to some X E E . If
(F,G) is a pairing of two vector spaces, the Mackev to polo^ -c(G,F)
on G
is the topology of uniform convergence on o(F,G)-compact, convex
F. The space F
subsets of topology 3
on G
is the dual of
if and only if
G for a locally convex
T belongs to the interval [ o , ~ ]in
the ordered set of topologies {5, Chap.IV, 41, no.1, ThkorGme 1, p.IV.2). Valdivia says that E
is a Mackey space if its topology coincides with
7(E,E11. In [501 Valdivia gives new proofs of some results of [171, and it is this paper that he follows in Chapter I, 93,l of [A]. He proves that if
E
is a Mackey space, E' is locally complete for t(E1,E), and the completion E
of
E
is barrelled, then E
is barrelled [A, Chapter I, 93,1.(7)1.
(An) an increasing sequence of closed, convex subsets of E such that U A =~ E. If E is a Baire space, then there is an N E N such that AN has a non-empty interior [17, Th.4; 50, Corol.4.1; A, Chapter I, 43, 1.(4)1. Let
E
be a barrelled space and
,.
The mathematical works of Manuel Valdivia
If E
5
is a metrizable barrelled space, and there exists a sequence
(A ) of bounded, balanced, convex, complete subsets such that E n span of UA,. then E is a Banach space.
A balanced subset A if
AA>B
is the
of a locally convex space absorbs another set B
for some A > O .
A set which absorbs all bounded sets is said
to be bornivorous.
E is:
A locally convex space
infrabarrelled (or quasibarrelled) if every bornivorous barrel is a neighborhood of
0,
bornolonical if every balanced, convex, bornivorous set is a neighborhood of
0,
ultrabornolo~ical if every balanced, convex set that absorbs all balanced, convex, compact sets is a neighborhood of
0.
E is a infrabarrelled if and only if every strongly bounded subset of E' is equicontinuous; it is bornological if and only if it is the inductive limit of normed spaces; it is ultrabornological if and only if it is the inductive limit of Banach spaces. One has the implications: barrelled
infrabarrelled.
bornological It is easy to give an example of a non-barrelled, bornological space: R ( ~ as )
a subspace of
t2.
In 1954 L. Nachbin and T. Shirota considered
independently the space & ( T I
of continuous functions on the completely
regular Hausdorff space T, equipped with the topology of uniform convergence on compact subsets of
T.
They proved:
1) & ( T ) is barrelled if and only if every closed subset of which every 2) &(TI
f~ P(T)
is bornological if and only if
{ 1 4 , Chap.8)).
T, on
is bounded, is compact;
T
is replete ("realcompact"
6
J. Horvath
The continuum hypothesis implies that there exists a completely rethat satisfies condition 1) but is not replete, and there-
gular space T
fore there exist non-bornological barrelled spaces.
It was of interest to find such spaces without using the continuum hypothesis. This was first
done by
Y. ~omura,and in
uncountable family of barrelled spaces the subspace G
1 Ei of an i~I (e.g. lines) and considers
[ 2 2 ] Valdivia takes a product E =
consisting of those
#
{O)
for which x.=O with the
exception of countably many indices. He proves that every subspace F
E containing G properly, such that the codimension of G
in F
of
is
finite, is barrelled, dense in E, and not bornological. The same result holds replacing "barrelled" by "infrabarrelled". Using a similar construction, Valdivia proves in [291 that if E is a separable infinite-dimensional Banach space, then E equipped with c ( E , E 1 ) can be imbedded as a dense subspace in a barrelled space that is not bornological. In his Silivri lecture of 1973 [271 he gives other imbedding theorems. Thus every metrizable barrelled space F
is a closed
subspace of some metrizable ultrabornological space E; and every barrelled bornological space F space E. Since T
is a closed subspace of some ultrabornological
He takes E = F(T), where
T
is F' equipped with r(F1 ,F).
is replete, an extension due to M. De Wilde and J. Schmets of the
Nachbin-Shirota theorem implies that
E
is ultrabornological. A different
construction shows that every bornological space is a closed subspace of some ultrabornological space. There are in [27] also representations as quotients. Recall that a space is semi-Monte1 if every bounded closed subset is compact; it is Monte1 if it is barrelled and semi-Monte1 {S,Chap.IV, 92, no.5, Dkfinition 4, p.IV.18).
Then every bornological space and every separable space is a
quotient of some complete semi-Monte1 space. Further representations as closed subspaces and as quotient spaces of ultrabornological spaces are contained in [331 for spaces which have families (UiIiEI of neighborhoods of
0 such that
n
U = {O) i i~I
and the cardinal of
I
is non-measurable
{14, Chapter 12). In [Sll Valdivia proves that if p.345; 26, 4.1.2, p.70) and
E is a nuclear space (17, 16.1.4,
F any separable infinite-dimensional Banach
space, then there exists a balanced, convex neighborhood V
of
0
such
7
The mathematical works of Manuel Valdivia
$ is
that
norm-isomorphic to
the subspace N quotient of
F; here
E
on which the gauge qv of
V
qv, which is a norm on E/N.
v
is the quotient of
E modulo
vanishes, equipped with the
For special F this result was
discovered by several authors t17, 21.2.5, p.483 and 21.11, p.517).
A
complete nuclear space E, whose topology is not o(E,EJ), is the projective limit of an arbitrarily given family of infinite-dimensional Banach spaces.
Answering a question of Dieudonne, Valdivia constructed in [ 5 6 ] a bornological space E
whose completion
k
is not bornological. Actually
E
is a space P(T) with an appropriate replete T, and it is a hyperplane
in
t. An ultrabornological space is both barrelled and bornological. In the
first edition of {5) Bourbaki asserts that it is not known whether the converse is false.
In [I81 Valdivia shows that if E
an infinite family of ultrabornological spaces
#
is the product of
{O), or the normed space
t i , where I is an infinite set of indices, then there exist two sequences (En) and (F ) of dense subspaces of E such that E n > F n > E for all n n+ 1 is ultrabornological, and each F is barrelled and bornologin cal but not ultrabornological. In [211 he gives examples of barrelled,
n, each E
bornological spaces E
that are not even inductive limits of Baire
spaces, and in [471 he shows that E
can be metrizable.
In his lecture I301 at the Bordeaux conference of 1973 (where I first met Valdivia in person) he produces a barrelled, bornological space that is not the inductive limit of barrelled, normed spaces. If R by D(K)
is an open subset of
IRn, and K a compact subset of R, denote
the vector space of all infinitely differentiable functions on
R
with support in K, equipped with the topology of uniform convergence of f E D(K)
U
D(K) K is equipped with the locally convex inductive limit topology, and its dual
D1(R)
and all its derivatives. The test function space D(R) =
is the space of Schwartz distributions which we equip with the
strong topology.
In [341 Valdivia proves that on
D'(R)
there exists a
coarser topology which is barrelled and bornological but not ultrabornological.
In [621 he shows that on both
D(R) and
D1(R)
there exist
coarser topologies which are barrelled and bornological but not the inductive limit of Baire spaces. For D'(R1
this is based in part on the
following observation: If (G ) is an increasing sequence of closed subn spaces of E' whose union G = UG is c(E' ,El-dense, then E is bornon logical for t(E,G). De Wilde's closed graph theorem (19, 935, 2.(2)} is used to show that under appropriate hypotheses E is not ultrabornological. is the strict inductive limit (5, p.11. 361
According to [35], if E
of the sequence of spaces E which have the topology r(En,EA), and if the n E' equipped with t(E1,E ) are ultrabornological, then E' equipped with n n n r(E1,E) is ultrabornological.
Dieudonnk proved that a finite-codimensional subspace of a bornological space is bornological [A, Chapter 1 , 93,2.(31, p.431, and in [I41 Valdivia proves that a finite-codimensional subspace of an infrabarrelled space is infrabarrelled [A, Chapter 1 , 93,2.(2),p.431. The situation is completely different for ultrabornological spaces.
In his Comptes Rendus
note [53] Valdivia constructs in any ultrabornological space, whose topology is not the finest locally convex topology, a hyperplane H
that is
not ultrabornological. Observe that H is another example of a barrelled, bornological space that is not ultrabornological. Valdivia gives an example in [21] of a countable-codimensional subspace F
of a bornological space E
in 1471 he shows that
F
that is not even infrabarrelled, and
may be dense in E.
However, in I161 he proves
that every countable-codimensional subspace of an ultrabornological space is bornological [A, Chapter 1,93,2.(5),p.451. To prove this, he uses the following theorem [16, Theorem 31:
If E
is the inductive limit of the
barrelled spaces E. and if
F is a subspace of E which is countablecodimensional in each F + E i , then F is barrelled and the inductive limit of the spaces E n F i
If now
E
is ultrabornological, then it is
the inductive limit of Banach spaces E. which are a fortiori barrelled 1
(observe that the same conclusion cannot be drawn if and the E. only normed). so in each
F + E i . Thus F
If F
E
is bornological
is countable-codimensional in E, it is
is the inductive limit of the normed spaces
F n E . and so it is bornological.
In 1921 Valdivia proves that if F is a dense, non-barrelled subspace of a Frechet space {5, p.11.26 and 66) E, then there exists an infinitedimensional closed subspace G
of
E such that F A G
= {O).
It follows
The mathematical works of Manuel Valdivia
that if E
is a barrelled space such that
then every subspace of
E' is complete for r(E',E),
E with codimension < Z N 0 is barrelled.
A locally convex space is said to be countably infrabarrelled (resp. countably barrelled)
if whenever
(Vn) is a decreasing sequence of
balanced, convex, closed neighborhoods of vorous (resp. absorbing), then V the space E
0
such that
is a neighborhood of
V
= nV
0.
is bornin Equivalently,
is countably infrabarrelled (resp. countably barrelled) if
in E' every /3(E',E)-bounded (resp. o(E1,E)-bounded) union of a countable
A local convex space E
family of equicontinuous sets is equicontinuous.
is sequentially barrelled (resp. sequentially infrabarrelled) if every o(E8,E)-bounded (resp. /3(Ef,E)-bounded)sequence in E' is equicontinuous. Grothendieck observed that, while the strong dual of a metrizable space is not always barrelled (19, § 3 1 , 7 ) , it is countably infrabarrelled, and it obviously has a countable fundamental system of bounded sets. He called spaces having the last two properties (DF)-spaces
{5,p.IV,58; 19
§29,3).
In [I51 Valdivia proves that a finite-codimensional subspace of a (I)%)-space is a (I)?)-space. (2)F)-space E sion of
F
More generally, if
is a subspace of a
such that for every bounded subset B of
E
the codimen-
F in the span of F u B is finite, then F is a (DF)-space.
On the other hand, in [21] he gives an example of an infrabarrelled (DF)-space which has a subspace of countably infinite codimension that is not a (I)F)-space.
In [441 Valdivia proves a very general theorem concerning the inheritance of some barrelledness-like properties of finite-codimensional subspaces. Let
8 be a cofilter basis on the space E, i.e. a collec-
tion of subsets of exists B
3
E
E such that if B 1 and B2 belongs to 8 , then there
33 containing B 1 u B 2
Assume that
B
consists of bounded,
balanced, convex, closed sets which satisfy U B = E, and that if
BEB
BEB
then
ABEB for all
a topology on E
A+O (i.e. B
is hornothecy-invariant). Let
TO be
that is compatible with the pairing (E,E1),and let
be an infinite cardinal number.
Assume that every subset of
a
E', which is
bounded for the topology of uniform convergence on the sets B e B , and a
which is the union of at most is itself equicontinuous. Let
sets equicontinuous with respect to F
To
be a finite-codimensional subspace of E
8 the collection of all sets B n F , where B varies in 8 . Then F every subset of F ' , which is bounded for the topology of uniform conver-
and
gence on the sets of
BF, and which is the union of at most
continuous with respect to the topology induced by
a
sets equi-
7 on F , is itself 0
equicontinuous. 9 , Y o and
Various choices of
a
yield the inheritance by finite-
codimensional subspaces of the property listed in the last column of the table below. We use the following notation: of
E, 8
1
sets of
T
is the original topology
is the collection of all bounded, balanced, convex, closed sub-
E, and
B2 is the collection of those Be13
1
which generate
finite-dimensional subspaces. B
a
To
B2
o-(E,Ef1
card E'
o-(E,E1
card E'
infra-barrelled
o-(E,E1
Ho No
sequentially barrelled
7
f l ~
countably barrelled
7
No
countably infrabarrelled
o-(E,E' 1
82
property barrelled
sequentially infrabarrelled
From the last row it follows again that a finite-codimensional subspace of a
(DY)-space is a
(D9)-space.
In analogy with (D3)-spaces, a space is said to be dual metric if it has a countable fundamental system of bounded sets and is sequentially infrabarrelled. Obviously every (2)F)-space is dual metric, and in [911 Valdivia proves theorems which yield dual metric spaces that are not (DY).
Equip E' with o(E1,E), and let by
a
be an infinite cardinal. Denote
the collection of all balanced, convex, closed subsets of
that every subset of cardinality at most studies the topology T
a
a
E' such
is equicontinuous. Paper [ 2 6 ]
on E of uniform convergence on elements of
Ba.
11
The mathematical works of Manuel Valdivia
If A C E
is the union of a family
card I is at most
The case a =
on A.
(AiliEI of precompact sets such that
a, then the original topology on E
NO yields a
coincides with
Ta theorem of Grothendieck on (B9)-spaces
(19, 929, 3. (8)).
E is a-barrelled if every o(E1,E)-bounded subset of B of E with card B S a
is equicontinuous.
Valdivia proves that if
exist a-barrelled spaces that are not
a < l , there
l-barrelled.
In 1971 C.L. De Vito (8) introduced (ab)-spaces, which sit between bornological and infrabarrelled spaces. In [31] Valdivia proves that every infrabarrelled (2)Y)-space is an (abl-space, together with some characterizations of (ab)-spaces.
In [321 he constructs an example of an infra-
barrelled (XITI-space that is neither bornological nor barrelled, giving an answer to a question of Grothendieck (15, Question 3, p.120).
Examples of
non-bornological, barrelled (TIT)-spaces can be found in [631, see Section 7. Grothendieck also asked whether there exist spaces with a countable fundamental system of bounded sets which satisfy the Mackey convergence condition but not the strict Mackey convergence condition (15, 111.1, Definition 3, p.105 and p.111).
An example was given by Seth Warner in
1958, and Valdivia proves in [791 that if E' equipped with
E
is a Banach space such that
o(E1,E) is not separable, then there is a topology T on
E, compatible with the pairing (E,Er)such that
E
equipped with
Y is a
(2)T)-space satisfying the requirements of Grothendieck. Further results on ultrabornological spaces are given in the related papers [40] and [421, which continue a line started in [30] to find conditions under which any infinite-dimensional Banach space is the inductive limit of spaces all isomorphic to the same infinite-dimensional space F. This is the case if F F
F
.
is a Banach space whose dual is separable for
This is also the case if
F
is a sequentially complete space
such that: 1) F contains a bounded, countable, total set, 2) in F' equipped with
o(F1,F) there is an equicontinuous, countable, total set, 3) if
the injective linear map continuous.
Since B(K)
u : F+F
has a closed graph, then u
is
as well as its dual satisfy these conditions,
every ultrabornological space is the inductive limit of nuclear Frechet spaces, and also of nuclear
(BY)-spaces. See (13, 52, pp.210-213).
In the note [38] Valdivia proves, among more general results, that if
E equipped with r(E,E1) is sequentially infrabarrelled, then it is infrabarrelled if and only if E' is quasicomplete (i.e. every bounded, closed set is complete) for the topology of uniform convergence on the balanced, convex, bounded, separable subsets of
E. The same result holds
if we replace "infrabarrelled" by "barrelled".
In [77; A , Chapter I, 95.2; 871 Valdivia investigates the third and the fourth link in the following chain of increasingly larger classes of spaces: Baire
+
unordered Baire-like
*
totally barrelled
Baire-like + quasi-Baire A
space E
+
suprabarrelled +
barrelled.
is suprabarrelled if given an increasing sequence
(Ln) of
subspaces such that UL = E, there exists P E N such that L is dense n P in E and barrelled. This class was introduced by W.J. Robertson, I Tweddle and F.E. Yeomans (27) under the name of (db)-spaces, and S. Saxon (31) showed that any metrizable (IY)-space (19, §19,5) is Baire-like but not suprabarrelled.
E is totally barrelled if given a sequence (Ln) of subULn = E, there exists P E N such that LP is barrelled and its closure i has finite codimension in E. For unordered BaireA space
spaces such that
P like, Baire-like and quasi-Baire see { 2 5 ) , Definitions (9.1.191, (9.1.1)
and (9.1.91,respectively. The above quoted references give inheritance results, e.g. to products and to countable codimensional subspaces. A separable, infinite-dimensiona1 Frechet space always contains a dense subspace that is suprabarrelled but not the inductive limit of unordered Baire-like subspaces. For a similar result see (31, Theorem 4, p.77). In Valdivia's joint paper [931 with P. Perez Carreras it is proved that every non-normable Frechet space E
contains a dense subspace which
is a metrizable (LZ)-space, inductive limit of a sequence (F ) of Frechet n spaces. If E is reflexive (resp. Montel. Schwartz (15, 111.4, Definition 5, p.117; 17, 10.4, p.ZOl)), then the
Schwartz).
F are reflexive (resp. Montel, n
The mathematical works of Manuel Valdivia
Baire-hyperplane spaces [68] are those in which every union of a sequence of closed hyperplanes has a void interior. Every metrizable (ZY)space is a Baire-hyperplane space. Let
E
be either an infinite-
dimensional, separable Frechet space or a locally complete inductive limit of infinite-dimensional, separable Frechet spaces. A construction similar to one given in [771 shows that there exist two dense subspaces F of
and
G
E such that G is ultrabornological and a hyperplane in F, and F
is barrelled and not the inductive limit of Baire-hyperplane spaces. Answering a question of V. Klee and A. Wilansky. J . Arias de Reyna proved that assuming Martin's axiom, every separable, infinite-dimensional Banach space contains a meager, dense hyperplane. Valdivia proves in [951 that, still assuming Martin's axiom, every separable, infinite-dimensional Baire space contains a dense, meager hyperplane.
Assuming the continuum
hypothesis, he proves that in every infinite-dimensional Frechet space E there exists a subspace L
such that given a countable set P c E , there
exists a closed, separable subspace G ties: G
contains P, and
meager subspace of
LnG
of
E
with the following proper-
is a dense, countable-codimensional,
G.
It is well known that a product of Baire spaces is not necessarily a Baire space. and
In [97] Valdivia finds such examples inside the spaces co(I)
eP( I), 0 < p < m.
exists a family
More precisely, assume that card
I >No. There
(Xi)iEI of dense subspaces in any of the listed spaces
such that X; is a Baire space for every m e H , i e I , but X. x X is 1 K not a Baire space whenever i + ~ .The case p = 2 was considered earlier by J. Arias de Reyna {I}. In the note [98] published in the proceedings of the 1984 Campinas conference, Valdivia answers a question of L. Nachbin by showing that if the topology of a space E seminorms p
is defined by the collection COS(E)
such that the canonical surjection E +E/~-'(o)
of all is an
open mapping, then COS(E) is not necessarily a directed set. He also proves a three-space theorem: If E
is a non-barrelled, metrizable space,
then there exists a closed subspace F
such that neither
F
nor
E/F is
barrelled. Grothendieck {16, Chapter 111, Section 3, Exercise 8, p.199) introduced the subspaceI(:?!.
of
em( I
generated by characteristic functions
14
J. Horvath
of subsets of the set I, i.e. the space of functions on I which take only a finite number of values. relled, normed space.
He asserted that it is a non-complete, bar-
In [70; A, Chapter I, 97,2, pp.133-1361 Valdivia
I
considers a tribe dl of subsets of em(l) 0
and the subspace !,:(I,&)
of
generated by the characteristic functions of sets belonging to 8.
He proves that ~:(I,B) is suprabarrelled, from where Grothendieck's assertion follows. The proof is preceded by a long list of interesting propositions, the first two of which are parallel to those in [691, where I = H
and
dl
proved that
consists of all the subsets of H.
~?;(I,B)
J. Arias de Reyna ( 2 )
is not totally barrelled, and more precise results
and M. L6pez Pellicer (12). A sequence of elements (xn in a space E is said to be subseries
were given by J.C. Ferrando
converaent if for every strictly increasing sequence (n ) of integers the k converges; the concept figures in the statement of the series Z x k "k Orlicz-Pettis theorem (10, Chapter IV). Given a subseries convergent sequence (x ) in E, let T : E' +el be the map which associates with n 1 the WEE' the sequence (<xn,w>); it is continuous if E' and e are 1 ,!:I, respectively. The equipped with the topologies cr(E1,E) and cr(! image of the unit ball of
t
em0
under the transpose T
barrelled, normed subspace denoted ( 7 ) introduced the space
that
Zl<xn,v>l< m
G'
E[(xn)l.
= G1(E)
:
em0 +E
D. Bucchioni and A. Goldman
of all linear forms v
1 1 for all subseries convergent sequences
Valdivia proves in [691 that E
spans a
equipped with r(E,Gi)
on
E
such
(xn).
is the inductive
1 1 , and answers questions of Bucchioni and n Goldman. He also gives conditions which ensure that every subseries conlimit of the subspaces E[(x
vergent sequence is bounded multiplier convergent (10, Chapter IV, Exercise 4, p.29). Returning to the tribe B
on
I of [70], let H(8) be the space of
finitely additive bounded measures on
A.
Let now
T
be the topological
isomorphism from H(8) onto (em(l.B))' which with each measure p asso0 ciates the linear form x +p(A), where xA is the characteristic funcA tion of A E ~ . Let (dl be an increasing sequence of subcollections of n dl such that Udn = dl. Using T Valdivia proves the following results: 11 There exists
PEN
such that if McH(8)
bounded on 8;2 ) There exists P E N
is bounded on dl
such that if
then M is
P' (pn) is a sequence in
The mathematical works of Manuel Valdivia
H(d) and (p (A)) is a Cauchy sequence for every n weakly convergent in H(d).
AEB then (pn) is P'
3. BIDUALS AND REFLEXIVITY
In [20] Valdivia proves, generalizing a result of R.C. James, that if E
is a real, quasi-complete, non-semi-ref lexive {5, Chap. IV, 92, n02,
Definition 2, p.IV.15) space, then there exists a certain o(E1,E)-closed, convex subset A
of E' not containing the origin, and a balanced, convex,
bounded, closed subset B upper bound on B.
of
E
such that no u e A
attains its least
In [241 he uses the result of James to show that in
any infinite-dimensional Frechet space E there exists a balanced, convex, bounded, non-closed subset B gauge of
such that its span EB, equipped with the
B {5, Chap.11, 92, nO1l, p.II.22) as norm, is complete, i.e. a
Banach space. Further examples of such sets B
are pointed out in the
final remark of [421.
Under what conditions is a finite-codimensional subspace F Mackey space E
is sequentially complete for c(Ef,E), i.e., E is closed.
of a
itself a Mackey space? Saxon and Levin proved it when E' has "property S", and
F
In [371 Valdivia proves the analogous result assuming F
separable instead of closed.
In [441 he shows it under either of the
following conditions: 1) E is separable and in E' equipped with c(E1,E) every strongly
bounded Cauchy sequence converges; 2) F
is dense in E
for the topology of uniform convergence on
strongly bounded subsets of 3) F is closed and
E';
E' equipped with
c(E1,E) is locally complete: in
this case F may have countable codimension. In [441 he proves furthermore that if
E
is the strict inductive
limit of a sequence of Frechet spaces, or if E is an infrabarrelled space, and
F
is a dense subspace of
E, then F
E
is a Frechet space and we equip E' with
of
E' are Mackey spaces if and only if E
o(E,E11.
(2)F)-
is a Mackey space.
If
r(E1,E), then the hyperplanes is sequentially complete for
Grothendieck (15, Question 8, p.121) asked whether, if E is the strict inductive limit of the sequence (En), is E" the inductive limit of the Ei?
In [661 Valdivia gives a negative answer to the question by
showing that if E
is a Banach space whose dual
E'
is separable for the
strong topology, then there exist in E' equipped with z(E1,E) two sequences with
(Fnl and (H of subspaces such that each Hn can be identified n Fi, the inductive limit of the Hn is a dense subspace of E'
distinct from E', and the bidual of the inductive limit of the Fn is E' itself.
R.C. James showed in 1950 that there exist Banach spaces which have finite codimension in their biduals; such spaces are said to be suasireflexive. In [801 Valdivia proves that if
E
is the locally convex
direct sum of a sequence of quasireflexive Banach spaces, then E Krein-Shmulian property, i.e. a convex subset of
has the
E' is o(E1,E)-closed if
its intersection with every r(E1,E)-closed equicontinuous set is o(E',EIclosed. This follows from the fact that the o(E1,E)-closure of any convex subset of
E' coincides with its sequential closure. The proof involves
the subspace of
(E'l
*
formed by the linear forms on E' which are bounded
on equicontinuous sets. Article [I001 is dedicated to prove that if the Banach space E
is
not quasireflexive, there exists a separable non-quasireflexive closed subspace F
such that
A space E
E/F
is not quasireflexive.
is weakly compactly generated
{lo,
p.228) if there exists
a balanced, convex o(E,E1)-compact subset whose span is dense in E.
It
is weakly countably determined if there exists a metrizable, separable topological space T and a map
@
from T
into the collection of all
such that E = U @(tl and given a sequence t€T (tn) in T which converges to t e T and a neighborhood V of @(t),
r(E,E' )-compact subsets of
E
there exists an N E N such that @(t ) c V for n t N . Every weakly n compactly generated space is weakly countably determined. Let
F
proved that
be a closed subspace of the Banach space E. E
In [I121 it is
is weakly countably determined (resp. weakly compactly
The mathematical works of Manuel Valdivia
17
generated) under either of the following conditions: 1 ) E/F and
is separable
F is weakly countably determined (resp. weakly compactly generated);
2) EU/F is separable and E/F
is weakly countably determined (resp. weakly
compactly generated). In what follows I denote by with
the closure of A C E
in E" equipped
r(E",E1). A decreasing sequence (A of subsets of the space E is n U of 0 there is an m c N such
quasibounded if for every neighborhood that
AmcmU.
In [I131 Valdivia proves that a metrizable space E
is
weakly countably determined if and only if there exists a sequence (U ) of n neighborhoods of 0 such that for every X G E there is a quasibounded
W
(U with X G 6 c E. If E is a Frechet space, the "J j=l "J can be chosen balanced, convex. Motivated by this fact, Valdivia says
subsequence U n that
E
is weaklv countably convex-determined if there exists a sequence
(Un) of balanced, convex neighborhoods of
0 such that for any
XGE
(U ) with x G 6 c E. "J -j=l "J The density character d(E) of the space E is the smallest cardinal
there is a quasibounded subsequence
c
such that there exists a dense subset A
w
be the first infinite ordinal and
is d(E).
p
of
E
with cardA = c. Let
the first ordinal whose cardinal
A family {Pa: w < a < p } of continuous projectors on the
such that Pao Pp = P B ~ P a = Pfor a s p , a is the identity, d(P (El) S card a, and for any limit ordinal a the a PU closure of U P (E) coincides with P (El, is said to be a resolution of B a B 0 the sequence xn 11 converges to 0. A subset A of E is hyperprecompact if there is a sequence hyper-
Section 2).
convergent to the origin whose balanced, convex, closed hull contains A. A closed, hyperprecompact set is hvpercompact. If
A
is hypercompact, then there exists a hypercompact, balanced,
convex set B c E
EB is a Hilbert space and A
such that
compact in EB, see also (13, p.211).
is hyperpre-
Using a construction based on this
result, Valdivia proves the following theorem: Assume that the infinite-dimensional Banach space E
has a dense
subset G with cardinal a. Let (EiIieIbe a family of infinite-dimensional Banach spaces such that card I equals the cardinal of the set of all in-
G, and assume that E! equipped with o(Ef,Ei)
finite countable subsets of
1
i~ I.
is separable for each
Then for each
i e I there exists a hyper-
U A. = E, each Ei is ieI form a directed family of subspaces, and E
precompact, balanced, convex set A . c E 1
isomorphic to E the E Ai ' Ai is their inductive limit.
such that
-
E:
There is a similar result for the case when the
are separable for
E- , where is the closure of A. in E. Ai i As a consequence Valdivia obtains the result that if (Fi)iEI is a family B(Ef , E i ) In that case !'E
1
of infinite-dimensional Banach spaces and card I L 2 x O , then TI Fi is not i~I an infra-Ptak space. Let
E
closure of
be a Frkchet space, A A
in E" for cr(E",Et1.
E", and assume that
E
the following results:
a bounded subset of Denote by
G
E
and
the span of
A" the E U A" in
has finite codimension in G. Then [49] contains 1) The balanced, convex hull of
A
belongs to G
(this generalizes the Krein-Phillips theorem {19,§24,5.(4))); 2) For every x e A" there exists a sequence in A cr(E",E1). of
E
3 ) If
converging to x with respect to
G*E", then there exists a separable, bounded subset M
EUM"; 4) If F
such that
G
is a proper subspace of the span of
is a subspace of
G
containing E, then E' is complete for
Valdivia considers furthermore a subspace F E + F , assumes that
F
is separable for p(E1',E')
of
T(E',F).
En such that
and obtains the
following conclusions: 1 ) E' is barrelled for r(E1,E"); 2 ) For any
E" =
35
The mathematical works of Manuel Valdivia
subspace L
of E" containing E
to x
the dual
E' is complete with respect to
there is a sequence in E which converges
r(E1,L); 3) For every x e E"
with respect to o(E",E1).
To explain some of the results of the related papers [ 5 5 ] and 1781, the best is to use a slight extension of the terminology used by Valdivia in [781.
A subset
A
of a topological space T
continuous function on T
is bounded on A .
if every closed hyperbounded subset of
T
is hvperbounded if every
The space T
is an A--
is compact. Thus one of the two
Nachbin-Shirota theorems quoted in Section 2 asserts that &(TI is barrelled if and only if
T
is an A-space.
One of the early theorems of [551 states that if the locally convex space E
is a hyperbounded subset of
is quasicomplete, and if A
equipped with o(E,E1), then A consequence, a Banach space E
is relatively o(E,E1)-compact. As a is reflexive if and only if every o(E,E')-
continuous real-valued function on E
is bounded on the closed unit ball.
More generally, a quasicomplete space E
is semireflexive if and only if
every o(E,E1)-continuous real-valued function on E bounded subsets of Denote by
A
*
E
is bounded on
E. the o(E'
*
,El)-closure of
ACE
in E'
*
.
The proof of
the preceding theorem rests in part on the following remark: If hyperbounded subset of
E
equipped with o(E,E1), then
* A *
continuous on E' equipped with o(E1,E), i.e. each X"E A on every separable subspace of
A
is a
is separably is continuous
E'.
In 1551 the following condition is introduced: (CHI Let
A
be a convex subset of the space E. For any sequence n (Hn) of closed hyperplanes of E, such that A n ( n H.) + + for n e H , one has
j=1
n Hnf+. neN
Then: 1) (CHI implies that a bounded, convex subset of
A
*
is separably continuous; 2 ) Let
E; then A
only if (CHI is satisfied; 3) Let
J
A
*
A
be
is separably continuous if and
be a bounded, convex subset of
E;
36
J. Horvath
then (CHI implies that A is p(E,Ef)-bounded; 4) Assume that E barrelled and that
A
is a bounded, convex subset of
c(E1,E); then condition (CHI, where E
is replaced by
is infra-
E' equipped with E', implies that A
is relatively cr(E1,E)-compact; 5) Assume that E is quasicomplete for the topology of uniform convergence on @(E8,E)-compact subsets of be a bounded, convex subset of E; then (CH) implies that
E'; let
A
A is relatively
o(E,Ef)-compact. It follows from 2), 5 ) and the first remark concerning A
*
that if E
is quasicomplete for the topology of uniform convergence on ff(E',E)compact subsets of equipped with
E', then every convex, hyperbounded subset of E
o(E,E1) is relatively compact.
In [78] many conditions are given which imply that certain subsets of the space E, in particular hyperbounded, closed subsets, are @(E,Ef)compact. Several of these (Theorems 1,3,4) involve the smallest subspace S(E') of
*
E
II
equipped with @(E ,E) which contains El, and in which every
separable, bounded subset is relatively compact.
Denote by dl1 the collec-
tion of balanced convex, separable, compact subsets of S(E1), by collection of all balanced, convex, c(E',E)-compact
subsets of
dl2 the
E', and
= {A1 +A2; A1 E dl
A E dl2}. Assume that E is complete for the 1' 2 topology of uniform convergence on the sets belonging to dl. Then: 1) Every let
dl
c(E,E1)-closed, hyperbounded subset of E
with o(E,E1), let
A
E
is o(E1,E)-compact; 2) Equip
be a closed subset of
E, and assume that every
bounded, increasing, pointwise convergent sequence of continuous functions on E
converges uniformly on A; then A
is compact
Similar results are given in terms of the smallest sequentially C
complete subspace T(E') of
E which contains E' .
7. SEQUENCE SPACES
In this section and the next I can be brief since Chapters 2 and 3 of [A1 treat, among others, most of the results obtained by Valdivia in the two areas. [321 gives a condition on an echelon space A [ A , Chapter 2, Q2,2, p.2121, {19,§30,8) so that its K6the a-dual hX [ A , Chapter 2, §1,1,p.1731, {19, §30,1) equipped with
r(hx,A ) has a dense subspace E
equipped with the topology induced by
such that
E
/3(hx,hl is infrabarrelled but not
37
The mathematical works of Manuel Valdivia
bornological. (1)Y)-space Let
This is used to construct an example of an infrabarrelled
that is not bornological (see Section 2).
h be an echelon space, and assume that equipped with t(h,hX) it
is a Montel space.
If h is not a Schwartz space, then in the paper [631,
published in the Proceedings of the 1977 Campinas Meeting, Valdivia constructs in hX equipped with
z(hX,h) a dense subspace that is barrelled
but not bornological (see Section 2).
Conversely, if
E is a Frechet-
Schwartz space, then every barrelled subspace of E' equipped with z(E1,E) is bornological, cf. [A, Chapter 2, 92,4, p.2371. In [671 Valdivia proves that the echelon space h
equipped with
t(h,hX) is a Schwartz space if and only if h has no quotient that is 1 topologically isomorphic to t (cf. [ A , Chapter 2, 92,3.(14), p.2261). Also every separable Frechet space is topologically isomorphic to a quotient of a Frechet-Monte1 echelon space.
Most of the results of [731 and 1791 can be found in [A, Chapter 2, §2,4, pp.227-2371 and are augmented there with (23): Let
E
be a Frechet
space. Assume that in E' equipped with c(E1,E) for every bounded set A there is a balanced, convex, bounded set B > A cides on A
with the weak topology of
subspace of
E' equipped with
If h
Eb.
such that c(E1,E") coin-
Then every infrabarrelled
/3(EJ,E) is bornological.
is an echelon space which equipped with t(h,hX) is a Montel
space but. not a Schwartz space, then Valdivia proves in [73] that there exists in hX equipped with t(hx,A) a dense subspace which is an (LB)-space but is not locally complete, and another dense subspace that has countable dimension, is infrabarrelled but neither barrelled nor bornological. Valdivia also constructs non-bornological, infrabarrelled (2)F)-spaces (see Section 21, and proves that infrabarrel led subspace of Let
A
h
is a Schwartz space if and only if every
hX is bornological .
be an echelon space, equip hX with /3(hX,A), and denote by
p(ll the subspace of
hX which consists of all limits of sequences in K (lN)
that converge in the Mackey sense. Then [79; A. Chapter 2, §2,4.(21), p.2351 the following are equivalent: 1) h equipped with
t(A,hX) is quasi(1) is normable {IS, 111.1, Definition 4, p.106; 17, 10.7, p.214); 2) fp
complete; 3) hX satisfies the Mackey convergence condition.
If these con-
is the strong dual of a non-complete
ditions are not satisfied, then h
(2231 -space.
In [I031 Valdivia proves that if the echelon space h space but not a Schwartz space, then A
(ellH; furthermore if H closed subspace G however, h
is a Monte1
has a quotient isomorphic to
is any separable Frkchet space, then h
such that
H
is isomorphic to a quotient of
is a Schwartz space not equal to K',
has a
G.
If,
then there is a sequence
(An) of nuclear echelon spaces with continuous norms such that
h
has a
of rapidly decreasing sequences a = (a,)
(i.e.
quotient isomorphic to nAn.
8. FUNCTION SPACES AND THEIR REPRESENTATION AS SEQUENCE SPACES
It is the space a
-
such that lim m' a = 0 for any r E N ) , equipped with the seminorms m mwo m p (a) = max r ( a I , which figures in these representations. For the spaces m m of distribution theory I use the notation of Laurent Schwartz {33), for the spaces of ultradifferentiable functions [82] I will refer to Meise's expository article {23).
Here are some of the isomorphisms proved by
V a l d i v i a (most o f which were found independently by D. Vogt { 3 7 } ) :
H [61; A , Chapter 3, $1,121: &(R) = a , &'(Q) [61; A , Chapter 3, $1,131: D(R) D(R)
-
a ( ' ) ,
= (a')
Df(R)
(H
.
(41)~. In the proof that
is not an infra-Ptak space [571 the fact that it has a closed
subspace isomorphic to a(H) plays an essential role (24, Satz 4.5.12, p.78). [65; A , Chapter 3, §1,14.(71,(8),p.3881: Let
D(n) = {f E & ( [ R ) ;
supp f c [-n,m)) with the topology induced by
&(R),
and
DL
with the inductive limit topology. Then 2)+ = aH x a (HI ,
[75; A , Chapter 3, §1,15.(22), p.3981: D(K) that
=
U D(n) n
a. Earlier it was only known
D(K) is isomorphic to a complemented subspace of a.
39
The malhemalrcal works 01 Manuel VaMivra
176; A, Chapter 3. 41.16. (8).(9). p.403):
Let
r
be a closed cone In R"
Dz = (f t G(R"); supp f c z
z t R" let
For each
with nonempty lnterlor and vertex at the origin.
r) equipped with the topology lnduced by
+
Dr = U DZ with the lnductlve limit topology. Then
&(R"), and let
Z
D;
(aN)(N'.
a ( ( 0 1)
(N) )H,
Let (Mn) be a sequence of strictly positive numbers such that
1821:
Mo = 1.
(5 M
~ Mn+, - ~ and
Z ( M n - l m n )< a. and let
1
be the closed
unit cube of R". Then for the spaces of ultradifferentiable functions of Beurling and Rounleu type
1831:
(23. 4 . 2 Definition. p.201) we have
In his thesis Crothendleck proved that
010'.
a ? of
to a complemented subspace of
OM Is lsomorphic
and asked whether the
Isomorphism proved by Valdivla Is true.
186; A. Chapter 3. 41.22. pp.433-439. 42.6,pp.461-4653: Let
V
be a non-
compact differentiable manifold. countable at Infinity. Then ea(v)
m
oN, D(V)
D l N , O concerning Denote by functions f orders
m
a
.
-
gm(v) gm( '1 )
a ( ' ) ,
: .
Dm( I 1'.
Chapter 3. 4 2 of [ A ) contalns many results
g m not published In articles previously.
B(Q) the vector space of all lnfinitely differentiable on the open subset
Q
of R" whose derivatives
a c Nn are cont lnuous and bounded. and equip
subspace of
B(Q)
pa(f) =
aaf
of all
B(Q) with the topo-
laaf(x)l. Let Bo(Q) be the xcn consisting of those f which vanish at the boundary of
logy defined by the seminorms
Q, l.e. given
-
Drn(v) gm( 1
G(V) = (H)
a e N n and
c>0
SUP
there exists a compact subset
K of Q
40
J. Horvath
such that
laaf(x)l 6 c
for X E R ~ C K . The space B1(R)
introduced by
P. Dierolf consists of the restrictions to R of those f EB(IR") which vanish outside R. Clearly Bo(R) c B1(R) c B(R). Properties and representations of these spaces are given in [84] and [A, Chapter 3, $1, 18-21, pp. 413-4331. If R = Rn, then Bo-a$co and B1 = B = a 6 t m . If R + R n Valdivia of cubes as follows. Let a E ien and letQ, be the n collection of all those cubes {XER ;a < x < a +l,l<j
VL) " iUk = k! for e v e r y r 7k. and claim t h a t (2, A s ~ u r n et h a t thlv I S n o t t h c c u e T h c n t h e r e is r have
>
and z E
~ 1 ,
-
win'
win'
(7, +
LVk " [I,.
V k )$ i l l t \Ve
Regularity properties of (LFJ-spaces
with xi E
f w,('),i = 1 , . . . , k
-
1 , xk E CVk,CfZIlail 5 1. Therefore
which contradicts t h e choice of W k . We put U = rEl; then U is a ~irighborhoodof zero in E and U
wLk', since for rn 2 k we have CVir) C wL"') c Wk c Vk. We choose for a , T
Z
TO,and
UL
> TO,ok L TL. 'I'lier~
TO
c Vk for
suclr t h a t z,
-
all k ,
.I, t f f
IIence 2 x , @ Uk lor all k, wllicl~cor~tradictsl11c assumptions. For the following two results, i.c. conil)lctcness of acyclic (LI:) spaces ant1 r.eg11larity as its conscqucnce, set: I'alamodov [14], Corollarirs 7.1 ant1 7.2. T h e o r e m 3.2 An acyclic ( L F ) s p a c e is con~plete.
I'ROOF: We awurne t h e U , in 'L'licorc~n2.10 a1)solutcly co~lvcxand open and wc: Insy icssllrne U,, IJ,, = E . 'I'hen according to I,c.r~rr~~it 3.1 tllerc is p and for every T, V a o 1 T and x,," t (T, V ) n I/,,. 0l)viously t l ~ i stlrfines a Cauchy net i l l U,, wit11 rcs1)cc:t l o tllc topology of E , Irence because of ' I ' l i c ~ o r c2.1 ~ ~1~with rcspcct to t h e topology of I:'L loo. An (LI?) space E is callrd rrgular if evcry bounded set in E is contairlcd in soriir b;L and bounded there; it is called seqr~ci~linlly i.ctinclive (scc 1:loret [GI) i f evcry convrrgcl~t scqucncc is contained in some Ek arld cor~vergcntthcrc.
+
C o r o l l a r y 3.3 Ail acyclic (LI") space is qirn~iro7nplcle,r-egula~and sequer~linlly1.c1r.n~live.
PROOF ('o~nl,lctcncss implies cluasicon~~)l(:trrims and t l ~ i s ,by (:rotl~cr~~(lieck's larlol.iz;~ ti011 tl~c:o~.c.m, regl~larity.ljc.rausr of r.cgul;t~.ilya cor~vcrgentscqucncr is c.oritair~c.(lin sonlc F;,, and bountlcd tl~c.re,I~c.llcrcontitir~ctli l l so111t:XlJ,,, U , as i l l 2.10. 'I'l~c: result k)llows from 'I'lirorcrn 2.11. D e f i n i t i o n : l ' h c ( L F ) space is said t o llavr closcd local ncighborl~ootlsi f it a t l ~ n i t sa defining s1)rctrurn L . Fly2 -+ . . . s u c l ~tliaf (.very FATk has a brrqis of absolutely c-o~~vc\x r~cigl~borl~ootls of zrro wl~ic-11 a1.c c.losc:tl i l l I:'I wit11 respc:cf, t o tl~c:topology i~~tluc.c.tl I)y I:. R e m a r k : ( I ) I f k,' 11as closc.tl local ~ ~ c i g l ~ l ) o r I ~ ot ol ~( les lwc ~ 111ay~ S S I I I I I C1l1af. {.I. c : I l : ~ l l ~ ,I ~} is cIost.(l i l l F,'k with r.c'sr)~(:tt o tl1(3to1)ology irldt~cctlI)y I S l l ~ l l+~~. (~I x I I ~ . ~
1
-
We define T,, = x 2 = 2 2 k , nand have x,, E t:,. We show that x , 0 in E. (1 (1 be any co~itinuousserninorm in F . T l l c ~ cexist increasing sequences rn(l), Cl suc11 that llxll 5 C111x111,7n(l) for all x E El. For 1 rn(1) we put m = n ~ ( l 1) and obtain for 71 2 in
>
Hence llxl,ll have:
--+ 0
for
12
-
m. Next we show that x ,
+
ti 0 in every Ek.For 12 > k
we
Therefore ( x , ) , does not converge to 0 in Ek. Obviously condition (1 1 ) in I'ropositio~i 3.6 arid condition (Q) are cquivalcnt. 'I'herefore we have shown
T h e o r e m 3.7 If E has closed local nezghborhoods and is sequentially retmctllre t h c i ~~f satzsfies condztzon ( Q ) . The following t,heorem we coultl, due to Lemma 3.4, also formulate as a complete ccluivalence in the spirit of Theorem 2.8. We prefer the following version. T h e o r e m 3.8 If the (I,F)-space E is adjustable and has closed local rleighborhoods then the followii~g are eqni.ualei~t:
( 1 ) E is acyclic
(2) E is sequei~tiallyretrr~ctive (3) Condition (Q).
P R O O F : This follows frorn Corollary 3.3, Theorem 3.7 and Theorem 2.8.
4. We are now stuclyi~~g consecluences of weak acyciicity on the topological proprrtic.~ of E. First we give a description in terms of the dual spaces. We cor~siderthe tlual spectrum &* : E ; + Eb . . . of (DF)-spaces. Setwise by rlatural idcntificat,ioii we have E' = l i p E; and the identical map Ei lim E; + EL is continuous. Since E
-
- .--
is barrelled the bounded sets in E L and EL coincide, hence also the bour~dedsets in 15; and lim E; and these coincide with the ec~uicontinuoussets in E'. We refer to tllc~n'is +1,ounded sets in E'. L e m i n a 4.1 E is weakly acyclic if n,nd ordy ifProjl E* = 0.
PROOF: This follows from the definition of Proj' E* (cf. [25]) and the fact t h a t wcitk acyclicity is equivalent to ni being surjective. L e m m a 4.2 If E zs ulaakly acyclzc thcn bornologrcal.
EL
=
IyE;
topologzcolly arlrl thzs spacc rs
-
PROOF l3y [24], Theorem 5.6 and our Lemma 4.1 the space lim E; is homological. This and the remarks before Lemma 4.1 show that thcn Ei = l i p EL. L e m m a 4.3 I f E is weakly acyclrc, the11 for every bounded set B c E therc zs k and cln Ek-bour~dedset M C E L , slich that B zs contazned an thc E-closure of hd.
P R O O F : The polar B0 is a neighborl~oodof zero in lim E; by Lemma 4.2. Tile rcst~lt +follows from the bipolar theorern. 5.13, 5.14 show, this does not irnply that every weakly acyclic E R e m a r k : As Tlieore~r~s is regular.
73 4.4 I f E is
acyclic
&
E is rcgrrlar, rcgrrlar, re-
aiid PROOF: To show show that is is regular let let BB C EC be E bounded. We choose an Ek-boundcd set M according M to Lemma 4.3, 4.3, which which may be assumed absolutely convex convex and Ek-closed. Ek-closed. By reflexivity of E h it is weakly weakly compact in inE k , hence in inE. E. M M is E-closcd and Bc B Mc c M Eck is Ek-boundcd. The assumption and regularity imply that is scmireflexivc and therefore, since barrelled, also reflexive. reflexive. From Lemma 4.2 then follows that follows EE == is complete. The following following be parallel will will Proposition 3.6 and its proof. However However thcre is a characteristic change, cf. cf. also (221, Proposition 4.4 aiid aiid Proposition 5.9.
PROOF:We may assume that y l , . ...,.., are linearly independent and that we have . ..,.t, E E,, E such that yk(z,,) = &,,, = for = 1,. = . . .,.u., w e put
21,.
cc uo uo
Pz
= = yu(z)zu, Qz =Qz 2 =-2P-t
for for z EzE E,. Theii P Pand
4.6
If
(z,,),, to 0 0
EL, k 2 k 2
Q are Q projections in E,,and
docs
i m Q = X. = We have
property
0 0
(WQ)
is
does
aa
74
D. Vogt
We lllay replace (En), by a subsequence ( E n ( k ) ) k and and so assume that for evcry k :
S 35 E El
v 7 ~ ,
> S (Ilzllk+l,n
:
II~llk,k
+
11~111,k-1)
11 l l n ( k ) , r n by a subsequence 11 lln(k),nL(i) (I3)
.
k = 2 , 3 > . .. For fixcd 12 we choose inductively sequences x k , , E El, y k , , E We choose x z , , according to (13) with k = 2, n , S = 2'+'" such that 11z2,n112,2 = r = 2". Ihen we choose yz,,, E Eb,z, 1 1 y 2 , n l l z , 2 I 1 such that Assumc that xz,,, . . . ,~ k - 1 ,yzVn,. ~ ~ . . ,~ k - 1 , being ~ chosen. Put l
Xk,n
= {x E El :
= 0 for v = 2 , .
y,,,(x)
. . , k - 1) .
Since y,,, E EL,, C E;,k-l the space X k , , fulfills the assumption of Lemma 4.5. IIer~ce (13) is satisfied on X k , n We choose S = 2kt2'177 and find x = z k , , E X k , + such that
> 2kt2nn( I l x l l k + l , n
II~llk,k
and
11x11k,k
scqucnces
+
II~lll,k-1)
5 1 such that y ( x ) = 2". We obtaill such that
= Z7'+'. We choosey = yk,,, E ( ~ k , n ) k = 2 , 3,...
11xk,nllk+l,n Illk,nllk,k
in
E l , ( ~ k , n ) k = 2 , 3,...
I
i2-k-nf1
=
2n+1 0 for k
~ r , , ~ ( ~ a , ~= )
[I~ll;,~
in
1
,
yk,n(~k,n)
12-k-n+l
I
I
=
2".
We proceed now as in the proof of Proposition 3.6. We dcfinc x, = C;=2x k , , and have x,, E E l . Using the same ~ ~ o t a t i oas n in the proof of Proposition 3.6 the relevalit inequalities become llxnll
I
llxnllk,k
2 2n - !2-"
C1+1:2-~
for n for n
>m > k.
This shows that 2"xn corlverges lo zero in E but 2 P x n does not convcrge to zero in evcry
Ek. As arl irrllnediate consequence we have
Theorein 4.7 I j E is regular then it satisfies condition (WQ).
5.
We sl-~allnow apply the previous rcsults to the case of sequence spaccs. Let a j , k , , ( j , k, n E IN) be a matrix with onne negative extended real valued elltries and the following properties
for all j , k , n . Moreover we assume that for every j and k there is that for evcry j thcrc is k such that a3;k,, < +cc for all n. For 1 I p < +awe set
7z
with
a,;k,,
>0
and
Regularity properties of (Lo-spaces
EL = {x
< +M for all n
= ( x i , z h . . .) : /Jxll;,, = J
1
We set
Er
= {z = ( z 1 , z 2 , . . . : x k , , = sup xjlaJ;k,,
< +m for
all n )
,
J
= {x t E r : lim 1zjlaj,k, = O for all n )
.
J
All these are F g c h e t spaccs wit11 h~ndanicritalsystcrns of scrninorms (11 Ilk,,),,. \VC= set EP = Uk E: equipped with tlrc inductive liriiit topology of the dcfining s p c c t r u r ~El ~ -+ Ez 4 . . . of Frkchet spares. For these spaces one can give a fundanicr~talsystem of semir~ormsin a very collcrctc form (projective dcscriptiorr, cf. Bierstedt, Mcisc and Summers [ I ] ) . We denote Ily d X the set of all sequences a = ( a l ,a z , . . .) wliicll satisfy estimates a, Cka,,L,,(k) for all j, k and suitable C:k,n(k). By D X wc dcr~otethe sct of all systems A = (a(k))k,a ( k )= (aIk),r r y ) , . . .) which salisfy ujk+l),a:"' ) i i n f , = l , ,k C,aJ , (,I for all 1, k with suitable C k , n ( k ) .
Proposition 5.1 A jundameillal system ojseminorms in EP is given by thc sc1ni7lornls
11~11. IIxII.
= (
c I ~ ~ J P, ~ ,n PE )d X~ ,
jo1. 1
I p < +m
1
= SUP I
XJ~~J
J
9
a€dX,
llzllA = infsup lzJlajk) , A E D X , k
jorp=O for p = +m .
J
P~oor:.Since these arc obviously conlir~uoussemir~ormson EP, we haveonly to sliow t l ~ a t every conti~iuousseminorni p on EP can be estimated by some 11 (1. or 11 l l A rcs1)c:clivcly. We have p(x) ) C~.llrll~,,(~) for all k with suitable Ck and n ( k ) . Wc may choose thc sequence CI. increasing so rapidly that lilnk C ~ a , , ~ , , ( k=l +co for all j. For 1 5 p < m or p = 0 we put
and define for x E EP : J
0 ,. 1hen x =
i f I. is the smallest ilu~nberwit11 a, = 2kCkaJ,rn(k) otherwise.
ELx ( ~and ) C k l l ~ ( ~ ) l l ~5, 2-kllxllo, ,(~) hence
or p = +m we put
A = (nj"') whc1.e
76
0. Vogt
We fix a: E E m , k E IN and define for 1 5 v
z, 0
if v is t h e slnallest number with a?) = 2''Cuaj,,,,(,) otherwise.
II( ch for every k tllerc are i z ( k ) and Ck such that
U
c {n: :
ln:,la;k,,,(k,
I Ck} .
3
'rhcn we have for n. E B
Lemma 5.12 IJProj' &: = 0 tl~eilE1 zs co~rtplete,zf l'roj' &? = 0 t l ~ e ~1:'r r . clrld I F c ~ rr con2pltle.
PROOF:According to (241, 55, u ~ l d c rthe respective assumption, the space E" c c l ~ ~ i l ~ p c ~ t l with the strong topology b(E",E!) is cornl~lete. Therefore, because of Lenllna 5.1 1, 11' and E r respectively are complete. Le~nrna.5.2 a.nd a standard a r g u n ~ c n t(scc I 0 witli Ig(z)l p for all a compact subset Ii of z E Ii. IIence the map which sends each f t o f g gives a local inlbedding of A,(ntfL') into I. Thus we get that I e C3(Cd) by our decomposition method (1.3. Corollary). IIo\vever there is a better result in this direction. Let A' be a Stein space such t h a t O ( X ) is isomorphic t o some A,(a). If A4 is a closed sublnodule of C3(.Y)Vor some p, then A4 cx C3(cd), d = dirnX. We reler to [3] for this and also t o [2] for other results in this direction.
>
cd
Next we consider spaces of CW-functions. Let Ii be a compact subset of R7' with noriempty interior, D ( K ) tlle space of Cm-functions witli support in I i , &(I{) the space of Whitney-differentiable functions. Our purpose is to show how some results of Valdivia [16] and Tidten [14] can also b e obtained as applications of our decompositiotl results. Let L be a rectangular region whose interior contains Ii. Theti the sequence
is exact, wliere J ( I i , L) = {f E D ( L ) : D mf
lrc
= 0 for evcry a } .
Some applications of a decomposition method
89
Tidten [14] proved that J(li,L) has property R and so it is a complemented subspace of s. For ally rectangular region M we have D ( M ) .v s [12]. We fix a rectangular region 0 L1 C L \I 0. If ?L is the composition of h with t h e restriction map from II(EL) into H ( 0 ) we have
and so
: A,(P(cu))
-+
H ( 0 ) is a local imbedding.
If E is a nuclear Frkcliet space which satisfies DN and R a n d aEt h e associated exponent sequence we can assunie that a" is equivalent t o (- log dn(Uktl, Uk)) for each k (cf. [ 5 ] , $1). From 2.1. Proposition we know that A ( H ( 0 ) ) C A,(P(cuE))' for any open subset 0 of EL ([13]; 3.3. Proposition). Let us assume Al(aE) is nuclear; i.e. (@f) E ll for ~ ) , we every 0 < q < 1. We find some R, > 1 such t h a t d,(Uktl, Uk) = O ( T L - ~ R ; ~ where write cu for aE to simplify t h e notation. Let Q be a compact subset of 0. Then we can find allother compact subset Ii c Q such that t h e associated linking m a p has diametcr dn = O(R-0") where 1 < I? < and /3 = P ( a ) . This follows from the estirriatcs ol~tainedby Borgens, Meise a n d Vogt [8]; 3.3. Lemma and 4.1. Theorem. So we have proved the following which is a n improvement of ([13]; $ 5 ) .
a
2.2. Proposition. Let E be a nuclear Fre'chet space which satisfies DN and R, with the associated exponent sequence a . I f A l ( a ) is also nuclec~ra n d 0 c EL is open, we have
One remarltal~leproperty of P ( a ) is that it is always stable [7]. Hence from 1.4. Theorem we get that II(A,(cu)b) is isomorphic to a subspace of I I ( O ) , provided H ( 0 ) has DN and 0. If f2 is a complemented subspacc of s, then II(EL) Y H(A,(a)b), which means
Some applications of a decomposition method
91
that H ( E I ) has a basis, altllougl~we do not know if E has a basis. We refer t o [13], for these and some otlier consequences of our results in the context of infinite-dimensional holomorpl~y.
53. SPACES OF FUNCTIONS ANALYTIC IN A REINHARDT DOMAIN Zaliarjuta [22] proved that tlicrc exists a coiltinuum of pairwise nonisorriorpl~icspaces in the cla55 of spaccs of functions wliicll are analytic in co~llpleteReinhardt dorllains in d 2 2 Let D c be a complete I{einliardt tlomain.
cd,
\.lie know that tlle associated exponent sequence of O ( D ) is ( n ' l d ) . D is completely determined by tllc furlction
+ +
defined on tlie simplex C = ( 0 E R" : 0 , . . . Od = 1 , 0, 2 0 ) . For n = ((711 . . . n d ) E N d , let 1 1 ~ 1 = i ~ , . . nd arid 0 ( n ) = ~ L / I T LTlle I . set T ( D ) = { U E C : h ( 0 ) < +m) is convex. nleasurc. \.lie liave already proved in [5] Let m de~lotetlle (d - 1)-tlin~ensiorialLel~esgr~e tliat m ( s ( D ) ) = 0 if and orily if O ( U ) is isomorphic to O ( G d ) x F for some quotient space F of (?((Ed). Tliis case corresponds to A ( O ( D ) ) = A ( O ( ( E d ) ) .Our aim now is to complement this result as follo~vs:
+. +
3.1. T h e o r e m . Let D be a con~pleteRelnhclr.(lt rlomnzn rn
cd.Then one and
only one
of the follon~ing is true:
Further a) l~oldsif ( ~ n doillg ~fr,z(.rr(D))> 0 and 6) holds J and only if 7 i ~ ( s ( D )=) 0 h o o f : R y what we hake alleddy piovetl in [ 5 ] (cf. ' r l i e o ~ e ~ i1.5. i arltl Lemma l.4.), we only have to prove that n z ( w ( D ) )> 0 irnplics ( a ) . To sii~lplifythe notation t o some extent we talie d = 2. Lct 0,,,,,, = n / ( i n 1 2 ) , 112, n E A', .F = { ( I z , ~ ): On,, E s ( D ) } . We know tliat O ( D ) is a subspace of C3(A2). 'The set
+
is a bor~nded subset of O ( D ) . IIerc wc treat O ( D ) as tlie Kothe space X ( A ) where A = { ( i ~ , , , , }(I:,~,, , = c ~ ~ ) { ( I1 L 1 2 ) / 1 , ~ ( 0 ~i~lld , ~ , ~( )1} ~ is~ )a, sequence of fuilctioris on [0,11 increasing to h [22]. 'I'lle step space X(A)IF is isomorphic b y a diagonal transformation
+
92
A. Aytuna, T. Terzioglu
to a step space of O(A2) rx Al(&) and so it is a power series space of finite type. T o finish t h e proof we will show that X(A)IF has t h e same diametral dimension as A1(&). For this we need a somewhat technical result which we prove first. Let 9 : N -t R be a non-decreasing unboullded function. By abuse of conventional notation we set
L e m m a . Let y and $ be as above and there is c E N such that cp-'(t) 5 c$-'(t) for all Then there is n , E N such that $ ( n ) 5 cp(cn) for all n 2 n o .
t 2 2,.
Proof: Let 91 < cp, < . . . b e t h e range of p. Choose any j such t h a t if then to (pk-1. We have p ( j ) = PI;. Suppose i is such that
(p-'('PI;),
1 with a,,, ua,, and so get for IL rz,
Some applications of a decomposition method
where
We now proceed with the proof of t h e theorern. We first choose a closed interval I c T ( D ) with m ( I ) > 0 and arrange things by using Dini's theorem so that h - h, 5 p-l on I. Let Uk be t h e I-Iilbertian ball defined by hk and set $k(n) = d n ( B , Uk)-', ( ~ k ( n = ) exp(fi/k). We first estimate $;'(et) from below. Since
we have #{(n,rn) : On,,
E
I, 12 + i n 5 t k ) 5 $ i l ( e t ) .
By a result of Zaliarjuta [22] (Lernma 2) there is a constant B t2 -k2172(I) 2&
-
> 0 such that
Btk 5 $il(et)
(3)
for all large t. On the other hand p;'(et)
= n if and only if
J;C 5 tk < m.I-Ience
Using this and (3) we obtain lirn inf +-l(et) - ~ y k (el)
t
4 1 ) > 0. 22 a
wliicli ~nc,ountledsets in H, we call apply a result of cle Wiltle (see e.g. Perez Carreras and Bollel. (131, 8.2.27) to obtain
(6) + (4): Let B be a closed absolutely convex I>ou~~tlecl subset of S ancl let SB clenote the canonical normctl space geaerat.ecl by B. Since Ss is n Rtulnch space, (6) together with Grothendieck's factorization theorenl implies ( 4 ) . -... (4) (5): For fixecl n c IN we use ( 4 ) to tincl k F IN, k .. n,so that. R,, 11 Bk1-1 H" =: D. Since D is bou~~cled in t l ~ e(DFSl-space S ,we c-tn~G t l t l ti1 1 :, k so that D is containetl in B1, YO that XIB,,, ant1 X intluce tlie snliie topology "11 B, ant1 so that Bk is relatively compact in X B , . Then we have
(see e.g. Perez Carreras ant1 Bonet [13], 6.1.12). (4) + (2): In the terminology of [13], 8.3.22, co~itlitio~~ ( 4 ) nlealls that H is a "large" subspace of X. Hence (2) follows directly fro111 (131, 8.324. (2) i (1): This was show^^ by Valtlivix [ICi], P I . O ~2.. Here is a~iotllerargu~nelit: Since A' is a (DFS)-space, H is a yuauil)arrelled (DF)-spttct. sntisfyi~igthe strict Mackey contlition (see e.g. (131,6.1.39 am1 5.1.31 ). C : o ~ ~ a c ~ly,i ~Hc is~ ~ bo~.nological t I)y Elierstetlt and Ronet [I], 1.3(a) ancl 1.5(6). ~) ltmrt of a n cmbcddtng 2.5 Le~nnla. Lct E = ind,, En bc thc ( I f a ~ i u d o r rriduci~t~c spectrum oJ locully coiitles spuceu. [,el I i br u uctbuptrcr8 o j E Jor. mhrch H,,:= tl TI En is dense in En for each 71 E N. T h e n E and id,,, H,, ~ I I ~ U Cthe C s a m e lopology o n H . This is a particular case of n result of Valcl~viato be fou~lcle.g. in (131, 6.3.1. 2.6 1'rol)osition.
Lct {Xn,jn,,,)be ait i ? ~ i i ~ i c t i t~wp e c i n i ~ onf Banach spuces loath compact c.inbedtlings. Let X := intl,, .A',,, lct Y br u uubspacr of X u i ~ dp.ui Y,, := Xnn Y,ic c IN. Tltcic thc followirlg coicdiiio~ttsU?T cq~i;ocilciit: ( 1 ) ind,,,
(2)
-
Y:" is a topologicul subspact- of -.Y
ufleI\yfn = P"
(3) Y iu bornolugicul for the topology rrrduccd by .\'
( 4 ) oil Y thc topology indltcccl by A' coirrcidcs
rrtitlt tlrc otlr of incl,,
17,.
J. Bonet, R. Meise. B.A. Taylor
104
*
is a (DFS)-space, this follows from Propositiou 2.4. PROOF.(2) (3): Since (4): This is easy to check. (3) (1) + (4): Lemnla2.5 inlplies that i~ltl,,, Y,, is a topological linear suhspace of illtl,,, 7;" a~ltlhence of S . (4) G. (1): Fro111 (4) ancl Lelii~ila2.5 we gel t.llal or1 Y the topology of X and o f in&,, F : coincide. Si~lcethe t.opo1og.y of illtl,,, F:" is finer t.l~a~l the one intlucetl by S,Bierstedt, Meise a~itlSulll~ners(31, 1.2, ilnplies (1).
2.7 Remark. ( a ) Let F be a I;i.Pchrt.-Schwartzspace x~ltlIrt ( Pk)r.in be nli ecluicoutinuous sequence of cont.ini~ouslinear operat.ors on F such t.hnt. lin~~,..,..f i x = x for all r tz F . If H is any linear subspace of Fx cont.aining UI.o~.~nclecl in 2. Now assulile that S(C/) :1 3'(E ) , i.e. tliat assertioi~%.3(1) l~olcls.The11 cleilote by r t l ~ e topology of El, = S and by ~o tlie topology of Z. Next. note that. by Proposition 2 . 1 the illclusion Inap (9,TI,,,) 5: is conti~~uous. Hrwe we llxvr TI^ = q I I , . Since T I , is finer than rIz and since (C is clense in Z we get from Bierstedt, Meise and Suinnlers 131, 1.2, that 71, = 712. C!onsec~uently,Proposition 2.6 ant1 ( * ) iinply t.hat ( X I ,r l s l ) is bornological. However, by our choice of E we get from Pr~posit~ioil 2.9 that ( X I;rlsl ) is 11ot bornological. Fro111 t l ~ i scoiltraclictioil it follows that tSheiissel.tio112.3(1) does i~ot. hold.
-
3
On the range of the Borel map
In this section we use toheresults of t.he previous one t.o i11vestigat.cwliich secluence spaces are coiltained in the range of the Borel mal> BN acting 011 :.,,,,(lKN) alld 3.1 Definition. Let w be a 11011-qurrsianldyt.ic weight. Then we define (a) P,(x
+ i y ) :=
1 v. (a:
(b) K " , ( J ) :=
dt. .I;'= W t'
-- t)'
-1- y"
(lt. y # O
3.2 Reulark. ( a ) P, is continuous ancl l~nr~nonic in the open upper and lower half plane and satisfies w 5 Pw. ( b )K , is a weight sat.isfyi~lgw 5 tiw. 111 general riw ueetl i ~ o tLe non-q~~asianalytic, see Uxanlple 3.8 below.
The range of the Bore1 map
107
3.3 Lemma, For a non-quasianalytic weight u, and a tueig1)t u the follo.wing coi~ditions are equivalent:
(1) Pw(z)= O(u(2)) as
2
tends to i.t~fi9rity
(2) riw(z)= O(u(2)) as z tplrds to iil$fi:liity.
PROOF.Ry the argunlents of Csrlesoil 151, p. 198, we have m u P,,,(t) = P,(.iy), for all y > 0. I.l=v
Next note that
On the otlier hand
Hence the equivalence of (1) and (2) is shown. 3.4 Lelnnra. Let w be a no,&-guusianalytic,weight ~ n Ad : lRN + [0, oo[ be continuQ Y:=, (h(In1a) ous and positively homogeneous. Put 1P := ( I t ( 1 m ~ ) ~ W ( : ) ) ~ ~ L rnP,(~)),~~yand p(z) := I1111 21 ~ ( 2 ) The.1~ . .we h ~ u t .
+
+
+
( a ) AQ is closed k A,.
( b ) For each f E AQ thew exists a seq,u.ence (fii)kr:R. ~ T A L Y with f = l i i ~ i ~f k- .i ~ n A,,. 0
('c) AQ is the closure of Ap in A,.
PROOF.(a) Since A, is a (DFS)-space, it suilices to show that AQ 1'1 A,, is closed for each n E IN (see Floret ancl W l o h [9]). Fix f E A,,, ant1 ( f , ) , e ~ in AQ so that f j -+ f in A,,,,. Then ( fj)3El\. is bounded in A,,, lmlce bliere exists il > 0 such that for each j E IN: I f,(z)l 5 Aexp(n1Imel nw(2)). (*I Since f, is in AQ we have: I f,(z)l 5 A, exp(A(1mz) n,Pw(t)). (**)
+
+
J. Boner, R.
108
Melse, B.A.
Taylor
From (*) and (**) it follows as in the pro01 of [lo], 2.2, that tliere exist C > 0 and k E IN, k 2 n so that for all j E IN
. f, -, f uniforully ou the co~l~pact subsets of Hence ( f j ) l E ~ yis bounded in A Q , ~ Since we get that f , -,f in AQ,k+,.Hence f is in Ao. (b) For j E IN define
cN,
wliere A, and B, > 0 are choosen in sitcl~a way tllat. w, is coatinuous, increasing and satisfies 1.1 (6) and w, 5 w. Then w, -, w and llence e,, P,. Note also that P, = O(log). Using It(Inl2) + P,, as the funct.io~is9,in Taylor [15), Thm. 1.) it follows that for each f F AQ,, there exists a sequence ( f r ) ~ . ~inl \ All.,,, so that fn.-,f in A,,:,,,. (c) This follows trivially from (a) ancl (b).
-
3.5 Corollary.
Let w, h, IP and p be as itr 9.4. Then
rs bornologtcal tn the topology tttduced by A , ( @ ~ ) .
PROOF. By Leninia 3.4 the closure of Au. in A, equals the space AQ, clefinecl ia 3.4. Hence Lelllnln 3.4 (b) i~ilpliesthat for A ' = A, aticl Y = A U . ,cu~~tlitio~i (2) vf Propositio~~ 2.6 is satisfied. Therefore, 2.6 (3)shows that AM,is bornological for the topology induced by A,. 3.6 Theoretn. Let w bc a )ton-quasia~~ulylic weight ant1 /el o be u rueighi. Then the followtng assertions are equtcalcnt:
(2) P,(z) = O ( o ( r ) ) as :teuds to infinity (3)
K,(z)
= O ( u ( i ) ) as :tends to infinity.
PROOF.Under the itletitifications mentioned i t ) sect.ion I , we have that BI, : A ( w , N): = A, A, - q,,(lRN); is the natural inclusion lilap, while the adjoi~itof the inclusion S : A(a, N ) + A(w, N ) eclitals t.lw i~~cliisiot~ A, 11,. Thcrefc~reC:orollary 3.5 t.ogetlier with Corollary 2.3 shows that. ( 1 ) is eqi~ivalenttts: (* For each set B in -4, wliich is I~ounclecli l l A,,, R is boii~~tletl in A , .
-
--
me range of the ~ o mmap l
109
In view of Lenlma 3.3 it autticcv llierefore lo show that ( 2 ) is ecluivaletil to (*). (2) =. (*): Apply the Phragmen-Linclelof principle as in Meisc mlcl Taylor (10). (*) + (2): To argue by contrrrcliction, cwvu~ilethat there is R sequeilce in Q: wit11 lini,-, = m. Without Ions of generality we can assulllr thst Inlo, >. 0. Then note
%
that by tlie proof of [lo], 2.3. (claim), tliere exists a sequence ( f,),er\. of polynomials, hence in A,, so that for suitable C , D . 0 mlcl for all j e IN we have:
If,(:)l Heiice (
1 let w , be clefillecl by
t
w,(t) = ~liax(O,(log et )* 1.
The11 an easy calculatio~lshows that. tcw,(t) = ( n -- 1 ) w,.
.
,(t).
Let w and T be non-quasialit~lyticweights, assume that 6 , 5 w ant1 let I< be a colllpact convex subset of lRN with non-elnpty inherior. T l ~ e the l ~ results of tile present section together with those of Meise ancl Taylor [lo] respectively Bonet, Meise and Taylor (41 illlply that t 11e following Ilolcls: he a fanlily of colltilluous fu~ictions oil Ii whicl~sat isties f(,,, 1I; E c ' K ' ( ~) Let (f,, and ( f,l,, I n )la) = f, 1 ,, for all a E .:NI K IC Then 3.9 R e m a r k .
sup sup 1fu(r)l e x p ( - " u p * ( -14 -)) ,i~l\/ rt1;
..
co for all in F IN
7 t2
inlplies the existence of y 5 f,,,(lFtN) satisfying 9'"'
Ilc=
sup sup I f,,(x)l exp( xEIi
i ~ l ~ p l i the e s existe~lceof y
f,,
I
for all a t I N:,
cp'(inlnl)) c: ax* for some ,tiz
4;
IN
111
f i , { ( R N sat.isfying ) (*).
Ackllowleclgeille~~t: The research of the first named author was partially supportetl by DGIClYT ProyecBo no. PS 88-0050.
References [ l ] Inet.: Barl.cllecl Lt~cally C:t>~lvcxSpil.ces, Nort.11-Hollantl Matllelllatics Stuclies 131 ( 1987) (141
R. A .
Taylor: Analytically unifornl spaces of illti~lit~ly tlifkrent.iable fi~nct.ions, Conim. Pure Appl. Mat.11. 2.1 ( I971 ), 39-c) 1
[15] B. A. 'raylor: On weight.etl ~)t.,lyllorllialal~pl.oxilliat.it,~~ 01' nl1t.il.r rullct.i~)11s, Pacific J . Math. 36 (1971), 523-5119 (161 M. Valclivia: Algunas propricclncles tlc los c.spncios i~scalonaclos,Hev. Hcal. Acacl. C:i. Matlricl 73 ( 1979), 389-1399 1171
class of cluasibarrcllccl i Dl')-sl)aces wllicli are not bornological, Math. '2. 136 (1974), 249-251
M. Valdivin: A
[Id] D. Vogt: AILexalnple of a lluclear l~'r6chetspace wil.llout bllc boiu~dcclapproxin~atio1.1property, h4at.h. Z. 182 ( 1983). 265-267
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Progress in Functional Analysis K.D. Bierstedt, J. Bonet, J. H o ~ a t h& M. Maestre (Eds.) O 1992 Elsevier Science Publishers B.V. All rights reserved.
Biduality in Frecliet and (LB)-spaces Iclaus D. Bierstedt and Jos6 Boiiet Dedicated to Professor. Alr~nr~cl I'cllrlrrllc~ OIL the occc~s~on of his Goth bzl-thday
Abstract. Let E be a Frechet space (resp. an (LB)-space il~tl,,E,,) and II a topological subspace of E (resp. H = i~ltl,,If,, for a sequence of norm subspaces If,, of En with II, C fIn+l). We give necessary and sufficient conditions that 6 is canonically (topologically isomorphic to) the inductive bidual (I/;): or (even) the strong bidual 11; of I f . This characterization is applied to weighted spaces of holomorphic functions, weighted inductive limits of spaces of holomorphic functions, spaces of differentiable functions with Holder conditions and other examples i n distribution theory, holornorphy and Kothe sequence spaces. locally collvrx (;ll)l)rc~viatc~tI: 1.c.) spacc ( E ,r ) has a basis of al>solutely If a bor~~ological convex l~oundedsubsets whicll arc co~lrl);rc.t.wit.l~rospcct. t o a wealter 1.c. topology i ( a condition which is callctl (Bl3C) I)c:lo\\~),it is t11t. il~ductivcdu;rl F/ of t h e complcte space
F' =
{{L
E*;
( L I B?
- ~ ~ I I ~ ~ I for I I I( XO~ IIC II lI \~ o r ~ ~ ~set t l c tBl
c E}.
This result is due t o h'l~~jicir (scc [ I I ] or [25]). 111 our (plc1)ari1t,ory) Section l., we are interestccl in 1.c. propert.ics of t11(% 1)rctl11al1;' i r ~ ~ prove tl (C!ol.oll:rry 2.) tl1a.t I: is barrelled and E tlre strong dual 1'; of I;' wl~c~l~c~\.c~r T illso Ilas ;L 0-11~~ig11hor11ood 1)a.s~of absolutely convex i-closcd sets ( c o ~ ~ t l i t i o( (~IrN ( : ) ) . \\/l~ilcfor (1,13)-sljac.c.s I':, F' clearly is a Frdclict space, the 1.c. propcrt,ics of tllc. ~)rc~lu;ll I;' ; ~ r (~lrailrly . i ~ ~ l p o r t n nint the case t h a t E itself is Frdchct,. In fact,, a 1"ri.cllct s1);1(.(~ I,' s;llislicts I ~ o l l(13I3C) ~ ;r11(1 ( C N C ) if and only if it is (topologically isonlorl,l~ic to) t11(~s t r o ~ ~t lg~ ~ i 01' l l a co11111l('t.(' I)ar~,ellrd(DF)-space 17, and a Frdcllot space E is c l r ~ a s i ~ l o r ~ ~i111c1 ~ n l silt,isli(~s )l(~ (1%13(:)allti ((:NC) i f a ~ l t lo ~ ~ ifl yit is t l ~ c strong dual of a b o u ~ ~ t l ( ~ trc~t.ri~cI.i\.c~ lly (l,l3)-s1);rcc~ (C:orollirry 5 . ) . Section 2. of t11c: article c o l ~ t i ~ i~~, I lI (s' ~ ~ l i l Ii Y ~ 'l S I I I ~ \\:(: S : c ~ I ~ I ~ ; ~ c ~(by c L .tllc ~ z (co~lditions : (i) and (ii) of Thcorem 6.) wllc~llor ;I ~ol)ologic;~l suljsl);~cc11 ol' ir I)ornological 1.c. space E with (BBC), tllc restrictioll 111irp1)illgI? : 1" + 11; is a to1)o1ogicaI isorno~~pl~isrn onto, i.e. E is c a ~ l o ~ ~ i c at.ll(> l l y i ~ l t l r ~ c t i \I)itlual ~o (11;): of 11, i r ~ ~ 1lc.11cc tl tllc strong bidrial II: wllenevcxr (CNC) lloltls. 111 t.11(, c;rsc3of ; I I I (I,l3)-sl)aecs /I = intl,, E,,, t h e hypotllescs have to b e lnodifietl in order to irvoitl t l ~ otopologic;~lsr~l)sl):rc~: p r o l ~ l c ~for r ~ inductive lil~lit~s; it turlls out that /I = illtl,, I/,, wit11 ( I ) i111tl (11) of ' l ' l ~ ( ~ ) r c7.~ lI ~ T ~ I I S ~11rc~ssari1y , he a ~ ( i i ) i l l 'l'11eorc1116. (rc~sp.(11) in topological sul~spaceof E. 111 I I I ~ I I CI ~~ X ; I I I I ~ ) I ( ~ Sco11(1it,io11 T l ~ c o r c7.) ~ ~is~alrcatly sat.islic~tl,;III(I ~ I I C ~ I o111j. I c o ~ l ~ l i t i(i) o ~(rcsp. ~ ( I ) ) , wllicll i ~ ~ v o I \ ~ c s
114
K.D. Bierstedt, J. Bonet
tlic approximation of l~ou~rtletl w t \ i l l F; I ' I ~ I I Ibou~rdrds ~ ~ l x eof t s 11 in t h e topology ?, must b e verified to obtain t h e I ~ i d u ~ ~ loi lt y11 alitl E . Tile results in Section 2. arose from an arlalysis, and a sweeping generalization, of tlie functio~ial-analyticpart of tlie ~ L ~ ~ I I I I I Ci Il lI ~[S]. S I-Icrlce our first applicatioris again a r e weighted spaces of Iiolo~norphicfu~lctions(Subsection 3.A.) and weighted inductive limits of spaces of holorriorphic f u n c t i o ~ ~(Sul~sc~ctio~l s 3.B), where the arguments of [8] can b e utilized t o obtain t h e bidualitics IlI+'(C,') = (III-Vo(C))t a.nd V I l ( G ) = ((VoH(G))b):, ~ n a i n l yfor sequences IY resp. V of radial weigl~tsor1 certain l)alanced domains G c C N . Another example, t h e canonical bidualily Ak,"(Cl) = Xk'a(R):of spaces of k-times continuously differentiable f~ulctionson all opcrr sct (2 c IllN wliich satisfy IIijlder conditions with exponent (Y E ( 0 , l ) on each c o m l ~ a c ts r ~ l ~ sof c t 62 is worltetl out in Subsection 3.C in some detail. At t h e end, in Subscctio~i3.11, \\.c ir~clutlco t l ~ c rexarr~plcsfrom distribution theory (the result 8 ( R ) = B o ( R ) t of I)icroll', Voigt [ l o ] ) , i ~ i f i ~ ~tli~rle~rsional itc holornorpliy (Ilie space IIb(U) of liolomorpl~icl ' u r ~ c t i o ~of~ sI ~ o u ~ ~ t ltype c t l ~ L I results I ~ of Galindo, Garcia, Maestre [14], [Is]) and from I).
ill
F (as cornpi~retlwith t h e
But any 1 E F' helollgs to ( I i n 1,')' I'or s o ~ l ~0-~lctigl~l~orl~ootl c~ 11 in L;. 13y (BBC), there exists B E Bo wit11 1 E ( D o n F)" = 1)'" = U , \vl~icllscrvcs to sllow t h a t J : 13 + F' is surjective. Fro111 now 011, 1r.c \\!ill il:\. /I wit 11 1"' a1gc~l)raicnlly(via J). i c l c 3 1 1 t
As B varies in Do, t h e corresl)o~lcli~lg s ~ t sI)* H I I ~ I lIeOdescril~eI~ascsof 0-rleigl~borhoods in tllc sl,ace I: ant1 I~ascsof t11c c ' c l ~ ~ i c , o ~ ~ t ~('1,s , i ~ ~ itl ~l oI,", ~ ~~b. c : s ~ ) ~ c t i v ~InI yview . of this,
116
K.D. Bierstedt, J. Bonet
it clearly follows froin (+) that J is a topological iso~norpliismof E (= indB EB = the bornological space associated with E ) o ~ ~ F/ t o = intlu Fb,. This finishes the proof of Theorem 1.
2. By (BBC) each bounded subset C' of E = 1;; is contained in an some B E Do and hence, by equation (+), in tlie polar of some 0-neigl~borhootlB* = B0n F in F . Thus C
1;1' = indu 1;;;, 11l~1stbe a regular inductive limit. of the proof of Corollary 2., put G := ( E , i)'c F and let Uo be as in (CNC).
is equicontinuous on F , and
For the rest Each U E Uo is u ( E , C;)-closed a11tl i L fortiori u ( E , F)-closed, whereby U = U*' (tlie first polar again taken in F ) . As the ecll~icontinuousset U* is /?(I;:E)-bounded, U must already be a 0-neigl~borliootlill P L = (P', B(I:', F ) ) , and we have proved F: = FL. At this point, the last assertion of ( a ) implies I;' cluasil~arrelletl. Clearly, a space E satisfying I ~ o t l(1313C) ~ and (CNC) must be quasi-complete. (This follows from the closed neigl~l)orl~ood conclition (CNC), together with the con~pleteness consequence of (BBC), cf. I O for each I I E Uo, for111a I~xsisof tlic T-1,ountled sets in E , and each B((Mu)ri) is absolutely corlvex a ~ ~i-closetl.) tl 'L'l~r~s, if (CNC) lioltls and if (E,?) is a semi-Monte1 space, then we n ~ u s talso 11ilvc(131K). The first consequence at this poillt, is hlt~jica'scol~ll)let.c~~ess tl~eorc~ll for (1,B)-spaces (sec [23]) wliich was insl~iretll ~ ya t.l~eol.c~r~ of ~ ~ ; L I I ; I C I I - D ~ X I011 ~ Idual ~ CBanach ~-~~'~L~ spaces (cf. Waelbrocck [30, Prop. 11 nntl Ng ["GI). It is stated here for the sake of cornplctcness. 3. Corollary (Mujica). I,cl I I = ( I ) , T ) 6 c (111 (1,11)-bl)nce, E = ind,, E,, for an zizcrea.szng sequeizce (En)nEN of UIIZ(ICIL .zl)(~c(.zii11111 ( O ~ L ~ I I L ~L ~L L~ , ~~ CL CS ~ ~ Oa IrZ~Sdsuppose: ,
(*) Tltere exzsts a 1.c. IInusdo~lfSlopology i 5 each En zs f - c o m p a ~ l ,I L = I,',. . ..
T
or? E sl~cltIltnt
file
closed
711111
ball 13, o j
( a ) T h e n t;' := (11 E E * ; , i ~ p , ,is ?-~ort/ittliolisfor e(ic1111 E iV}, e~ldoruerlruitli the topology of unifornt conuergrnce 011, the scls I),, ( 1 1 E N ) , is (1 I;i~t:'chetspace ( i n fact, n closcd subspnce o f E [ ) sucll [hat llle evnluc~liott trtc~ppir~g J yielrls (1 lopological isorr~oryltismo f E onto F:, ant1 llcncc 15 irlllst be cor~rl)lt/r.
( b ) If, in addition, (CNC:) holtls, ll~crtF is tlisli~t~gc~islretl, clrld E is topologically isomor-
Biduality in Frechet and (LB)-spaces
phic t o the strong dnal
117
FL.
PROOF.Grotl~endieck'sresult 011 t h e "nl~llostregularity" of countable inductive limits of (DF)-spaces (cf. [22, $29, 5.(4)]) a i ~ t l( * ) ccrtai~llyimply that ind, En is regular, and hence ( n B n ) n E Nis a funda~nentalsequence of b o u ~ ~ d esets d in E . Therefore, condition (*) implies (BBC) (and the d e l i ~ ~ i t i ool' n I" agrees wit11 t h e previous one). In view of I., t h e proof of (a) is fil~islletlIly ~ i o t i n gt h a t F is metrizable in t h e present case and that t h e inductive dual of a i~lctrizallle1.c. space must b e complete (cf. [22, 529, 4.(2)]). (b) is a simple consequence of ?.(I)). We add tlie following
4. Remark. If, undtr- ihc c o ~ l d r / t o i ~ofs .?.(i1), i~itl,,E,, quasznor~r~able, nnrl / h e conue,ae h o l t l ~( 1 , corll.
LA
6ou11~ltdly retractzve, the12 F is
Clearly, I~oundedrctractivity ol' t l ~ ercy!,ulas i~~tluc:tive liinit, I< = ind,, E,, means precisely that E satisfies the strict R1iicl;ey collvcrgellce c o n t l i t i o ~ ~Tlius, . 4. follows from Bonet [9] (a Fr6chet spacc F wl~oscil~tlucti\,c:tlr~alI:,' is secli~c~~tiitlly retractive must b e quasinormable) and [27, 8.3.351 (if b' is clrlasillor~nal)le,I$ = I;b' satisfies t h e strict Mackey convergence contlitio~l). For a non-distinguisllcd Frdchet space (:, I? := G: is a conlplrte (LR)-space which satisfies (*) for i := a(G1,G), but does not silt.isfy (CNC:). (If G: llatl a basis of 0-neighborhoods which arc, absolut,ely convex iind rr(C1,C;)-closetl, tllcl~G \vould be distinguished in view . ) all (1,13)-sl);1ce 6 wit11 (*), it lnay actl~allyturn o u t t o b e of the bipolar t l ~ e o r e r ~ ~For rather hard to clicck wl~ctli(:r(CNC) I~oltls. hlujica [23] ~ ~ o t that c s (CNC) is satisfied if for each 0-neigl~borllood U i l l L?, t l ~ c ~cxist.s c a sequence (\/lL),LEN of absolr~telyC O I I V ~ X ?-closed 0-neighl~orlloodsI/,, in I? wil,l~I 1. v := I l ( r ~ = ) rLI,I I)clol~gsto C,"' n R(I7) c I) ~1. Now assume that (i) Ilolds, 1t.t u E 1,'snt.isl'y 0 = l l ( i ~ )= 1111, ant1 take x E l3 arbitrary. (i) yields a bountlctl sct C: C 11 \\fit11.r E p. Ijy (131jC:), \\re can then find an absolutely convex 1)oundctl and i-t:ompilct sct 13 c L: wit11 C c B anti thus, a fortiori, 7 7 c B. By definition, E F is i - c o n t i n ~ ~ o0 1~1 ~I),s alid since 11 va~iishes011 C c 11, we also get u ( x ) = 0. Thus IL = 0 (ill I;'), w l ~ c r c l ~1I y is ilijcctive. '1'0 sho\v t h a t R is open onto R ( F ) , fix a bountled set D c l? (or, cclr~i\ral(~~~tly, a O-~~eiglll~orhood B0 n F in F ) . By (i), thcrc is C C 11 I)ountlctl wit11 11 c I f ,u E C' n R(1Y1), tllcre exists u E F with v = R ( u ) = I L I , ~ant1 I I,u(c)l = I I L ( ~ [or ) I all c E C,'. Ily ( I j n C ) , t,hcrc is ari absolutely convex Iloundcd ar~cli - c o ~ n p ; ~ cs ~t ,~ l j s iI1 t t of I c o n t , a i ~ ~ iC ~ l, gand thus also Now lul a on C , h' C C and ,u E 1;' is i - c o ~ ~ t i ~ i u o011 u sI ) ; 11enc:e l i ~ l 5 1 011 B, 11 E B0 n F and v = R ( u ) E R ( B On F ) . \Ye II~L\JC s11o\v11C* n IZ(17) c R(Ilo n I:), urhicl~clearly implies that R is open 0nt.o /?(I").
c.
>
c.
il~il~.y. \\I(' ~ ~ l r lc'xtcnd sl v t o a n elernent u
E
F.
120
K.D. Bierstedt, J. Bonet
Given x E El we know by (i) that t l ~ c r cis a n al)solutely convex bounded set C c H with x E C. By (ii), v is actually i - u ~ ~ i f o r n continuous ~ly on ( t h e absolutely convex set) C , and since it takes its values in tllc co~npletespace R or @, v lias a uniquely determined ?-uniformly continuous extension Cc to P u t ~ ( 3 : ):= Ec(x). It is a m a t t e r of routine t o check that u(x) is indeed well-defi~led(i.e., i~iclepentlentof C ) . Thus, we have defined u E E* with U I H = V . It r e ~ n a i n st o verify t h a t the restriction of u t o each bounded subset B of E is ?-continuous (so t h a t u E F'). But for any such B, (i) yields a n absolutely and since v was extended t o Ec on in a convex bounded set C c II wit11 B c ?-uniformly continuous way, 111ust again be i - c o n t i ~ ~ u o u s .
c.
c,
Remark. Let ( E , T ) be a bori~ologrccil1.c. sl)clce wllzch has properly (BBC) (wzth respect L condztzons (i) t o a I.c. topology i 5 T) a i ~ d11 (1 tol)ologzc(~lsub~pnceof E U ~ ~ I C Isatzsjes and (ii) of T l ~ e o r e ~6.n ( a ) Then II must be q~~c~sibni.~~ellcd.
(b) IJ, in addition, ( C N C ) holds, 11
i.s
c~lsoclis/ir~gz~ished.
PROOF.(a) Let T d e ~ ~ o tt le~ cfines^ 1.c. t.ol)ology on L' :'vl~icl~coi~icitleswith i on all bounded sets. Now (i) also means that for e a c l ~boundctl s u l ~ s e tI3 of E , there is a n absolut,ely convcx bounded set C c I1 wit11 I3 C := t h e closure of C in ( E , T ) . Since ( E , T ) is bornological, (B13C:) clearly i1111,licsi T ; OII t h c other hand, for t h e canonical predual F of E , 1;' = (I?,?)' I~olds,JYIICIICC o ( E ,1 7 ) T.
Thus, {7~E t l ( F ) ; q,,(v)
5 6 ) c Il({.11 E 1"; ~ J , ( ? L ) 5 E } ) , ant1 R is open onto R ( F ) .
Conversely, let I2 b c i~~jcctivc! a1111o1)o11o11t0 I?(/*'). Tlien for e ~ c hn E N thcrc are
122
n u n ~ h c r sN = N ( I L )E N , N
K.D. Bierstedt, J. Bonet
2 1 1 , and
e = ~(11)> 0
~ ~ 1 tllat ~ 1 1
Suppose t h a t (I) fails; there cxists I L E N with 13, $ 1)2C1, for each m 2 11. Choose > rnax(N(rz), ~(11)-') and b E B,, \ 1)2C,,,. 1Ii~l111-Baniich yieltls rL E (E,?)' C F with Iu(x)I 5 l / m for all x E C,,,, I)rrt lu(0)l > I . Now u := ll(t1) = ul,, belongs t o R ( F ) and satisfies
172
<
)I 1. In the rest of the proof, we will assumc c o ~ ~ t l i t i o(I) ~ l ancl show that then (11) holds if and only if R is surjective. A g a . i ~ tlic ~ , implicatio~r R surjective + (11) is trivial, and we assurne now t h a t (11) holds. Tlrc~lan arbitrary 11 E /I' must b e extellded t o some u E F . This can be done exactly as i l l the proof of G . : By (II), vl,~,, is ?-uniformly continuous ( n E N ) and has a, u~liquelyclctcr~ninc~cl i - r ~ ~ r i f o r mconti~iuous ly extension to G. In view of (I), one can easily cl~cckthat tliesc. ext,e~~sions are coherent and combine t,o yield a well-defined 7~ 6 E* wl~oscrestrictior~to each I L = 1 , 2 , . . ., is ?-continuous.
n,,,
Altcrnat,ively, it is also possil)le ( a ~ ~not t l t,oo Ilartl, using, say, [27, 8.1.121) t.o prove t h e follovving 8. L e m m a . Let 7 (re.51). i )d c ~ t o l c{ / L C J1r1cst 1.c. to/)ology O I L E (resp. II) P U I L O S ~restriction s 7. T l ~ e (I) i ~ irr~plics = .i. l o each B, (resp. C,), I L = I , ' ) , . . ., c o i ~ ~ c i ( l c,uiitl~ By (11) any v E 11' is actr~ally?-continuous orr 11; llclice 1,cmina 8. implies ti E (Il,.flr[)'. By use of the I~lahn-Banachtllcorcm, one can find .IL 6 (IS,r)' with trl,, = v. But then PL~B,, is 7-continr101is for each 7 1 , i ~ ~tl111s ~ t l21 E 1;1 with R ( u ) = v. \4'e will now assume (I) ant1 (11) ancl itlcntify 11; wit11 F (via R ) and I;;' wit11 E (via J ) . Then (Hi): = E holds canonically. To concluclc tllat in tlris case (If,7') is a topologic;~l subspa.ce of ( E , T ) , it is clcarly ellougl~to observe that a.ny (LB)-space II is a topological subspa.ce of (11;):: Certainly H is a tol~ologic;ilsubspace of its strong bidual H i := (IIL);; hcilcc (/I;): already induces a strorlgcr tol)ology 011 11. On t h e other hand, in view of t h e continuity of t h e e ~ n l ) e t l t l i ~I1 ~ g+ I/[, tile c a n o ~ ~ i c mapping al H -+ (Hi): has closed graph, and thus must b e continuous by Grotlie~~diecl;'~ closed graph theorem (see [17, Thborkme 13, 2.1); i.e., (11;): also intluccs a \\'calier topology on 11. 9. Corollary. If, ill the s i f u u f i o r o~ j 7 . , (I) a ~ (IT) ~ d holrl, i h r i ~ind, 11, bovl~dedlyretractive i n ~ p l i e sF quasinormable. 111 illis cnse, 13 is (cnrto~licnllylopologically isoli~orphicto) llle sti-or~gdual of F and the stl*or~gb i d ~ ~ of n l 11, r~,rzrliirtl, E,, i s bou,~,rledly~.etractive as well. This is a simple consccluence of 7.(11), 5.(tl) ant1 4 .
Biduality in Frechet and (L8)-spaces
3. Exaiiiples
A. Weighted Fr6chet spaces of lioloiiiorpliic fuilctioiis
>
In the sequel, C denotes an open sul,set of gN ( N 1); the space I I ( G ) of all holomorphic functions on G will be endowcd with tllr, tol)ology of unifor~ncollvergence on the compact ~ ~strictly positive continuous subsets of G. For an increasing sequcncc I V = ( I U , , ) , of functions w, on G, the weiglltctl sl)i\ccs of holonior pllic fullctions are definccl by I I W ( C ) := {f E II(G); for car11 11 E
JV,]I,,(f ) := supzEGto,(z)l f(z)I < m ) and
IJT/Vo(G) := { f E I I ( G ) ; for c>ach11 E liV, to,,f vanishes at m
011 G).
Under the topology T given Ily tile scclucwcc ( I ) , , ) ~of~ 110r111s, ~ ~ E := H W ( G ) is a Frkchet space; its topological suhspacc 11 := IIIVo(C:) is closctl, l~criccitself a Frkchet space. topology to E . Then E has a Let ? 5 T denote tlic rcstrictiorl of tllc co~~ipact-ope11 of al)solutc~lyconvex i-closed sets, viz. 0-neighborhood base I& = (Il,r)T,EN
u, = {f E I I W ( G ) ; plL(f)I 1/11) = { f E E ; I f(z)I 5 ~ / ( I z T u ~ (forz ) all ) z E G), n = 1'2, ... Put
v := {C> 0 co~ltill~ior~s on C;:
for ctacll
11
E h\J, sup ro,,(z)C(z) < m) zEG
and for arbitrary 6 E I/,
D, := { f E IICV(G);
If ( ~ ) l
< v(2)
f ~ l i.~ 1 1z
E G ) , Cfi := DE n IIl*V0(G).
Clearly ei~cliU , is al,sol~~tely col~\~cx, T - I ~ O L I ~ I ~ and C ~ ?-compact by Montel's theorem. Since a sct B C E is 1,011ntlctl if nrltl orlly if U is col~t.ainctlin sornc B, (cf. [7, Proposition 2.51, together with [Ci, O.2., l'rol)ositio~r]i~ntl15, 4.2]), LS = IIIH/(G) satisfies (BBC) and (CNC). 13y Corollary 5.(b), 11 is (1ol)ologically isomorpllic to) tlie strong clual of the complcte barrcllcd (Ill?)-spac(~
F
= (11 E
P ;t i l ~is ? - C O I I ~ ~ I Ifor L Ieacll ~ I I SI)ou~ltlctlset R
in E),
eridowed with t,he topology P(F, E ) Ncxt we show that condition ( i i ) of 'l'lrcorcm 6. is always satisfied for II = IIT.Vo(G). In fact, by tlie IIahn-Ba.nacli ant1 llicsz rc,prescant,at,ioi~tl~corcms(cf. [28]), for each 1 E II' tlicrc are n E IN ant1 a bountlctl Rat1011 rllcasurc / L oil C \vit.ll
Now the restrictio~l of 1 to ~ L I I arbitrary I,ou~ltletl sul>sc:t C of H 11111stclearly be ?continuous by the inner regulari1.y ol' / I . 'l'l~~rs wc 11avcprovotl:
lo.
Proposition. ( ( l I l l / o ( f ~ ) )=~ )IIll'(C;) ~ lroltls cc~itor~rcally if (111d o r ~ l yif for rach
124
K.D. Bierstedt, J. Bonet
bounded set B C /IbV(G) there zs u n absolutely convex bolrndcd set C an HWo(G) such that B zs contazned an the closure of C Z I L llbV(C) ~ 2 1 1 1re51)~ctto the compact-open topology .i. (In thzs case, IICVo(G) nlust be dzstznguzshed.) Proposition 10. generalizes ( t h e isonlorpliic version of) [S, Thcorern 1.1 and Corollary 1.21 t o weighted Frdchet spaces of holomorpllic functions. 111 [8, Section 2.1 t h e case of radial weights on tlie unit disk D c Q, the complrx plane 0' or on some related domains G c C N was discussed. To a large part, this discussion also applies in t h e present (more general) setting. We will now give some details for entirc functions of one variable.
~ For t h e time being, take G = Q and assume that all t h c weights in W = ( w , ) , ~are radial, i.e. to,(z) = zu,(IzI) holds for all s E C , 12 = 1 , 2 , . . . By t h e argument of [ G , 0.2, Prop.], it will then be n o loss of gcncrality t o suppose that also alLwcigllts u E V a r e radial. Finally, pk denotes t h e function pk(z) = z k for z E C , k = 0 , 1 , . . ., and Pk:= span(p0, ...,pk} for k E N o , P-, := (0). For fixed k E N o , pk E I I W ( C ) (resp. p k E HIYo(C)) trivially irnplies t h a t H W ( C ) (resp. EItVo(C)) contains Pk. Since all tlie weights arc radial, one can easily check (e.g., see t h e indirect proof in [S, final part of Section 2.1, wl~ichi~lvolves'l'aylor series expansion ahout 0 and the maxirnurn modulus thcorclll) that
11. Proposition. [J11(Err the p t ~ s rtt r c~s.~lr~nl)troi~s, the Jollo~uzr~g propertzes arc cqazvale~tt: (i) For each bounded I? c 1Itl~'(C) there is (LIL (~bsolutely(:ouvex b o ~ ~ i ~ dCc dc HbV0(@) s ~ r c hthat B is co~ltailledi r ~ (closui~ein (III+'(C), i)). (ii) For each 6 E V, U , is c o ~ ~ f ( ~ i tit1 l c d/he i-closure of C,. (iii) J'or
arbitrary k E IVo, 1 ) E ~ IIIIf(Q") ~,npl?es E III/I/,,(~).
PROOF.Trivially (ii) + (i). A~itli l l view of tllc. p r c c e t l i ~ ~ren~arks, g (i) + (iii) because pn E IIW(Q)' \ HI.Vo(Q') woi~ltlimply lIIVu(Q') C 'Pk-,, ant1 since t h e last space is i-closetl, we would get a cont~adictiolit o (i).
c Pk for some I;, and thus in order to show Finally, (iii) talccs care of tile cas,. 11II/(Q') t l ~ a (iii) t + (ii), it sufficcs to prove tl~at.(ii) l~oltls\vllcweve~,all \vcights 70, arc (radial anti) rapidly decreasing (i.c., pk E III,I,'(@) or, ecluivalcntly, p k E lll,lfO(C)for all k E N o ) . For this pr~rposefix f E B, ant1 let j;. tlcllote the f u ~ ~ c l i of,.(z) n = f (1.2) for z E C , 0 < 1. < 1. We have f, + f ~~nifornily or1 co111pi1c.ts~ll)sc-~,s of ( I as'1. -+ I-. Since v is radial, tlie nlaximurn motlulus tl~coretni~~ll~licts t l ~ i ~fort illiy 2 E @ ' t,llerc is ru E Q , ICYI = 1, with c e C C',). At this point, it relnair~sto verify that f, E IIIVo(Q') for 0 < 1- < 1 (and h e ~ ~ (f,), For this one can follow t h e p ~ o o fof [S,Esamplc 2.21 allnost literally.
125
Biduality in Frbchet and (LB)-spaces
It is obvious from 10. and ll.(iii) t h a t for a sequence W of radial wcights on 67,
i.e. I I W ( @ ) is reflexive. On the ot11c.r Iialld, we have:
12. Corollary. If lV 1s a11 I I L C ~ ~ ~ ~ ,he(luetlct ,LIL~ of I-adzal mpzdly rlecrenszng wezghts on C, then ((HlVo(67))b)b = IICV(Q') holda ca~~orrrcally. In fact, t h e argument after [S, 2.21 de~~ioristrates t h a t this corollary remains t r u e for an increasing sequence of rapidly decreasillg weights w on CN wliich are radial in t h e sense that w(Xz) =
W(Z)
for all :E Q ' ~alltl c v c ~ yX E @ with [ X I = 1.
Similarly, following t h e same lnct l~otlof proof (cf. [ S , Exa111l)le2.1]), we also get (in a n even simpler way) ( ( l l I l r u ( l ) ) ) ~=) ~ IllV(D) for any increasing sequence 1.V = (lo,,),, of radial weights on t h e unit disk D which satisfies l i ~ n ~ , ~ , ~tu,(z) = 0 for 11 = 1 , 2 , . . . Antl this result carries over t o related types of balanced (bounded) domains C C C N a ~ radial ~ d weigl~tswliich vatlisli a t aG, cf. t h e rcmarks after [S, 2.11.
B. Weighted (LB)-spaces of holomorphic fuilctions In part 13., V = ( v , , ) , , ~d~~ ~ i o t ci L~tIc:cri~;~si~~g s S C ( I I I C I ~ C Cof strictly positive continuous functions v,, on a.n oprn sul,sct C; of P N ,;11,c1
E,, := Il.u,,(G') = { f E 11((:); 11.1'11,,
=
S L I ~ , ~u,,(z)lf C
11, := I I ( v , ) ~ ( C ) = { f' E ll(Cr');,u,,f' vallis11c:s a t
(z)I
< m),
m on
G)
are t.11~ wcigllted Danncl~spaces (cntlo~vc~tl wit.11 II.II,,) of 1io10rnor~~hic futict,ions on G which are associatetl with V, 12 = 1 , 2 , . . . 1111c' closed 1111it1)illl of I?, (resp. H,) will b e detloted by 113, (rcsp. C,,). 1Vc are i~lteresl.c~tl i l l tllc, (LEI)-spaces
, (0, y)
+
126
K.D. Bierstedt, J. Bonet
Then F must b e distinguished, wl~enceV I l ( G )= 1%. By t h e same argument as in A. abovc, one can see t h a t condition (11) of Theorem 7. is always satisfied for H = V H o ( G ) ,and thus we obtain:
13. Proposition. ( ( V o t l ( G ) ) b )=: V I I ( G )holds canonically ifanrl only i f f o r each n E h' there are m n and M 2 1 ~uilllB, c MC,,, ( c l o s ~ l ~i nt ( V I I ( G ) , ? ) ) .In this case, we = V I I ( G ) whenever V is ~*egularlydecreasing. have ((VoH(G))b)b
>
As we did in A., we now turn t o the case t h a t V = (V,), is a sequence of radial weights on G = (C. T h e following proposi1,ion (in wliicli closures a r e always taken in (VII((C),i ) ) is parallel t o 11. arid can b e proved in exactly the same way.
14. Proposition. For a decrcasz~~g scquence V of mdicll ~uezghtso n Q , the Jollotuzng condztzons are equzualent:
( i ) For each n E h' there arc in (ii) For n E
N
there zs In
>
12
arid A f
> 11 a ~ c that l~
2 1 w ~ t hB,, c
a.
Jbr all k E &, 11k E Iiv,((T) zrnplzes
pk
E
II(vvn )O(G). Under t h e assumptions of 14., it is clear t h a t
(for t h r last equality see the elid of the proof of 7 . ) , i.e. VII(Q') is reflexive. O n the othcr hand, if all v, are radial and rapitlly dcc1ea5ing o11 ff, [8, 2.21 directly shows t h a t (even) B, c C, holds for every 17 E LM, wlic~ice: 15. Corollary. IJ V is a clecrcasiilg scquet~ceof 1,adial m p i d l y dccreasing weights o n (II, then ((VoH(Q))b): = V H ( C ) 110111.5 canor~icnllycmrl VoH((C)is a topological subspace of V I i ( C ) . IJV is, i n addition, .vegulnvly decrcn.sing, ule also hnve ((VoH(Q))b)i = VII((C).
Corollary 15. r e n ~ a i ~true ~ s for a decreasing sequence of rapidly decreasing weights o n C N which are radial in the sense explained aCter 12. Actually, t h e examples of [8] again allow to deduce biduality results for weigl~tedintluctive lilnil,s of spaces of holomorphic functions on D (resp. 011 certaii~halanced hounded dolllaills G C C N )if t h e weights v, in t h e dccreasing sequence V a r e radial a ~ i t lvanish nt OD (resp. radial in t h e sense after 1'2. and vanish a t DG).
C. Spaces of differentiable fulictiolis with Hijlder conditiolis Let k E No, 0 < cu < 1, and Ict 12 denote an ope11 subset of R N ;fix a basis of compact subsets of R wit11 li,, C I:,+,, 72 = I , ? , . . . In this section we investigate t h e following algebras of Ck-fi~nctions011 R wl~osepartial derivatives of order k satisfy a IIijlder corldition with exponel~ta on cach compact srlhsct of 12:
127
Biduality in Frechet and (LB)-spaces
~ " ~ ( 0:=) { f E C k ( 0 ) ; for each c o ~ n p a c tIi
c0,
~ " ~ ( 0 := ) { f E ~ ' " " ( 0 ) ; for CLLCII c o ~ l l l ~ a cI< t c O and each Iim U U f ( x ) - D L i f ( 2 / ) = 0 as 1" - 2/10
I.
- y ~-+
with 1/31 = k,
o wit11 r , y E 10.
Both spaccs are Fr6cliet under t h e topology givcn 11y t h e sequence of seminorms y..(f) := max
(z
O 0. For ciicl~1 E N clioose a C"-function 11.1 on RN wl~ichis identically 1 011 Iil ali(1 II;IS its s1111port illside I?~+,,and take a sequence ( E I ) I E Nwith supp(/zl)+ u ( o , ~ E ~c) I?~+,( I E AV) ant1 li111El = 0. I-"
Next, for a nonnegative Cm-ful~ctior~ pol1 l l Z N wit11 sup11(/)) c B ( 0 , I ) and JnN p ( ~ ) d x= 1, put pi(") := E ; ~ ~ ( I . / E(n. ~ )E E N a ~ l t l1 E bV). IJinally, for arbitrary f E Ak'a(R) tlefine
for each f i E Cm(R) c Xk."(0) = I1 wit11 hul)l)(f i ) C blll)l)(hl) + B(0,E ~ c ) 1 E N ,ar~tlit is ~ c l l - l i ~ l ot ~l ~ni ~I; t -t / i l l ('"0). T ~ I I I Y the , closure of t h e set
Thus
128
K.D. Bierstedt, J. Bonet
in (E,.i)contains B , and it r e n ~ a i n st o prove that C is bounded in Xkla(R). For that purpose, fix the s e ~ n i n o r mp,. Since B is compact i11 C h ( R ) a n d hl -+ 1 in this space, it is easy to verify (e.g., using IIorvitll [l8,4, $7, Exercise 6 and 4, $10, Proposition 21) that C must already be bounded in C"(R, i.e. lplsk
sup ID'f1ix)l "Elin
I M < oo
independent of f E B and 1 E N.To sllow that also supgEc q,;,,(g) 1 E N ,/3 with 1/31 = b and x , y E Ifirlc!d orr (111;)-spaces. We prove that t l ~ c yarc of uniformly b o ~ ~ n d ctype. d 'l'lrc spacc of all thcsc furrctiorls is a 1:rdchct space with its natural topology. Sorl~rconseqllcnccs and related results are obtained.
In t,his pa.per we plwvc t h a t tlre classical rcsult of Grot.hcndictck [ l o ] \vhich s t a t e s t h a t on a (1)I:)-space any linear mapping whose restriction t o eaclr boundcd subset is (uniformly) cor~tinuorlsis necessarily continuol~srerriairls valid for G-l~olomorphicmappings. From this start,ing point we show t h a t t h e spacc 3-1 b ( E ) of all holornorphic functior~sdefined on a (DF)-space I:' which a r c bour~tlctlO I I ljo~rr~tletl scts entlowcd wit11 t h e topology of t h e uniform convcrgerrcr on bor~ndctlset.s is a Frticlrct spacc.
We also prove (,hat every f E 'H b(l:')(I: being a. (Dl7)-space) is, as in t h e trivial case of normed spaces, of urliforrnly boundetl type, tl1a.t is, tlrcrc is a convex balanced 0-r~cighbourhoodV in E such t h a t J E
3-1 b(lF)-spaces:
A locally convex spa.ce E is said to be a (DF)-space if a ) it has a fundamental sequence of boundcd sets arid 6) i f every subset of E t h a t is t h e inter~ect~ion of countably many closed convex balar~ccd 0-ncighbourhoods a n d t h a t absorbs cvcry bountled set, is a Oneighborlrhood. We refer t o [13] for t h e general theory of locally convcx spaces ant1 t o [7] for t h e gcrlcral theory of infinite dimensional holornorphy. If E and il
I: arc locally convcx spaces,
is a subset of E and 3 is a family of I"-valued mappings defined on A , we will call
(in a natural way) 3 equiuniJor.mly continz~or~s on A if for ca.ch 0-r~eighhourhoodW in F , there exists a 0-r~eighhourhoodV i r ~E such that if x , y E A and x J(x) - J(y) E 1.11 for cvcry
f E F. As ~ ~ s u aifl , f
:
A
C
E
+
-
y
E V then
F , we will denote
II J [In:= SIIPIII J ( x ) I1 :
x E A). We will tlenot,~by 3-1 b ( E ;F ) t,t~espace of a.11 holornorl,hic rnappir~gsfrom E into
wliich are boundcd on t h e bounded subsets of 15, enclo\ved wit11 t h c topology
~b
F
of t h e
uniform cor~vcrgcnccon bounded subsets of E. l'ropcrties a.hout these spaccs when E is a Banach spacc have bccn st,udicd in [I], (31, [9], [ l I ] and [12]. We bcgin with an auxiliary result.
1.Lemma.- Let L: be a locally convex space an$ It.! C, D , U and 1.V be convex balanced subsets of E such that 21.V
c I f . Then
I'roof
U ) n (.! +ILV) t l ~ c ntherc a r c c E C', d E D,u E U and + to. Since x = ( x - 211) + w ar~tlx - w = d E D, wc only need t o chcck that x - w E C + I f , hut this is obvious berarlse Let x belong t o ( C
tu E W such t h a t x = c
+ +
311
and x = d
137
Holomorphic mappings of bounded type on (DF)-spaces
2.Proposition.- Let E be a (111~)-spaceand let
sequence of bounded convex balanced .sets in E:. If
be an increasing Junda~nental
t"
is a Uanach space and
F
is a family
of mappings defined on E with values i r ~I: such thnf.for each bounded subset R of I;: the family { f i n : f E F} is equiuniforrnly continuous, then there is n 0-neighbourhood Z in E such that ill c Z and
In particular
F
C 3-1
* ( E ;I:), if all mappings in
F
are C-holornorphic.
Proof: By assumption { f i A n : f E
F} is
a r ~cql~i~~niforrnly c o n t i r ~ t ~ o ufamily s on A,, so
t l ~ e r eexists a convex balanced 0 - n c i g t ~ b o ~ ~ r h oVnP1 o d in E s11ch t h a t i f x , y E A, and
x - y E 1/,,-1 the11 11 f ( x ) - J ( y ) (1 < 11%" for C V,,ln = 1 , 2 , .. . suppose l.i+ We dcfine R1 := A l and 13, := (Eln-]
72
= 1 , 2 , .. and f E .F
+ Vn-l) nA,
for
11
2 2.
. Obviously we can is an increasing
sequence of subsets of E. Since evcry f is uniformly continuous on t h e convex balanced set A, we have that j is hol~rldctlon A,, for each n . First of all, we prove that
Indeed, glverl x E B, there exist bn-l E lj,-l and then x
= v,-1
- /),-I
E V,-,,
z
E Vn-l slrch that x = hnPl +v,-l E A, ar~tlh,,-1 E B,-I c An-] C A,. Thus 11,-~
hence
II
f ( x ) II
II
f(bn-I)
II + I1 f ( x ) - J ( h z - I ) II
5 II f
IIH,-~
1
+ -.2"
Now we put
I t is clear fronl the dcfil~itior~ of (111:)-spate that Z is a 0 r ~ c i g l ~ b o r ~ ~ li l~l o15.o dWe claim that Z
c UF=O=, B, for evcry p
E N.
Let 11s stlow by indr~ctiorlthat
P. Galindo, D. Garcia, M. Maestre
138
If12=1, thenAl
+
= B1
+
$Vl.
Given n >_ 2, if we suppose t,hat (2) is true for n
(
I
+
-
1, then we have
1 Vn-I) n (An 2
-
Now, if we apply Lemma. 1 by taking C =
+ 51 Vn)
U = Vn-l, D
= An and
W=
Vn, we
have
Therefore (2) is true. Now the claim follows frolr~
For each z E Z there exists
nt
E N such t h a t z E B,, hcncc according to ( I ) ,
and applying a trivial inductiorl
11 f
lz
5I /
la,
+
for p = 1 , 2 , . . . and f E 3, 2p
and the cor~cl~lsion follows. This proposition cxt,ends t h e classical result of Grothcndieck nler~t.ioneda t the very beginning in t h e following way: "IIf is a mapping frorr~a (DF)-space E into a Ranach space I: whose restric:t,ion 1.0 each bountled subset of E is uniformly continuous then J is continuous on E." (Giver1
t
> 0, choosc 171 E N snc11 f.hat 112"
0 such that B C pT6u(U,)J where F A denotes the closed, convex, balanced hull of A. (b) For each complete locally convex space F , the mapping
is a topological isomorphism, when L ( G w ( U ) ; F ) is endowed with the topology of uniform convergence o n the bounded subsets of G w ( U ) .
Proof. Since the evaluation mapping
is a topological isomorpliis~n,we can prove (a) by imitating the proof of [16, Lemma 4.21. And (b) is clearly a direct consequence of (a). 2.3. PROPOSITION. Let E and F be locally convex spaces, with F complete. Let U be an open subset of E , let U = (U,,) be a countable, open cover of U , and let ( f , ) C 'Hw(U; F ) . The.n the set U; fi(U,) is bounded i n F for every n E lli if and only if the family (T,,) c L ( G m ( U ) ; F ) is equicontinuou~.
The proof of this proposition is similar to t,he proof of [16, Proposition 2.51, and is therefore omitted. 2.4. PROPOSITION. Let U be an open subset of a Banach space E , and let U = (U,) be a countable, open cover of U , with each U, bounded. Then E is topologically isomorphic to a complemented subspace of G m ( U ) .
-
Proof. Under the hypotheses of the proposition, the inclusion mapping U E belongs to 'FIw(U; E). By Theorein 2.1 tllere exists T E L ( G m ( U ) ; E ) such that T(6,) = x for every x E U . Then the proof of [15, Proposition 2.31 or [16, Proposition 2.61 applies.
Linearization of holomorphic mappings of bounded type
153
Let U be an open subset of a locally convex space E, and let U = (U,) be a countable open cover of U . Let Bn denote the closed unit ball of 'Hm(Un),and let G m ( U n ) denote the Banach space
wit11 the induced norm. As pointed out in the proof of [15, Theorem 2.11, a theorem of Ng [17] guarantees that the evaluation mapping
is an isometric isomorphism. Clearly the FrCchet space 'Hm(U) can be represented as the projective limit of the Banach spaces 'Hm(Un)by incans of the restriction mappings
Consider the dual mappings
One can readily verify tliat R:,zLE G m ( U ) for every u E G m ( U n ) ,that is, the mapping Rn is continuous for the topologies a ( ' H m ( U ) , G m ( U ) ) and a('Hm(Un), G m ( U n ) ) .Thus if
denotes the restriction of 12; to G m ( U n ) ,then the mappings Rn and S n are dual to each other with respect to the dual pairs ( ' H W ( U ) ,G m ( U ) ) and ('Hm(Un),G m ( U n ) ) . We will see that, undcr suitable l~ypotlieses,the (DF)-space G m ( U ) can be represented as the inductive limit of the Bariach spaces GW(U,,) by means of the mappings Sn. We begin with the following proposition. 3.1. PROPOSITION. Let U be a balanced, open subset o J a Banach space E , and let U = (U,) be a sequence of balanced, bounded, open subsets of U such that U = UF='=,Un.
Then: (a) The mapping R n : ' H m ( U ) -+ H m ( U n ) has a a('Hm(Un), Gm(Un))-denserange. (b) The mapping Sn : Gm(IJn)--+ G m ( U ) is injective.
Proof. (a) Let f E 'Hm(U,,). Then, with the notation of 115, Proposition 5.21, the sequence of Cesbro Inearis (a,,,f') converges to f iri ('Hm(Un),7,). Since, by [15, Proposiwe conclude that ( a , f ) converges to f for tions 4.7 and 4.91, ('Hm(lJn),7,)' = GX(IJn), the topology a('Hm(U,), G m ( U n ) ) . Since each a,, f is a polynomial, and since each Uk is bounded, wc co~icludethat a , f E ' H m ( U )for every m E N ,thus proving (a).
Since (b) is a direct consequence of (a), the proof of the proposition is complete. Thus, under the hypotheses of Proposition 3.1, we may regard each Gw(Un) as a vector subspace of Gw(U).
3.2. THEOREM. Let U be a balanced, open subset of a Banach space E , and let U = ((in) be a sequence of balanced, bounded, open subsets of U such that U = Ur=lIIIn and pnUn c Un+l, with pn > 1 , for every n E N . Then Gm(U) = ind Gw(Un), and this inductive limit is boundedly retractive.
The key to the proof of Theorem 3.2 is the following le~nrna. 3.3. LEMMA. Let U be a balanced, open subset of a Banach space E , and let U = (U,) be a sequence of balanced, bounded, open subsets of U such that U = UrZlUn and pnUn C Un+lJ with pn > 1 , for cvcry n E N . Then Gw(U) and Gm(Un+l) induce the same topology and the same uniforin structure on the closed unit ball of Gw(Un).
Proof. Let Ak and Bk denote the closed unit balls of Gw(Uk)and 3-Im(Uk),respectively. By a lemma of Grothendieck [lo, p. 98, Lemma], to prove Lemma 3.3 it suffices to show that for each e > 0 there exists a = ( a k ) ,with a ~>, 0, such that
Let f E B,,+l. Since pnUn C U,,+l, tlie Cauchy integral formulas yield the inequalities
2 N sufficiently large. On the other hand, since Uk C ck Un+l, and this is less than ~ / for with ck > 0, for every k E N ,we have that
for every k E N. Set a = ( a k ) , preceding inequalitics show that
a k
=
f ~ E l A c r for , every k
E
N. Then
and (3.1) follows directly from (3.2).
Proof of Theorem 3.2. Clearly Ur=lGw(Un)C Gm(U), and the inclusion mapping ind Gm(Un)r GM(U) is continuous.
the
Linearization of holomorphic mappings of bounded type
155
O n the other hand, by Tlicorem 2.1 and by [15, Theorem 2.11, we have the canonical topological isomorphisn~s Gw(U)I, = R w ( U ) = proj 'Hm(Un) = proj Gm(Un)'. Thus, by polarity, each bounded subset of Gw(U) is included in the closure in Gw(U) of a bounded subset of some Gm(Un). Since Gm(Un+I)is complete, it follows from Lemma 3.3 that and thus each bounded subset of G m ( U ) is included in some Gm(Un) and is bounded , the spaces Gw(U) and ind Gm(Un) induce the there. Thus Gw(U) = U ~ ! l G m ( U n ) and same topology on each bour~dedsubset of Gm(U). Since we already know from Theorem 2.1 that Gm(U) is a (DF)-space, a theorem of Grothendieck [8, p. 68, Thdorkme 31 guarantees that the identity mapping G m ( U ) +ind Gw(Un) is continuous. This completes the proof. An cxarnination of the proof of Lemma 3.3 yields the following result. 3.4. PROPOSITION. Let E a ~ l dF be Banach spaces, let U be a balanced, open subset of E , and let U = (U,) be a sequence of balanced, bounded, open subsets of U such that U = Ur=lUn and pnUn C Un+lJ w ~ t hp,, > 1 , for every n E N . T h e n 'Hm(U; F ) is a quasi-norinable Fre'chet space.
4. THE APPROXIMATION PROPERTY
To begin with we state the followi~lgconsequence of [15, Theorem 5.41. 4.1. THEOREM. Let U be a balatlced, bounded, open subset of a Banach space E . Then Gw(U) has the approximation plape~.tyif and only if E has the approximation property. With the aid of Theorcm 3.2 we can easily extend this result to Gw(U). 4.2. THEOREM. Let U be a balanceil, open subset of a Banach space E , and let U = (U,) be a sequence of balanced, b o ~ ~ n d e dopen , subsets of U such that U = Ur=lUn and pnUn C Un+lJ with p,, > 1, for every n E N . T h e n Gm(U) has the approximation property if and only if E has the approximation property.
Proof. By Proposition 2.4, E is topologically isomorphic to a complemented subspace of Gw(U). IIence E has the approximation property if so does Gw(U). If, conversely, E has tlie approximation property, then each Gm(Un) has the approximation property, by Tlieorcxn 4.1. By Theorem 3.2 we may apply a result of Bierstedt and Mcise [5, Satz 1.21 to conclude that Gw(U) = ind Gw(Un) has the approximation
property. 5. HOLOMORPHIC FUNCTIONS O N QUOTIENT SPACES In 115, Corollary 4.121 we gave an explicit description of Gm(U). By examining the proofs of [15, Lemma 4.6 and Theorems 4.5 and 4.111, we can sharpen the conclusion of [15, Corollary 4.121 as follows. 5.1. THEOREM. Let U be an open subset o j a Banach space E . Then G m ( U )consists o j all linear functionals u E Gm(U)' of the form
with ( a j )E 1' and ( x j )c U. Moreover,
where the infimum is taken over all such representations of u . 5.2. THEOREM. Let E and F be Banach spaces, and let T E L ( E ; F ) be surjective. Let U be an open subset of E , aud let 1f = a ( U ) . Let S : GW(U)4 G m ( V ) be the unique continuous linear mapping such that the following diagram is commutative:
Then S maps the ball { u E GW(U): 1 1 1 ~ 1 1 < 1) onto the ball { v E G m ( V ): [lull < 1 ) .
P r o o f . Since Sv o a E 'Hm(U; G m ( U ) ) ,[15, Theorem 2.11 guarantees that llSll = 116~o rill = 1. IIence IISUII < 1 for evcry u E Gm(U) with I I P L J I < I . If, co~iversely, v E G m ( V ) ,with llvll < 1, tllcn by Tllcorem 5.1 we can find (y,) c V and (a,) E l', with C z , (a,I < 1, such that 00
v=
CCiJsyl. 3=1
Write
yj
= a ( x j ) ,with x, E U , for every j E
If we define
N.
Linearization of holomo~phicmappings of bounded type
then u E Gw(U), llull < 1 and
5.3. THEOREM. Let E and F be Banach spaces, and let .rr E L ( E ; F ) be surjective. Let U be a balanced, open subset of E , and let U = (U,) be a sequence of balanced, bounded, open subsets of U such that U = UF=lUn and pnUn c Un+l, with p, > 1 , for every n E N . Let V = x ( U ) , and let V = (V,), where V, = .rr(U,) for every n E N . Let S : Gw(U) -+ G w ( V ) be the unique continuous linear mapping such that the following diagram is comnautative:
u 2 , v
1. 1 6v Gw(U) --% Gm(V)
611
Then S is surjective and open. Each bounded subset of G w ( V ) i s the image under S of some bounded subset of Gw(U).
Proof. For each n E N we have a commutative diagram
Let B be a bounded subset of Gm(V). 13y Theorem 3.2, B is included in some Gw(Vn) and is bounded there. By Theorem 5.2, there is a boundcd sct A in Gw(Un)such that B = S n ( A ) . Thus B = S ( A ) ,with A bounded in Gw(U). In particular we have shown that S is surjective, and therefore open, by an open mapping theorem due to Grothendieck (see [9, Introduction, p. 17, Thd.orL:me B ] )or [lo,p. 200, Thdor&rne21).
6. HOLOMORPI-IIC FUNCTIONS ON PRODUCT SPACES
6.1. THEOREM. Let E and F be Banach spaces, and let U and V be open sets in E and F , rcsycctively. Then: (a) 'Hw(U x V ) is isometrically tsonlorphic to 'Hw(U;' H w ( V ) ) . (b) Gw(U x V ) is isorneti~ic(~11y isomorphic fo Gw (U)&Gm ( V ) .
Proof. ( a ) Clearly the mapping
defined by j ( z ) ( t ~=) f ( z , y ) for all z E U ant1 y E V , is an isometric isomorpl~ism. ( b ) O n one hand the mapping
is holomorphic and 116, @ 6,ll = 116,11 116, 1 1 = 1 for a11 x E U and y E V . By the universal property of G w ( U x V ) (see [15, Theorem 2.1]),there is a continuous linear mapping
with IlSll = 1, and such that SS[,,,) = 6, @ 6, for all x E U and y E V. On the other hand the mapping
( u , v ) E G w ( U ) x G w ( V )+ u x v E Gw(U x V ) defined by ( u x v ) (f ) = u ( v o fl) for every u E Gw ( U ) , v E G w ( V ) and f E 'Hw(U x V ) , is bilinear and Ilu x vll 5 IIuII llull for a11 u E G w ( U ) and v E G w ( V ) . By the universal property of tensor products, there is a continuous linear mapping
T : C w ( U ) & , G w ( V ) + Gw(U x V ) , with IlTll 5 1, and such that T ( u @ v )= u x v for all u E G m ( U ) and v E G w ( V ) . Hence T o S(6(,, ,)) = S(,, ), and S o T ( 6 , @ 6,) = 6,@ 6, for all z E U and y E V. It follows that S is an isometric isomorphism, with inverse T . 6.2. T H E O R E M . Let E and F be Banach spaces, and let U and V be balanced, open
sets in E and F , respectively. Let U = (U,) be a sequence of balanced, bounded, open subsets of U such that U = Uz=lUn and pnUn c Un+l, with p, > 1 , for every n E fir. Likewise, let V = (V,) be a sequence of balanced, bounded, open subsets of V such that and unVn c V,+,, with u , > 1 , for every n E N . If U x V denotes the V = Ur=P=,Vn sequence (U, x V,), then: (a) 'FIw(U x V ) is topologically isomorphic to 'Hw(U; ' H w ( V ) ) . (b) Gw(U x V ) is topologically isontorphic to G"(U)&,Gw(V). Proof. (a) Clearly the ma.pping
defined by f l ( ~ ) ( = ~ ) f ( x , y) for all x E U and y E V , is a topological isomorphism. (b) A glance at the diagram
shows that the mapping
belongs to 'Hw(U x V; G w ( U ) & , G w ( V ) ) . By Theorem 2.1 there is a continuous linear mapping : cW(u x V )+ G ~ ( u ) & , G ~ ( v )
s
Linearization of holomorphic mappings of bounded type
159
such t h a t S6(,, ), = 6, 8 6, for all x E U and y E V. O n the other hand, since Gm(U) = ind Gm(Un) and Gm(V) = ind Gm(Vn), t h e bilinear mapping ( u , v) E Gm(U) x Gm(V) -,u x v E G m ( U x V) defined by (ux v)(f) = u(v o f")for every u E Gm(U), v E Gm(V) and f E 'Hm(U x V), is separatcly continuous. By a theorem of Grothendieck on (DF)-spaces (see [8, p. 66, Corollaire] or [lo, p. 226, Corollaire I]), this bilinear mapping is actually continuous. From now on the proof proceeds exactly as t h e proof of Theorem 6.1.
7. HOLOMORPHIC MAPPINGS OF BOUNDED TYPE Let U be an open subset of a Banach space E. If U # E and x E U, then du(x) denotes tlie distance fro111 z to E \ U . Since IdU(x) - du(y)l 5 llx - yll for a11 x , y E U, we see t h a t du is a continuolis function on U. If U = E , then for convenience we define du(x) = co for every x E U. A set A c U is said to b e U-bounded if A is bounded and infZEa d u ( x ) > 0. If F is a locally convex space, then 'Hb(U; F ) denotes the locally convex space 'Hb(U; F ) = { f E 'H(U; F) : f ( A ) is bounded in F for each U-bounded set A ) , equipped with the topology of uniform convergence on all U-bounded sets. If F = C, then we write 'Hb(U) instead of 'Hb(U; 6').T h e rnernbers of 'Hb(U; F ) are called holomorphic mappings of bounded type. It is clear that Wb(U; F) = 'Hm(ZA; F) if U = (U,) is any fundamental sequence of open U-bounded sets. 111 t h e next proposition we give a fundamental sequence U = (U,) of open U-bounded sets which satisfics the hypotheses of all the results in the preceding sections.
7.1. PROPOSITION. Let U be a n open subset of a Banach space El and let (U,) be defined by l ~ n = { x ~ U : ~ ~and x ~ d~u (<x )~> 2z- " 1 . Then: tal oJ open, U-bounded sets. (a) (U,) is a f 1 ~ 7 ~ d a n ~ e nsequence (b) If U is balanced, then each U,, is balanced as well and, furthermore, there exists (pn), with pn > 1, such that pnUn C U,+, for every 72 E N. T h e proof of this proposition is straightforward, and is left t o t h e reader. For each
nE
N we may take pn = 1 + 1/122".
Theorem 2.1 yields Galindo e t al. [7].
ill
particular the following result, which improves a result of
7.2. PROPOSITION. Let U be an open subset of a Banach space E . Then there are a complete, barrelled (DF)-space G b ( U ) and a mapping E Ftb(U; G b ( U ) ) with the following universal property: For each complete locally convex space F and each mapping f E 'Hb(U; F ) , there is a continuous linear mapping T, E L ( G b ( U ) ; F ) such that TI o &iU= f . This property characterizes G b ( U )uniquely up to a topological isomorphism.
Theorem 3.2 yields in particular the following result, which improves a result of Galindo et a1 [7]. 7.3. PROPOSITION. Let U be a balanced, open subset of a Banach space E , and let (U,) be a fundainental sequence of open U-bounded sets such that each U, is balanced and pnUn c Un+l, with p, > 1 , for every i z E IN. Then G b ( U )= ind G W ( U n ) and this inductive limit of boundedly retlnctive. It is clear too that Proposition 3.4 yielcls in particular a result of Ansemil and Ponte [2] and Isidro [12]. Theorem 4.2 yields in particular the following result.
7.4. PROPOSITION. Let U be a balanced, open subset of a Banach space E . Thcn G b ( U )has the approximation property if and only if E has the approximation property. Theorem 5.3 yields in particular the following result 7.5. PROPOSITION. Let E and F be Banach spaces, and let R E L ( E ; F ) be surjective. Let V be a balanced, open subset of F , let U = R - ' ( V ) ,and let S : G b ( U ) -+ G b ( V ) be the unique continuous linear rnaypil~gsucli that the following diagram is commutative: U
6,
A
V
3
16v Gb(V)
1 Gb(U)
Then S is surjective and open. Each bounded subset of G b ( V ) is the image under S of some bounded subset of G b ( U ) .
Proof. Let (U,) be a fundamental sequence of open U-bounded sets such that each U, is balanced and p,U, c Un+', with p, > 1, for every i z E N. Let V, = x ( U n ) for every n E A'. Since U = a - ' ( V ) , it follows that each V-bounded set is the image under a of some U-bounded set. Whence it follows that (V,) is a fundamental sequence of open V-bounded sets. Thus Theorem 5.3 applics and yields the desired conclusion.
Remark. There are Banach spaces E and F , there is a surjective mapping L ( E ; F ) , and there is a convex, balanced, open set U in E such that, if we set V = r ( U ) , then the canonical mapping S E L ( G b ( U ) ; G b ( V ) )is not surjective. Indeed, if S were surjective, then S would be open, by an open mapping theorem due 7.6. R
E
Linearization of holomorphic mappings of bounded type
161
to Grothendieck (see [9, Introduction, p. 17, ThCorkme B] or [lo, p. 200, Thdorkme 21). Then, by a theorem of Grothendieck on (DF)-spaces (see [8, p. 76, Proposition 51 or [lo, p. 228, Proposition 4]), each bounded subset of Gb(V)would be included in the closure of the image under S of some bounded subset of Gb(U). Then the mapping g E 7fb(V) + g o n E 'Hb(U) would be an embedding, that is, a topological isomorphism between 'Flb(V) and a subspace of 7fb(U). But this is 11ot always true, as a cour~terexampleof Ansemil et al. [I] shows. Finally Theorem 6.2 yields in particular tlie followi~igresult. 7.7. P R O P O S I T I O N . Let E and F bc Banach spaces, and let U and V be balanced, open sets in E and F , respectively. T h e i ~ : (a) 'Ftb(U x V) is topologzcally isorr~orphicto 'Ftb(CI;7fb(V)). (b) Gb(U x V) is topologically isomorphic to Gb(U)6, Gb(V).
REFERENCES
[I] J. M. Ansemil, R . M. A r o n a n d S. P o n t e , Embeddings of spaces of holomorphic junctions of bounded type. Preprint, 1990. [2] J. M . Ansenlil a n d S. P o n t e , A n example of a quasi-normable Fre'chet function space which is not a Schwai.tz space. In: Furictional Analysis, Holomorphy and Approximation Theory, edited by S. hfacliado, pp. 1-8. Lecture Notes in Mathematics, vol. $43. Springer, Berlin, 1SS1. [3] K. D. B i e r s t e d t , A n introductioil to locally convex inductive limits. In: Functional
Analysis and its Applications, editcd by 11. IIogbe-Nlend, pp. 35-133. World Scientific, Singapore, 198s. [4] K. D. B i e r s t e d t a n d J. B o n e t , Biduulity in fie'chet and (LB)-spaces. In: Progress in Functional Analysis, edited by J. Bonet et al. North-Holland Mathematics Studies. North-Hollantl, Amstertlarn, to appear. [5] K . D. B i e r s t e d t a n d R . Meise, Bemerkungen uber die Approximationseigenschajt lokalkonvexer Fur~ktionenrauri~e.hlatli. Ann. 209 (1974), 99-107.
[6] S. D i n e e n , Con~plcxA ~ ~ u l y sin i s Locally Convex Spaces. North-Holland Mathematics Studies, vol. 57. North-IIolland, A~nsterdam,1981.
[7] P. Galindo, D . G a r c i a a n d M. M a e s t r e , IIolon~orphicmappings of bounded type. J. hlath. Anal. Appl., to appear.
162
J. Mujica
[8] A. G r o t l ~ e n d i e c k ,S u r les espaces ( F ) et ( D F ) . S u m m a Brasil. Math. 3 (1954), 57-123. [9] A. G r o t h e n d i e c k , Produits tensoriels topologiques et espaces nucle'aires. Amer. Math. Soc. 16 (1955).
Mem.
[lo] A. G r o t h e n d i e c k , Espaces Vectoriels Topologiques, 33 ediq8o. Universidade de SiLo Paulo, 1964. [ l l ] J. H o r v a t h , Topological Vector Spaces and Distributions, vol. I. Addison-Wesley, Reading, Massachusetts, 1966. [12] J. M. I s i d r o , On the distinguished character of the junction spaces of holomorphic mappings of bounded type. J. Funct. Anal. 38 (1980), 139-145. [13] P. M a z e t , Analytic Sets in Locally Convex Spaces. North-Holland Mathematics Studies, vol. 89. North-IIolland, Amsterdam, 1984.
North-Holland Mathematics [14] J . M u j i c a , Complex Analysis in Banach Spaces. Studies, vol. 120. North-IIolland, Amsterdam, 1986. [15] J . M u j i c a , Lineal.ization of bouilderl holomolphic mappings on Banach spaces. Trans. Amer. Math. Soc. 324 (1991), 867-887. [16] J. M u j i c a a n d L. N a c h b i n , Linearization of holomorphic mappings on locally convex spaces. J. hilath. Pures Appl., t o appear. [17] K . F. N g , On a theorent of Dixinier. Math. Scand. 29 (1971), 279-280. [18] R. A. R y a n , Applications of topological tensor products to infi7~ite dimensional holomorphy. Ph. D. tliesis, Trinity College Dublin, 1980. [I91 M. S c h o t t e n l o l ~ e rE-products , a12d coi~tinuationof analytic mappings. In : Analyse Fonctionnelle e t Applications, editd par L. Nachbin, pp. 261-270. Hermann, Paris, 1975.
Progress in Functional Analysis K.D. Bierstedt, J. Bonet, J. Howath & M . Maestre (Eds.) O 1992 Elsevier Science Publishers B.V. All rights reserved.
Spaces of Holomorphic Functiolis and Germs on Quotielits J. M. Ansemila-', R. M. Aronb and S. Pontca3l aDcpartamento de Anilisis Maternbtico, Facultad de Matembticas. Universidad Complutense, 28040-Madrid. Spain *Departnient of Mathematical Sciences Kent State University, Kent, Ohio-44242. USA
Dedicated t o Prof. M.Valdi,uia o n the occasion of his 60th birthday
Abstract Let E be a con~plcxlocally co~lvcxspacc, F a closed subspace of E and lct 11 bc thc canonical quotient mapping from E onto E / F . If U is an open subset of E and Ii is a compact subset of E , II ~nduccsa linear injective mapping H* fiorn X [ n ( U ) ] (resp. 3-I[n(I I 0 a~itlrz,
E N sucll that
for every f E X ( U ) . These topologics have 11ocii sttltlic!tl 1)y ii~iriiyatltl~ors.See [8]as a general reference for properties of
T,,
r,,
and
T*.
111
ultrabornological space [S, Prol). 2.3SI.
l)i~rtict~l:~r ~iici~t,ioii tlmt ( X ( 1 T ) ,T
~ is )
an
165
Spaces of holomorphic functions and germs on quotients
Given a co~lipactsubset I i of E , we will denote by X ( 1 i ) the vector space of all holomosphic germs on I~)~ISC
Es(.4) = nS1'X. \Vc slldl c d l a continuous lincar map T on a TVS Y , admissible if the map A + E r ( A )
is i~ s t d ~ l map e from F(C) into S(,Y). Tlle restrictive part of this tlcfinition is thi~ttlic
si1bs1mws ET(A) ~llustbc closed. The map T is said to be semiadmissible if thcrc is linear co~~tim~ous injection i illto mlother TVS -Ynrtlld an admissible map Tnon Xnfor which T7 = iT. Finally, we shdl call T decompo~ableif thcrc is a t~inpET : 3 ( C ) + S ( S ) , svl~osc rrulgc consists of T-i11wwitmt subspaccs, micl which is stable, dccoml>osal,leand si~tisfit-s the condition that a ( T I ET(F)) G F for evcry F E F(C). Examples. a ) Super-dccoml>osableoperators with no non-trivial divisiblc su1)sl)iicc.s on BCmachspnccs are culmisil>le[L N 11. b) Multipliers on commutntive semi-simple Banrtcl~algcl>rrlsarc seminclmis..il,lc [L N
21. This class of examplcs will bc cxpl;tinccl a ~ l dcxtcnclccl lnt'cr. S~tppwet11a.t S ~ l l dY tlrc t.o1>ologicd\ ~ c t spaces ~ r aucl that T i1.11~1 S iwe co~~tili~io~ls liucar maps on X m~clon Y, respectively. Supl>ose 8 : X
+
2' is linear, 1)1it not
neccswuily continuous. We sllrdl say tlictt 8 is au intcrtwiner (of S and T ) if SO = HT. More general concepts of i~ltertwi~~ing may be ancl havc been licnt.ionby 8(x) on Y. L L : h ~ 6. h ~ Supl>osc ~ thitt .Y and I' arc toj)ological vcctorspnccs, that; T is dcco~~~j~o.s;~blo on S,nnd tl~atS is aclinissible oil Y. Sapj>ose8 : ,Y + Y is ~ J i~ltertrviner I of S ;tl~cl
T. T l ~ c 8ET(A) i~ g E s ( A ) for c~fcr.y-4 E 3 ( C ) a ~ 11ei1ce ~ d 8 IJKS
Es)-loci~Jix;~blc
tliscoi~tiill~ities.
PIIOOF: Lct A C C bc givcn m ~ dtake X $ -4: siiicc u ( T I ET(.4))
-4, ( T - X)Ey-(=l) =
Eye(A), hencc ( S - X)BET(.4) = 8(T - X)ET(A) = OET(-4),fro111 which it follosvx, 1>y miuin~ctlityof E s ( A ) , tliat 8ET(.4) G Es(A). Sitlcc Es(.4) is closcd, tslirsccor~dc'li\il~~ follo\i~s.
185
Automatic mntinuity of i n t e ~ ' n e r s
T l r ~ o n ~ 7. h l Let X be an (F)-spacc and SIJ~CC.
Sup11clye T is r!ccomj~twnble on
B : S + 1- is
RII
'I be
ri
countably bo~inrledlygeneratrrl (F)-
-v r u ~ dthat
S is ~ d l l l i s ~ i b lon c Y. SII])~IOH(.'
intcl.t~virlr~. of S nntl T. Then tl1el.c is a IIOII-tl.iw"1polynomial 11 s11rh
that p(S)B is conti~~rro~ls. IVr lrnny
R ~ S I I I I I ~that
all roots of 11 FIJT cliticsl eigcnvall1c.s of
PROOF: By lemliia G, lemma 5 nl~plics~llldlicnrr thrrr i~ a fiiiitc s ~ Ft
6 ( B ) C E s ( F ) . Apply the stability lcmnia to tlie scqricnce T
-
C C for \vliirli
A, whcre X E F . This
yic-ltls a l~olyi~oi~lial p for wliicli 6 ( 8 p ( T ) ) = G(Bj,(T)(T- A)) for cvciy X E F . Siilcr:
B iiittrrtwillc.~S ailtl T this
~ I I C I I ~tliiit S
( S - X)p(S)G(B)-
X E F, I I ~propositioll 1. By b4ittiig-L(.ffler, therc is
t~
= ( p ( S ) 6 ( 8 ) ) - for c.vr*iv
tl(=nscs u b s l ~ ~ 1. c1c' ~C p ( S ) G ( 8 ) -
for wliich ( S - X)Tt' = 1.1.' for c3vrry X E F . Tliis illc;tlis tliat W C E s ( C
\ F), I)y tllc:
defiilition of tlic algebraic spectrrrl srll~spnct.~.Sillrc IY C 6 ( 8 ) C: E s ( F ) , wc (~btiuil tliat 1V C E s ( F ) fl Es(C
\ F) = E s ( 0 ) ,licncc
tlint W = {0), bc-cause S is LISSI~IC(I
ntlniissill(*.Cuil~rc~rirlltly, l)(S)G(B)= {O}. All i~ijectivcfactors S - X may b r rciilovc.tl frc~inp ( S ) ; \vlint is l ~ f will t still t~illli1iilat.cG(B). Tlius wc h a w o11txiilc:d u polynolllii~l 11.a11 of wllosr roots arc. t*igc.il\~alric~s of S , fur wliic~llp(S)B is colltiiiuuus. hIort.u,vc*r,sill(.(-
~J(S)B = tYl,(T), all ~ioil-criti(~al rigc-~l\'~~lrirs iiiay also I)(.
S C I I I O V ~ (fro111 ~ tliv
c ~ f1). 111 fii(.t. if X is lloii-rritirld, tllrii d i i i ~ ( S / ( T- X)-Y) < oc
11i1(1siirr(n
f;~ct,oi.iziltioLI
-1-is a11 ( F ) -
sl)iK(', fillit(: codiiiicl~sionof ( T - X ) S is cnor~gllto rlisurc thilt ( T - X ) S is rlosctl; nlorcover, T
-
X is nil opcli iilap of X oilto ( T - A S ) . T ~ ~ I ifI Hp (, T ) = i . ( T ) ( T- A) tlirii
flp(T)is c:ontinuor~soli S if cxsr if
i111tl
~ l l t unly l
if 81.(T) is coiltill~iousoli (T - A)-\- nntl this is tlic-
oiily if Bi.(T) is coiltiilru~ris(111
T I I I : O I I 8E ~(cf. I [L N 21). If T ib T is a rlrrc~ticl~t of
i i
rl
.V. Tllris oiily tllc rriticlrl rigcnviducs i ~ r (Ir-ft. *
colltin~lollsl i i l l - ~o~j .~ r ~ . a t o011l . nn (F)-sprlt.c- r l l l t l
t l r ~ r n ~ ~ ~ ~ ol~cl.r~tol. ~ o s i ~ l ~onl cnzl (F)-spr~cc-,lf S ih
;I
r.oz1til111011.s Ii~lc'ill.
c q ~ c ~ . i ~or1 t oan ~ . (F)-.q~acenl~tlS ih s e ~ l l i n c L ~ ~ i ~ rrritl~ ~ i b~.cspc.c.t l(~ to i111nd1ni.a-il~lrol,c-l~;~tor on n corr~~tnhly I~o~lntlrrlly grnt.rrltcd (FJ-rl~rrrc.,tllrn evr1-y intel.trvi11cl.for tllc pail (S, T ) i.q r o l l t i l ~ r l o if ~ l~~ n c 011l.v l if (S.T ) 11n.s no r ~ i t i r a lc i g c n ~ d ~ ~ c s . P I I ~ O I ?Srl111x)w : tlllrt (S,T ) lit~s~ i oc.ritic.iil c.igc*llvnl~rc.s...\ssrr~lr~first that S is
i~cliiiih-
186
K.B. Laursen
sil~lcand t,llat E" is locally l , o ~ ~ i l t l ( Lct ~ l . q Ije a cluot,icnt iliap n~ltlT-IIC tloc:oi~~l,osal)I(~ sucli that qT- = Tq. Then S(Bq) = (Bq)T? By tllcorcin 7 therc: is a p o l y ~ ~ o i ~ ip, ial of whose root,s are critical eigc~nval~lcs of ( S ,T^), for \vllich p ( S ) 6 ( 0 q )= ( 0 ) . T111ls
ill1
1j(S)Bq = Bp(T)q is contin~ioi~s.Since q is an opcli m ; ~ p ,wc col~clutlct,lii~tB])(T)is co~itiiiuoi~s. B y coilstruction of 11, all its roots arc eigcilvalues of S . Lct X 11c o~ic,s11c.11 root,. If d i m ( X / ( T- X ) S )
< m and if p(T) = r ( T ) ( T- A) then, just
t,lleorein 7, Op(T) is coiitinuol~son X if
;lilt1
as in t,lle 1)roof of
only if Bv(T) is cont~iiluouson ( T - X ) S if
a ~ i donly if Or(T)is coilti~nlo~ls on X . T h i ~ swc may removc from thc l)olyllolllii~lp ( T ) irll factors T
-
X for wliicli X is not
;I
criticill eigciivill~leof ( S ,T ) . n-itl~oiltt~lteriiigt . l ~ c ,
co~iti~luit,y of its 1)rotluct wit11 0. Siilcc, ~ v c ,ilsslurie t,liirt ( S .T ) Ilas
110
criticill ('ig(~ii~.i~l-
ut>s t,liis lllc,ails rclllovi~igall linear fact,ors of p(T) aiitl co~liplet,c,st 1 1 ~1)roof wlic~~i S is atliilissil~lc.
il'(.sliall 1 1 s ~t,llis to 11i1ntllc~ t,lle case wlle~iS is se~~~ii~dii~issil)lc.. Lc,t i Ilc a coiltilnloi~s 1111car i~ljectiona.nd S- a correspoiidiiig admissil~le111ap on a locally 1,o111itlctlsl)i~'.(~ for wllich S'? = i s . Since SYO = iBT, we may al)l)ly wl1a.t l ~ a sI)c>cnpro\rc~lso f i ~ r , irlitl
ol~tniiln poly~ioiliialI), ill1 of wl~osoroots arc: criticill cigcilv;\luc.s of (.ST T ) , for
n.11icli iOp(T) = p(S310 = Lp(S)H is contiiiuous, hc~icefor wliic.1~p(S)H is contilnlol~s,
1))- 1xol)ositioii 2. Lct q ( S )1~ tllc l>rotl~~ct of all thc: factors S is
i111
cigenvalut: of S , i.c:. for which S
-
-
X of p ( S ) for nliicl~X
X is not i~ijec:t,ivc:.Tllc:n collt,irll~it,yof p(S)B is
c,cl~~i~i~lnierosIndustriales, Avda. Reiria Mercecles s/n, 41012 Sevilla, Spain
This paper is dedicated to Don M a n i ~ e lValdivia o n the occasion of hi.9 60th birthday.
Abstract Let (-Y, Z, / I ) be a finite measure space, A a complete, solid lattice of scalar functions defined 011 ,Y,and E a normed space. We study barrelledness properties of the space A ( E ) of strongly measurahlc, functions f : X -t E such that 11 f (.)I1 E A. 1991 Mathematics Subject Classification: 4FA08, 46E40, (46G10) Keywords: Barrelled spaces. Lattices of irltegrable functions. Bochner integral.
In what follows, (X, C, 1 1 ) stands for a rneaslire space, where p is a finite, positive, countably additive measure definc,d oil a u-algebra C of subsets of X. Let i l be a complete, solid, locally convex lattice of measurable scalar functions q5 : X
-t
IR (as usual,
we identify fu~ictioristhat are equal p-a.e.). The topology of A
can be tlvfined by a family of continuous seminor~nsQ such that whenever q5 i5 ii1 A and
4 is a ~lieaslirablefunction satisfying
for every q E Q. I f we denote by
Idl(.)I
5 I4( )I then $ E A ant1 q ( $ )
< q(4)
the characteristic function of a measu~ableset Y ,
tlieri the 5et of all continuous linear mappings
*
T 1 1 ~5econd and third named autliors acknowledge support from La Conscjcria dc
Educacitin y Ciencia de la Junta de Aridalucia.
L. Drewnowski, M. Florencio, P.J. Pa~il
192
Examples of A are tlie classical Lebesgue spaces Lp(l') (1
cy . f )
< pn(f
> 0 then
) Therefore PDO;Y,, . fn)
X y . f E rD so that
-i
0. Q.E.D.
Theorem 2. Let A be a complete, solid, order-continuous locally convex lattice of rneasurahle functions defined over a finite measure space space. If' the measure
Y ,
(x,C, p ) . Let E be a normed
is aton~lessand A is barrelled, then A(E) is barrelled.
Proof. By T h e o r ~ m1, A(E) is quasi-barrelled so that, according to the Fact above, we have to prove that if M C A(E)' is a(A(E)', A(E))-bounded and (f,),
is a locally null
sequence in A(E), then sup{l(u., f m ) I
: m E
IN, u E M ) < +m.
Sup~>ose,on the contrary, that this supremum is +m. We start by making a sliding-liump type inductive const,ruction: (Stc~p1) Call X o := X; on account of our assumption, we have: sup{l(u,Xx., . f,)] : m E IN, u E M } = +m
(1)
196
L. Drewnowski, M. Florencio, P.J. Pa~jl
and, therefore, wc can find
ul
E M and in(1) E IN such that,
Apply the fact that A is order-continuous: we can find 6 > 0 sl~chthat wllencver
p(C) < 5 then
m(1) such that
Proceeding as in step 1, we can find a set S 2 c X I , sucll that
and J(112,t k z
. frn(2))I
> 2r
whrrc EL := -Yl \X2 is disjoi~ltwith I;. 111this way, we arc. ahlr to find a scqucrlce ( u , , ), E Ail, a disjoi~ltsequence (I;, ),, t S
and a suh~eciuence(f,,,(,)),, of (f,),,, such that
197
Barrelled function spaces
By (ii) in t,he Lemma above, the sequence y,, := lyn . f,(,), n = I , ? , . . . is locally 11~11 in A ( E ) and, by ( i ) in the same Lemma, ( N ( g , ) ) is also locally null in A. Let C 1~ a disk in A such that yc(N(g,)) -+ 0. Then we c a i take a subsequence (gn(k))ksuch that
Ckp c ( N ( y , ( ~ ) ) )< fm.
Since C is a Banach disc, because A is complete, we have that
C kN(g,(k)) is convergent
in Ac, and consequently i11 A. to some f u ~ l c t i o I~/!l E A.
Now, consider the function g : X + E defined pointwise by g(.) :=
Ckg,,(k)(.).
This fiinct,ion g is well-defined becalise the functions g,(k) have disjoint supports Y;,(k), and it is strongly measurable. Moreover, for t E S we have
tlil~sN ( y ) =
1/~ E h and it follows that g E h ( E ) . By ( 3 ) and the contirlliity of each
Sincc, &I is u ( A ( E ) ' ,A(E))-l>ounded, the scalar measures defined by
form a set wise l->oundt,clset, i.e. sup{lmk(E7)I: I; = 1 , 2 , . . . } Tllercforc,.
WP
< +w,
for all Y E
C
may apply tlie Nikodym Bo~~rideclness Thcorcm [I,1.3.11 to deduce that
i11 co~ltratlictionwit11 tllc fact that
Q.E.D. Remarks. Thi5 r t w l t was provrd in [13]f o ~.I a Iiijtlle fi11lctio11space, ant1 lllidcr IiloIr restrictivr, c o n d ~ t ~ oor1 ~ ~tlic s Irieasurc, sj,;lcc. a ~ i t ltllr ~ l o r ~ i l cspace,, d llaliicly S loci~lly
198
L. Drewnowski, M. Florencio, P.J. P a ~ j l
compact and E' having tlie Radon- Nikodyin Property. Tlle proof there ~ n a d ca stroilg use of the form of A(E)'. It is a hit surprising that E is not reclui~edto be barrelled. An unexpected co115e q11ence of Theore~n2 is that if E is not barrelled and the underlying IncXasurespace is atomless, then one cannot localize E as a quotient of A ( E ) .
REFERENCES 1 J. Diestel and J.J. Uhl, Vector Measures. Matllematical Surveys, no. 15. Amcricnn Matliematical Society. Providence, 1977. 2 J . DirudonnC, Sur les espaces dc Iiothe, .T. Analyse Math. 1 (1951), 81-115. 3 L. Drewnowski, M. Florencio and P.J. Paul, The space of Pettis integral)le fiinctio~is is barrelled. Preprint (1990). 4 I. Halperin, Function Spaces, Proc. Int. Sy~nposiurrlon Linear Spaces. Hebrcw Univ., Jerusalenl (l960), 242-250. 5 H. Jarchow, Locally Convex Spaces. B.G. Teubner. Stuttgart, 1981. 6 J . Linderlstrauss and L. Tzafriri, Classical Banach Spaces 11. Sprir~ger-\'c>rlag.
Berlin, Heidelberg a.nd New York, 1979. 7 J . A . Lbpez Molina, The dual and bidual of an echelon Kothe space, Collcct. Math. 31 (1980), 159-191.
8 G.G. Lorentz and D.G. Wertheim, Representation of linear funct~ionalson Kothe spaces, Canad. J. Math 5 (1953), 568-575. 9 A. ~ a . c d o n L l dVector , valued KGthe function spaces 111, Illinois J . Math. 18 (1974), 136-146. 10 P. Pkrez Carreras and J . Bonet, Barrelled Locally Convex Spaces. North Hollarlct. Elsevier. Amsterdam, New York, Oxford and Tokyo, 1987. 11 N. Phuong-Cric, Generalized IiGthe function spaces I, Proc. Cambridge Phil. So(-. 65 (1069), 601-611.
12 Ii. Reiher, Weighted inductive and projective limits of normed In) := {(xk)k>n E n X : (11 X k + n - 1 1lx)k~niE L ) and L(A') := L((X)k>i). It was Moscatelli in 1980 (see [ 3 2 ] )who discovered that "shifting device" t o construct twisted Frichet spaccs (i.e. Fr6chct spaces without a continuous norm which a r e not isomorphic t o a product of I.'rdchet spaces each of which has a continuous norm). So, each Frdchet space of Moscatelli type depends on t h e two entries i) and ii) and f ---+ X ) . If every a = ( a k ) k E N E L is t h e limit of its finite sectior~s
we write it as E ( L , Y
(which can b e equivalently stated in t h c form L = p ) , the dual L' of L , provided with its dual norm, is again a normal Banach sequence space. T h e following characterization of distinguished Fr4chet spaces of hloscatclli type has been obtained by the authors and C. Ferndrldez (see [12, 14, 251).
Theorem 1 Let E = E(I,, Y a ) If 1, =
X ) be a fie'chet space of h.lo.scatelli type.
and 1,' = @, then E is alulays disfinguished.
6 ) If L = 3 but I,'
# 3 or if I, # 3, then E is distinguished
if and only i f f is surjective.
T h c typical example for an I, in a ) is G,,whcrcas the typical cases in h ) arc 1, and I, f --t X ) t hc surjcctivity of J is cqriivalcnt t o
respectively. Ry [12, 2.101 for an E = / 13). +X
I.' -f X ) ---+
as in ii) then
c G"
= E(/,,
Y"
2 X").
I I I G will irtt, bounded sets with closure,
from which one derives -via a tecl~nicalcor1ditior1- the surjectivity of f (see [14]), i.e. t h e quasinormability of 11. c) If L is an arbitrary I3anach scquerlcc space such that y is not dense in L , a block c o ~ ~ s t r u c t i odue n to C. FernLnctez (sec [25]) shows that E ( L , Y X ) contains f f E(1,, Y + X ) as a coniplemer~tedsubspace. Thus, i f E ( L , Y -+ X ) is distinguished, one obtains with b) t h e surjectivity o f f and one is done. T h e above characterization of disting~~ishedncss in the class of Moscatclli t y p e Frdchet spaces enabled the authors and C. Fernindrz t o give a ncgative answer t o t h e following question of A . Grothcndieck from 1954 [27]: Is the hidual of every distirrguished Frdchet spacr again distinguished?
X with Y , X Banach be given such t h a t f is linear, continuous .Y) is distinguished by a ) in the theorem, whereas G" = E(l,, Y" 2 X") is not distinguished, as f and hcnce f L L In fact, let f : I.'
---r
and has dense range, but is not opcn. Thcn G := R(co,Y
are not surjective. For a comparison, we would like to mention how sorne other properties of Frbchet spaces (see the chart) behave with respect to the formation of t h e bidual: qllasinorrnability, being a quojection and having t h e density cordition arc all stable under t h e formation
J. Bonet, S. Dierolf
206
of t h e bidual (as can b e shown rather straightforward in all the three cases), whereas t h e property of having a continuous norm is not (see [20, 371).
Kothe echelon spaces
4
In order t o construct a Kothe echclon spacc, one again starts with two entries: i) p E 0 U [ 1 , m ] , a n index set 1 and ii) a Kcthe matrix A =
nEN 0
< a(.n) < 3 -
(jE
J, n EN )
and forms &(A) := { ( ~ ~ )E, K~Tl : Vn E N (aln)xi)3ErE l p ( l ) ) (where lo(1) = c o ( l ) ) , which is a Fri.chet space with respect to obvious norms. T h e distinguishedncss of echelon spaces has also becn complctcly characterized. It is well-known t h a t a ) Xp(A) is distiriguished for p = 0 or 1
1 a subspace of El' = for which F n .Y,!,'= {o). It follows that lhe quotient map E" -+ .J-: i one-t,o-one on F , hence d 5 dk and, therefore, card(En) 5 2"k. Rut t,his is absurd, since card(E1') > 2"" for all n; hence no such an F can exist.
n,,
hecardinal numbers d, = dens (EN/kerp,) satisfy '2dn d,,, for suitable rn = m(n).
k
Proof T h p first part follows directly from Theorern 1 applied to F = fl,, .Yrl. For the secoiid part, note that, we can always write, in the case indicated,
226
here W,
G. Metafune, V.B. Moscatelli
c Y . :-,
and S, are chosen as in the proof of Theorem 1 (cf. 171) with Rn : .Yl i & the cmonical projection and & acting only on .Xn $ Wntl , having chosen extensions y y of I? which vanish on ,Yi.R o n ~this, the second assertio~lin the proposition easily follows.
$:;
$;:
Remark 6. Note that Propo~it~ion 3 yields the surprising fact that, e.g.,
x (12)(N)
is the dual of a non-trivial preqnojection, while 1' x (12)(N) is not. Now we shall investigate those prequojections which are related to quojections of Moscatelli type. Recall that, if (.Y,) is a sequence of Banach spaces, if Y, is a closed subspace of ,li,for every n and if L is a normal Banach sequence space, then the quojection of Moscatelli type Q ((& ), (X, /ITn);L) is the quojection obtained by forming the projective limit of the Banach spaces
~ .\;, with respect to the (surjective) maps which arise from the identitlies of t h spaces and .Xn/k;, and the quotient maps .;Y, --+ Xn/Yn. Dually, we define the (LB)-space of Moscatelli type LB((,Y,), (I.,); L) as the strict (LB)--spaceobtained a s the inductive limit of the Banach spaces
There is a natural duality between these two classes of spaces, since they correspond vue to the other by duality. For properties of q~iojectionsand (LB)-spaces of the above type we refer to [ l i ] , 1121, [4], [5], [14] and 1161. Here we just recall from [17] that Q ((,Yn),(.Xn/Yn); L) is twisted if and only if Yn is not complemented in X;, for infinitely many 11, the same being true of the space LB((X,,), (Y,); L), for which twisted means not isomorphic t,o a direct sun1 of Banach spaces. We also recall that there exist both twisted quojections and twisted (LB)--spaces of Moscatelli type wit,h non-twisted duals (this was already implicitely contained in [17]). On the other hand, we have the following lemma, where t,he Banach sequence space L is assumed to be reflexive. Lemma 3. If E is a quojection (reap., &B)-space) of Moscatelli type for ulhich E' is
twi'sted, then all iterated duals E(") of E are twisted.
Prequojections and their duals
227
I'rool. Let E = Q ( ( S, ) , ( .X,,/ ) ; I,):then E' = LB ((.Y:,),(1;); L') . Supposing E' tnist,ed, by what we said above we niust have an increasing sequence ( n k )of positive int.egt>r-sfor which l,':, is not col~lplenleriteclin ,YA, (and hence Y;,, is not complemented ~ ) hence, ; if E" were in "I-, , SO that E itself is t,wisted). Now E" = Q ((.A'{), ( . A ' ~ ~ / YI,), not t,wisted, 1;' = I"j0 would be compleme~it~ed in .Y: for all 11, 2 no for a ~uit~able no. Then l',PoOwould be complerilented in A':' for R 2 110 and hence so woilld Y : , since the latter is complemented in Y,OOO. But :1' c Sk, hence Y,O would be complemented in for all n 2 110 and we would get a ~ont~ractiction. Thus E" is twist,ed. The same argument applies if we start with an (LB)-space of Moscatelli type i1.i) thereby gett,ing that if M' is t,wist,ed, t,hen so is M". Now, going back to our quoject,ion E, apply the lat,ter conclusion to the (LB)--spaceE' t,o obtain that alw E"' must be twisted! etc. )In
A'!.
Now Theorem 1 may be considerably strengthened as follows
Theorem 2. Let E = Q ((.7(,),(?i,/\',); L) be a quojecatron of Moscatellr type torth Y, separable lor all n suficrentely large. There exrsts a (countably n o r n e d ) prequojectron F u~11hF' = El tj and only rf 1; ru non-quasi-refiexrve jor znfin~telym a n y n. Moreover, in xuch a case there are a countably norn~edprequo~ecttonC: 714th G' = $, Y,' and erart ..equenres
Proof. 'rhe first part is direct consequence of Theorem 1 For the second part we write, with Wn C,!:I Theorem 1,
c .Y::-, and chosen as in
the proof of
where S,, is defined as in (F;), with jn acting only on .7i,$ R'n+l. E'rorn (19) it is then easy to see that the prequojection
is a subspace of F and that F I G = L (Sn/Yn). This establishes (17), since G' = $, Y,' (recall t,he proof of Proposition 2).
228
G. Metafune, V.B. Moscatelli
Now define maps
as follows:
---
liere yl E yl for j < n, w E CK,, (xk) E L ( x k ) ,4,: Sn .Yn /Irn is t.he quot,ient map and i runs through the set of non-negat,ive integers. Note t,hat the series in the right,-
I /I
0 , there are only finitely many indices A such that the diameter of Ax is greater than E. If X is compact the property above does not dcpcnd on tlic choise of d E Ma(-Y). T h e o r e m 3. If X is compact and A is shrinking then, for every p E Mht (X, A), the multihat Y = Mht (X, A ) is compact in the metric d = rnht p and the topology of Y induced by mht p does not depmd on the choise of p E M a ( X ) . Protrf. Assume that (y,) is a sequence of elements of Y. R o m Tlicore~n1 and Convergcilce Criterion it follows that each YA = S c ( A x , o x ) and, therefore, each hat X U Ax is a compact subsct of Y . Hence, if onc of thc hats contains infinitcly Inmy elements y,,, then there is a co~iwrgentsubsequence of the sequcnce (y,). Otherwise, we let z, = y, if y,, E X and z, = ox if y, E Crown(X, .AA, ox). Then d(z,,, y,) + 0. Since (z,,), whidi is a sequence of elements of X I ~ d r n i t sa convergent subsequc~icc, the corrr-sponding subacquence of (y,) is also convergent. We have proved that Y is compact. Assrime that pl ,pz E P ( X ) and dl = mlltp,, d2 = mht pz. Clearly, dl d2 = mht (pl p z ) , whence, as we liave just shown, the metric space (Y, dl d z ) is compact. For j = 1,2, the identity map h, from (Y, dl d2) to (Y, d,) is continuous, m d since the two spaces are compmt, is a homeomorphis~n.Hcncc tlic identity inap fro111 (1: d l ) to This complctcs the proof of the theorein. (Y, dz), rrlual to h;' o hl , is a l~omcomorpliisir~.
+
+
+
+
A ) satisfying the wsunptionv of Thcorein 3, the set Mllt (S, A) For tlie pair (S, will be rrgardcd as (compact) topological space equipped with tlic topology dcsril~cdi l l the Theorem. C o r o l l a r y 3. HAYis a compitctuln mlrl A is a shrinking family in X , then tlle operator mht , restricted to P r ( X ) , is a rcgrrlar. cxtc~nriorfor the j);u'r (Mht ( X ,A ) , X ) . Thc following l e ~ n m awill be applied ill tlie next section: A such that Mht L e m m a 3. If X is a colnpactu~nwit11 a shrinking fax~~ily AREP , then X E AREP .
(.Y,A )
E
Proof. Assuinc that W is a compiu:t,uln containing X . Taking into account tlic rissumption of tlie Lemma and Corollary 3, and applying Lemma 2.3 to tlic cliains of inclusions
X
c 14' c Mht (IV,A ) ;
S
c Mht ( X ,A ) c Mht (W, A ) ,
254
C. Bessaga
we conclude that the pair (X, W ) adnlits a regular extendor.
*
*
*
If A = [(Ax,ox)lxEA, where each Ax = { a x , o x }is a two-point subset of X, tlien the multihat Mht (X, A) is called a harp, the set X is the frame of the harp, the set,s S c ( A x , o x )are the strings, cf. Example 1.1. If the harp is regarded as metric space equipped with the metric d = lnht p, tllcn each string is well-spanned, i.c., hon~eomorphicto the line segment of length equal to the distance between the ends of the string. Even in the case where all strings are fixed at the same ends, t,hey do not touch each other in the middle points (see the for~liula (mht2)). Such an harp cannot be isonietrically eml~ecldedinto any Euclidean space, and therefore cannot be played by h ~ l m a nbeings. It will be referred to as a n angelic haq,.
4. A R E P does not imply A N R property. The Swiss Cheese couilterexaillple Before giving the precise definltiorl of the Swiss Cheese ohse~vethat tlic~c,i\ on(, sort of the cheese, very rnucl~appreciated by topologists the Cantor set Consider the 11-cube In and assume that [(Bk)llENis a countable family of pail-\vise disjoint closed Euclidean balls contained in R n . The compactum C = In \ UnENint B L is referred to as SWISSCl~eese Let An = bd B L , ok E B n for I; E N. Consider the sh~inkingfanlily A = [(an,on)lk~~. Proposition 4. There is a homeomorphis~nfi.0111 the rr~ultihatMht ( C ,A) o11to the n-cube In which is the ide~itityon C .
Proof. The required ho~nconlorpllisnlis obtained by pasting together the homcomorphisms of Example 1.2 with aid of the following elementary l s n ~ m a : Lemma 4. Let X and Y be con~pacta,A = [(Ax,ox)]xGAa shrinking family in X. If g : X -+ Y is a bijective nap such that the restriction of g to each Ax a n d the restriction to A'\ UxEAint AX are c o r ~ t i ~ ~ u othen u s , g is a homeomorphisni of S onto .
Theorem 4. T l ~ cSwiss Cheese C is an absolute regular extendor without being an ANR space.
Proof. The assertion: C E AREP follows immediately from Lemma 3, P r o p o s ~ t i o ~ l 4 and the fact that In is all absolute retract (Tietze's theorem). Since C is compact,
Linear extending of metrics
255
there is a cluster point, say y, for thc fitl~lily[il,,] of splicrcs, and therrfore C is not locally contractible to the point y. Hcncc C is not an A N R space (cf. Borsuk [2]). To establish tlie particular case of tlie Theorem, where C is the Cantor set it is enough to play with harps.
5. Open problenls
To keep in the fra~nesof Banacli spaces, we ask the questions about pseuclon~etricsfor coinpacta, although analogous qucstions can be asked in the case of general metrizable spaces.
Problem 1. Does every coinpact pair (X, A ) admit a regular exteridor? Do thcrc cxist bounded li~lcarextending operators from Pc(A) to P c ( X ) ? Problenl 2. Does S E AREP iniply Sc S E AREP ? Problem 3. Dcscribc all the edge vectors of tlie polylic.dra1 cone P({1, . . . , I T ) ) . As far as I know the qucstioii has been answered only for sniall values of I?.
Problenl 4. Let X be a co~npactum.Iiivestigate isomorphic l~ropertiesof tlie Baliacli space E(-') generated in the space C ( S x X ) by P c ( X ) . Does E ( X ) always have tlie approximation property or a basis'? By a Dugundji operator for n coinpact pair ( 2 ,Y ) we rnean a boundcd liiiear operator T : C ( Y )+ C ( Z ) such that T f 11- = f for f E C(17).
Problem 5. Given a compnctunl '4. Lct S = Sc A. Docs t l ~ copcrator sc : P c ( A ) + Pc(-Y) ad~ilitan extension T : C ( A x A ) + C(X x I Y ) which is a Duguridji ol>cratoi'? Problem 6. Lct A arid B be colnpacta witli tlie same Borsuk shapr (cf. [4]). If A E AREP , does it B '? Similar problerns to t,liose of linear cxtciiclillg of nletrics can be stated in tllc cquivariant context for: ( 1 ) spaces with grolip actions, (2) pointed spaces with group actioils fixing the basc point. a.nd A i l n i ~ , u( X ) the sets of all ad~nisLet X be a Banach space. Denote by Na ( S ) sible norins and admissible illvariant inetrics for X. The NCL( X ) is regarded as inetric space wit,li the Bana.cli-Mazur mctric:
C. Bessaga
256
for p, q E X'
Problem 7. Let Y be a closed linear subspace of X . Does there exist a [continuo~~s] linear operator F : Na ( Y )-+ Na ( X ) which extends norms? Problem 8. Does there exist a. retraction r : Mainv (X) + Na ( X )which is continuous in any sense?
6. Is the 'magic formula' magic ?
Here is the background of the particular proccclure of 'taking the rabbit out of a hat' of sect. 1. When I startcd studying linear extensions of mctrics, one of tlie first qucstioils was: ous for t l ~ ecircle to t l ~ o s c Does there exist a linear operator extending c o n t i ~ ~ ~ l metrics for the disc? Tlic disc is the cone over the circle. So, it is natural to begin with the si~llplestcone, the one over two points: x , y. Consider x a.nd y as the points of tlie plane R2 suc.11 that x , y, 0 are not colinear. The cone is then the union of two segments A and 4 joining, respectively, x and y with 0. The points of the cone A U B are of the form: rlr or v y , where P L , ~E I. Evely pseudometric p defined on the set { T ,y , 0 ) x { r ,y, 0) is a non-negative linear combination of the pscudonletrics po, p z , py, where po(r, y ) = 0 , po(O,x ) = po(O,y) = 1 and the other two are obtained by p c r ~ n ~ i t i nthe g symbols 0, z, y . Of course, it is natural to extend thcsc pscudometrics to the pseudonletrics d o , rl,, r l y for t,he whole cone A U B in the affine way, by the formulas: d o ( u x ,v y ) = do(ux,v x ) = do(uy,v y ) = 121 - 01; d,(ux, v y ) = 1 1 , d,(ux, v x ) =
IU
-
vI,
d Z ( u y v, y ) = 0 ;
Since the original three pseuclolnetrics arc linearly independent, tlie correspoiideiice 11 + d uniquely extends to the linear extension operator froin P ( { x ,y, 0 ) ) to P ( A U B), and this is exactly the 'magic formula'.
I must admit that finding this extreinely siniple path leading to the 'magic fol.lli~ila' took me several montlls. This is psycl~ologicallyinteresting: it illustrates how difficult it is to pass to NEW T'FIZNKZNG.
257
Linear extending of metrics
References
[I] C. Bessaga, 011linear operators and fu~ictorsextending pseudometrics, subniitted to Fund. Math. r Isomorphie der Funktionalriiurne, Bull. Inter. Polon. Sci., Ser [2] I 0 for every x E [O,l]. (3.21) Then, there exists a strongly continuous positive contraction semigroup
(T(t)), on defined by
0
1 ,
whose
generator
is
the
x ( l - x ) IZ (x) u " (x) A(u) (x) = -2-
operator
if
A:D(A)+ q[O, 11)
O<x):y E B ( X * ) n B ) for every x E A, U(X*,X)
nB(X*)
n A'
= (0).
T h e n there is a n o r m one projection P from X onto the n o r m closure of A w i t h lcernel(P)= B L . u(X*,X) Proof.- Let W := uF=ln B n B ( X * ) , E the norm closure of A, and R E the restriction mapping from X * onto E*. The condition (i) is also verified for every x in E and it implies that R E ( B n B(X*)) is u ( E * , E)-dense in B ( E * ) . T h e condition (ii) implies that is injective when restricted to I.V. Hence RE is a l ~ o r n e o m o ~ ~ p l ~ i s ~ ~ ~ ,J(XO,X)
between the compact sets B n B ( X * ) and B ( E * ) with the weak*-topologies. Consequently we have W n B ( X * ) = B n B ( X * )4 X 8 , X,) the Krein-Smulian theorem -u(X*,X) says that W is u ( X * ,X)-closed, and we have W = B . Hence
n
T ( ~ * ' ~ ) E'
=
(01
283
On weakly Lindelof Banach spaces
and by polarity X If we use again y in B', it follows B' has norm one.
=
E + BL.
+
the condition (i) we see that 11 x 11I: y E B ( X * )n B ) for cp(B n B ( X * ) )c A , and a ( A ) c 13.
e,uery x i n A,
P r o o f - If x E X we denote by V, a countable subset of F n B ( X * )such that
We shall proceed by recurrence. Assume that for a non negative integer p and every 5 p we have constructed the countable subsets Am c X, B, c F. We write Cm and Dm for the &-linear hull of A, and B,, respectively. We now define: m
and Ap+l := C pu { ~ ( f : )f E D~n ~ ( x * ) } . If we take
then the &-linear spaces A and B verify the required conditions. If we apply our lemma 1 to this pair of subspaces we obtain
Corollary 4.- Under the conditions of l e m m a 3 we have a n o r m one projection P o n X with P ( X ) = XI' 'I and kernel(P)= B L . We are now ready to prove our main result in this section:
T H E O R E M A.- Let X be a Banach space i n the class V . If y ( X , F ) is the topology of u n i f o r m convergence o n the bounded weaP separable subsets of F , t h e n X [ y ( X ,F ) ] is a Lindelof space. Moreover, every product space X [ y ( X , F ) l nis also Lindelof for n = 2,3, ... Proof.- Let {V, : y E r ) be a n open cover of X [ y ( X ,F ) ] . Given x in X , a basis of the filter of neighbourhoods of x in the topology y ( X , F) is given by the family of subsets
where S is a countable subset of F n B ( X ' ) and E > 0. We shall denote by B ( x , E ) the open E-ballaround x . For every x in X and y in r with x E V, we can find a countable subset S ( x ,y) c F n B ( X * ) and ~ ( ry ), such that W ( x ,S ( x , y ) , ~ ( xy ), ) c V,. In particular B ( x , ~ ( x , y )C) W ( X S, ( x ,y ) , E ( ~ , Y )C) '(/y
r
Given x in X and r > 0, we introduce the set r ( x , r ) of all those y i11 such that there is a countable subset S ( x ,y , r ) of F n B ( X * ) with B ( x , r ) c W ( x ,S ( x , y , r ) , r ) c V,. Observe that F ( x , r ) is non void for every x in X and r small enough. If r ( r ,r ) # 8 we choose one element y ( x , r ) E r ( x , r ) . We define the couritahly v;tl,icd map q from X into the subsets of F giver1 by TI(%)
= u { S ( T ,y
( ~r ), ,r ) : I?(%, r ) # fl and r E &+I.
Let cu be an element of r such that the origin belongs to If,, So a countable subset of F n B ( X * ) and E > 0 with ~ ( 6So,, E ) contained in V,. We set A. := p ( S o ) and Bo := So If we apply our lemma 3 we obtain countable &-linear subspaces A and B in X and F respectively, with
Ao C A C S,BOC B C F and such that (i)
11 z I[=
sup{]< x , y
>I:
y E
B ( X * )n B ) for every z in A,
:
On weakly Lindelof Banach spaces
(ii) p ( B n B ( X * ) ) c A, (iii) q(A) c B. Corollary 4 gives a norm one projection P from X onto the norm closure of A with kernel(P) equal to B L . We shall now prove that
which is a countable subset of the original open cover, and thus gives us the Lindelof property we want to prove. Indeed, for a given x in X, if P ( x ) = 0 then x belongs to , E ) c V,. B L and f (x) = 0 for every f in So, hence 3: E ~ ( 6So, If P ( x ) # 0, we find y E I' with P ( x ) E V,, and W ( P ( x ) , S ( P ( x ) , y), ~ ( P ( z )7>) ) C for some countable subset S ( P ( x ) ,y ) i11 F n B ( X * ) and e(P(x), y) > 0. Let 7- be a positive rational number with r 5 E(P(I),y). Since P ( X ) is the norm closure of A in X we find z E A with 11 z - P ( x ) I[< 1-12;then B ( z , r/2) C W(2, S ( P ( x ) ,y ) , 7-12) C W ( P ( x ) , S ( P ( x ) , y ) , e(P(x), y))
c Vy
from where it now follows that y E r ( z , r / 2 ) # 0. We take the cl~oserle l e m e ~ ~ t y(z, 7-12) E r ( z , 7-12) (not,e that it could be different from y), and we have
Since S ( z , y ( z , 7-12),r/2) C q ( ~ C) B we have f (z) = f ( P ( x ) ) for every f in S ( z , y(z, 7-12),7-12), Therefore P ( x ) E B ( z , 1.12) i~llpliesthat
which finisllcs thc proof of the Lindeliif property of X [ y ( X , F ) ] . Moreover, it is clear that every finite product of Banacll spaces in the class V is also in the class V from rn where the conclusion for finite products follows.
Corollary 5.- If I< is a Valdivia compact, i.e., Ii is a subspace of a cfrbe Ir ,with Ii n C ( r ) dense in I 0 the set, (2
E I :I< f , x i
>I>
E)
>I#
0)
is countable, from where it follows that {i E I :I< f , xi
is a countable subset of I and the mapping T from X * into C ( I ) defined by T ( f ) = (< f , xi > ) , € I is one-to-one and weak* to pointwise continuous, therefore B ( X * ) is a Corson compact space in the weak* topology.
287
On weakly Lindelof Banach spaces
A result of the same nature for the spaces C(Ii), where 'h is a Corson compact, is obtained in [AMN, th.3.51. 3. THE LINDELOF PROPERTY OF A DUAL BANACH SPACE WITH THE RADON-NIKODYM PROPERTY Let X be a Banach space. We proved in [ O W ] that X is a n Asplund space if and only if X* admits a projective generator defined on the norming slibspace X of X**. It is a n open problem to know if B ( S * * )with the weak* topology is a Valdivia compact, too. A positive answer for Asplund spaces with density character N1 was given in [DG] and [V5]. In any case, since we have the projective generator we can use the method above to prove the following:
THEOREM B.- Let X be a B a n a c h space a n d y ( X * , X ) t h e topology of rrniform convergence o n t h e bounded separable ~ u b s e t uof X. T h e n X is a n A s p l u n d space if and o n l y if X * [ y ( X * , X ) ]i s Lindeliif. Proof.- Let us suppose that S*[y(-Y*,X ) ] is Lindeliif. Then for any separable subspace F of X , the strong dual F*is a continuous image of X*[y(X*,X ) ] , so F*is Lindelof and consequently norxn-separable. This proves that X is an Asplund space, [St]. We suppose now that X is an Asplund space. We consider the multivalued map q5 : X -+ 2"' defined by q5(x) = { r E ~ B ( X U :< ) r , 7~ >=I1 x 111. 4 is 1111 to weak* upper semico~lt,inuousand co~npa.ctva111etl alitl X * has the RNP. The selection theorem of Jayne and Rogers [JR] gives us a first Baire class selector f : X + B ( X * )for the norm topologies, i.e. f ( x ) E 4 ( x ) for evely x in X, and there exists a sequence of to 1111 continuous fiinct,ions f,, : -Y + B ( X * ) such that 1111 - limn,, f n ( x ) = f ( x ) for every x in X. If we denote by
)III
the mapping ~ ( x = ) { f (x), f l ( x ) , . . . , f,,(z), . . .} , we obtain a projective generator in A", [OVl]. Hence for every pair A, B of &-linear subspaces A C X and B C X* such that (i)
11 f I/= sup{l< f , x >I:
x
A n B(-Y)} for every f
E
B, and
there exists a norm one projection P of X' onto the riorrri closure of B with kemcl(P) = A'. Note that for countable subspaces A ant1 B, wliicli is tlie case we are interested ill
Ilrlc, tllis follows flo~riJalncs' tl~corrtnas [FG] provrd. If wc use this map we can repeat the proof of lemma 3 above to prove:
Lemma 7.- Let X be a n A ~ p 1 , ~ nspace. d Let 11 be a countably valued m a p f r o m X * o n t o t h e subsets of ,Y. Let A. a n d Bo be countable subsets of X a n d X * , respectively. T h e n there exist &-linear countable subvpaces A a n d B i n X a n d X*, re.ppectively, with A0 C A C X , Bo C B C X * ,
and such that
(i)
11 f I[=
(ii) p(A)
sup{[< f , x >I: x E A n B ( X ) } for every f in B , C
B, and
(iii) q ( B ) C A. This lemma is the main tool in our proof (below) that X*[y(X*,X ) ] is Lindelof. With the same notation as in theorem A, for the open-cover {Va : cu E I?} in X * [ y ( X BX, ) ] we shall find a countable subcover by colistructing an adequate separable and complemented subspace of X*. We define as in theorem A , where now q : X * 4 2 X ; we select yo E I? with 8 E V,,, a countable subset So c B ( X ) , and t > 0 with W(O,So, E ) C Vyo. We write A. := So and Bo := p(So). If we apply lemma 7, we obtain countable &-linear subspaces A and B , A. C A c X , Bo C B C S*that verify the conditions above. So there is a projection P from X * onto the norm closlire of B along A'. Reasoning as in the proof of theorem A, we see that
frorn where the conclusion follows. The dual of an Asplund space always has an M-basis that can be constructed with the projectional resolution of the identity [G, prop.III.71, [V7].
Corollary 8.- Let X be an Asplund space. The following are equivalent: (i) X*[u(X*,X*')] is Lindelof. (ii) B(X**) is weal? Corson compact. (iii) B(X**) is angelic. (iv) For every f in B(X**), there is a sequence (f,) in B ( X ) which i.9 wealc' convergent to f . (v) For every f in B(X**), there is a countable subset D
c
B ( X ) s7~chthat f E
-(X.,X)
D
Proof.- It is the same as the one given in corollary 6. Indeed, (ii) =+ (iii) + (iv) + (v) is clear. (v) + (i) because under the assumption (2)) the topology y ( X * , X ) is finer than u(X*, X*'). ( 2 ) =+ (ii): Using the M-basis (x;,fi)iEI in A'*, the weak Lindeliif property of X * implies that the evaluation map
is weak* to pointwise continuous from X*' into C ( I ) . The equivalence (ii) e (iii) H (zv) ( v ) was obtained by R.Deville and G .Godefroy [DG, th.III.41. They also proved that these conditions are equivalent to the fact B(X**) does not contain any subset which is weak* Iiomromorphic to
On weakly Lindelof Banach spaces
289
[O,wl]. Here we are interested in the eciuivalence with the Lindelof property (i). In the next section we shall apply this equivalence to study the problem of Corson. 4. O N T H E P R O B L E M O F C O R S O N
As we said in the introduction, the following question was posed by Corson: Probleni - If a Banach space X is weakly Lindelof, is ~t true that X [ u ( X , X*)] x X[cr(X,X*)]must be Li~ldelof,too? The method of proof which we have used in the former sections for the Lindelof property of a given Banach space is very well adapted to the study of this problem. Indeed, if X E V then X [ y ( X , F ) ] x X[y(X, F ) ] is a Lindelof space (Theorem A). When could it be possible to cxtrnd this result to the weak topology? Our corollary 6 says that this is the case if ant1 olily if B ( x * ) i5 a weak' Corson compact. On the other hand, if X is an Aspluntl space, then X * [ y ( X * , X ) ]x X * [ y ( X * , X ) ]is also Lindelijf (Theorem B). Our corollary 8 says that the same is true for X * [ u ( X * ,X")] if and only if B(X**) is a weak* Corson co~npactspace. Fortunately, if X is any weakly Lindelof dual space, G.A.Edgar [E,prop.l.8] showed that X has the Radon Nikodym property. So we can prove the following result which gives a positive answer t o Corson p r o b l ~ mfor the case of dual spaces: T H E O R E M C.- L e t X be a d u a l Ban.ach space. T h e n X [ u ( X ,X*)] i s L i n d e l o f if, and O T L ~ ?i f~, ( X [ u ( X , X*])n iu Lindelof for 11 = 1 , 2 , ... Proof.- If X is weakly Lindeliif, X r~lustbe a tlual space with the Radon-Nikodym 11ropert.y. So we can prove the following result which gives a positivc arlswcr to C o ~ s o ~ ' ~ product of a finite number of Corson compact spaces is Corson compact, too. For every positive integer I L , Arn can be renorrned to have weak* Corson compact dual unit ball. Thus our corollary 6 says that X " [ u ( X n ,(Xn)*)]is a Lindelof space, and this space is ho~neonlorphicwith ( S [ a ( X ,X*)])" which is also a Lindelof space. The author has bee11 partially supported by DGCYT PS88-0083.
References. [ A P ] K . A l s t e r , R.Po1. 011function spaces of compact subspaces of C-products of the real line, Fund. Math. 107 (19S0) 13.5-143. [ A M N ] S.Argyros, S.Mercourakis, S.Negrepontis. Functional-analytic properties of Corson-compact spaces, Studia Math. 89 (1988) 197-229. [Ar] A.V.Arhangel'skii. A survey of C,-tlieory, Q. and A. in General Topology 5 (1987) Special Issue 1-109. [C] H.H.Corson. The weak topology of a Banach space, Trans.Amer.Math.Soc. 101 (1961) 1-15. [ D G ] R.Deville, G.Godefroy. S o ~ n eapplications of projective resolutions of identity, (1990) Preprint. [Di] J.Dieste1. Geometry of Banach spacrs. Selected topics. Springer Verlag. L.N.M. 485 (1975).
[El G.A.Edgar. Measurability in Banach spaces, Indiana Uxliv. Math. J. 26.4 (1977) 663-677.
[FG] M.Fabian, G.Godefroy. The dual of every Asplund space admits a projective resolution of identity, Studia Math. 91 (1988) 141-151. [GI G.Godefroy. Five lectures in geometry of Banach spaces. Seminar on Functional Analysis 1987. Notas de Matemitica 1. Universidad de M~ircia. [Gull S.P.Gul'ko. On properties of subsets of C-products, Soviet Math.Dok1. 18-6 (1977) 1438.1442. [Gu2] S.P.Gul'ko. The structxre of spaces of conti~iuousfunctions and their hereditary paracompactness, Russian Math.Surv. 34 (1979) 36-44. [JR] J.E.Jayne, C.A.Rogers. Bore1 selectors for upper semicontinuous set valued maps, Acta Math. 155 (1985) 41-79. [L] J.Lindenstrauss. Weakly compact sets, their topological properties and the Banach spaces they generate, Ann. Math. Studies 69 (1972), Princeton University Press, 235-276. [N] S.Negrepontis. Banacll spaces and topology, Handbook of set theoretic topology, Edited by Icunen-Vaughan, Elsevier Sc.Puh1. 23 (1984) 1045-1143. [OSV] J.Orihuela, W.Schachermayer, M.Valdivia. Every Radon-Nikodym Corson conipact space is Eberlein compact. To appear in Studia Math. [OVl] J.Orihuela, M.Valdivia. Projective generators and resolutions of identity in Banach spaces, Rev.Mat.Univ.Comp1utenseMadrid 2 Supplementary Issue (1989) 179-199. [OV2] J.Orihuela, M.Valdivia. Resolutions of identity and first Baire class selectors in Banach spaces, (1990) Preprint,. [PI] R.Po1. A function space C ( X ) which is weakly Lindelof but not weakly cornpactly gcnc~.atetl,Studia Math. 64 (1979) 279-285. [P2] R.Po1. On pointwise and weak topology in function spaces. Warszawski University (4)84 (1984). [St] C.Stegal1. The Radon-Nikodym property in conjugate Banach spaces, Trans. Arner.Matli.Soc. 206 (1075) 213-223. [TI M.Talagrand. Espaces de Banach faiblement I
6:
lias norrri 1. Thcn, because of tlie
>0
sucll that
11
zJ
IIAX
< AT, for
all j. L i e clairil that ( i ] ) is wealtly null. In fact, the scbquc.nce is clearly coiltni~icd ill (AX),(E;), wliosc tlual is (isolnorphic to) A X (E;*) = A(E:*).
If
11
E iZ(E:*),
with the notations of tllc Legiil~lirlgof the scction, we have
nrlicre
= (Sq,
11,
11
llliiJ,,
TI,,
I(=
il1i1)lii~stlicm tllat 1101 a
-
Sp,)(rl) co~ivc~gcs to O (recall tliat A = A,). In conseqllrnce,
0, and
SO
( z J ) is weakly nlill. Thc coild~tio~i < r,, zl
> >
6
Ii is not a Du~ifortl-Pcttisqct (and, a foi t ~ o r l ~iritlic~r , ii l~lil~t(~cl
relatively compact set.).
(h)
+
(a).
If Ii sati5ties I>(ii), thrlc is a sequcnce
collvcrging to 0 illid bllcli tliat
(6,)
of positive ~cmls,
Distinguished subsets in vector sequence spaces
Hypotllesis b(i) and Proposition l . l ( a ) show that 12
> 0. An
CZnzl I, o r , ( I < ) E 3-I(FA) for every
appeal to Proposition l . l ( b ) concludes the proof.
R e m a r k s 2.3. a ) If (AX), = A X (for instance, if A =
ep, 1
< p < cm,or A is
a reflexive
Orlicz sequence space), condition I)(ii) of throrem 2.1 is autolliatically satisfied. In fact, sl~pposcx E A(En) and z E Ax(E:), and let a , = sign(< r,,(x), r n ( z ) >), IL
= ((Y,,T,(Z))E AX(E:). Then
and 1jy ass~iinl~tioil, li~n~,,, S n 1 ( t ~=) 1 1 , for every
IL
in liX(E;).
I)) Condition I,(ii) of theore~n2.2 always implies condition b(ii) of tlicorelrl 2.1. 111 fact, suppose
.T
E A(En) and z E AX(E,*).Wit11 the sarne 11ot;ttions of part (a),
This yields tlic proof of part ( h ) of the ncxt corollary. c ) In the palticulitr case A =
el, contlitions b(ii) of
In fact, suppose that Ii c,cl~~iv;~le~lt.
2.1. Tllen, tlicic csists I),
>
C
O,(.rn)
theorcms 2.1 ant1 2.2 are
C1(E,) docs not satisfy
b(7z)
of theorcm
Ii :rritl a sul~secluenceof positlvc iiltcgcrs
< '11 < 111 < q2 < . . . ~ 1 1 ~ 1that 1
For c,\rc,ry p k
< < q k , let 11s choose a11 elt~iilrntzn E E7:such tliat 1) zit II= 11
< rr,(.rL),z,, > = 11
r,,(x)
11.
Tlicn
2
=
(2,)
E Qm(Ez)
~cI(E~)*
1 and
F. Bombal
It follows that condition b(ii) of theorem 3.1 does not hold. Tliis proves part (c) ot the following corollary. Corollary 2.4. Let ( E n ) be a sequence of B a n a c h spaces and FA = A(E,,).
a ) Let 3-1 and G be t w o classes belonging t o the set
{ K ,C, V P } (re.sp., { W , WC,
V * } ) . Then,, F;\ has property ('H, 6 ) if and only if every El, has propertg (31,G). In particular, FA is weakly sequentially c o ~ n p l e t e(resp., has t h e Gelfand-Phillips property o r P E ~ C Z ~ Tproperty L ~ ~ ~ '(SV f )if and only if so does every E,,. b) Let 3-1 = C o r V P . T h e n , FA h,as property ('H, W ) if and only if every En
I?,ILS property ('H, W ) . I n partic,ular, FA has the R D P * property if and only if so does every El,.
c) Let 'H ( ~ n d6 be a n y of the cla.sses V * , WC, W , V P , C o r XC. T h e n , !l(En) /I,II,J
property ('H,G) if and only if every En h,as property ('H, G).
I n partic,rrlar,
CI(E,,) is w e a k l ! ~seq~rentiallycomplete ( r e , ~ p . ,has the Schzir property, t h e DunfordPc/tis pr.oprrty, tlze 6 c l j ( ~ 1 ~ d - P h i l property, li~s Pelczynski's Property ( V * )o r tile R D P * - p r o p e r t y ) if and only if so does every El,. Remark 2.5. Corollary 2.4(c) is not true in grneral.
E,
=
111
fact, it sllffices to t,alie
K , the scalar field, for every n > 0. Then cvcry El, has property ('H, G),
w1l;ltever clioice of 'H a.nc1 G we made. B I I for ~ A = CP(l < p < m), A(E,) = Pp has ilcitller thr: Sclllu- property, nor the Dunford-Pctt,is propcrty. Since Cp(El,) contains always a conll)l(~lilc~it,cd copy of Cp, it never has these two properties.
3. C O M P L E M E N T E D C O P I E S O F P1 A N D
C,
I N A(E,,).
(\'*)-sets ant1 limited sets are esprcially usef~tlfor detecting co~riplrn~c~ntecl copics of
el and c,,
due to t l ~ efollowing result:
Lemma 3.1. a ) ([B4],[ E l ] ) A bounded subset of a B n n a c h space E is a ( V f ) - s e t if and o r ~ l y
if it does n o t contain a sequence ( z n ) cq1rinn1en.t t o the 11.971d basis of PI
ILTL~
. 0,
unconclitio~iallyCnuchy series ([Dl, Ch. V) in F A , as a
ii) Every sul)ssc~ue~lce of (n,,(lr,,,)) is not li~nitcd. iii)
11 ?T,,(IL,,,)11 2 E , for sonle c > 0 and [,very 17. E N.
From ( i ) it follows that
X n,(u,)
is also a wcakly unconditiorially Caucl~y
seric3s. Therefore, l,y (iii) and the Bessaga-Pclczynski selection principle (see [Dl, Ch. I)'), wc call assurnc (passing to a suhsc~quenceif necessary) that
( ~ , ~ ( t ( , , ,is) )
a basic scquencc:. But then, Corollary 7 of [Dl, Ch. V assures that (.rr,(~~,,,)) is
eqi~ivalrnt.to the unit c,-basis.
Hypothesis (ii) ant1 lcmma 3.l(b) conclucle the
1)roof.
Reillark 3.4. Note that we llavc provrd ;il)ovo thc following: If ( x k ) is a complemented c,-ba3i.q in F,,, there exists a subsequence ( ~ r , , , ) of (2k) and a.1~11 E N s u c h that (x,,(u,,)) i . ~a complemented c , basis of En.
Distinguished subsets in vector sequence spaces
Corollary 3.5. Let
/L
305
be a a - f i n i t e , purely a.tomic mea.ut~re and E a B a n a c h spo.ce.
T h e following assertion.3 are equ.iva,lent:
a ) E contain.,p a c o ~ r ~ p l e n ~ e n tcopy e d of c,. I)) For every p, 1 5 11 < m , Lp(p, E ) contain,^ a com.plemented copy of c , . c) There is a p, 1 5 p
< m,
Y I L C ~ I .that
L , , ( / LE, ) co7~ta.insa complemented copy
o f c,. Proof. Lp(p, E ) is isometric to PI,(E),anrl thc>orerrl3.3 applies.
REFERENCES [A] Ii. T. Andrc\vs, D,i~nford-Pettis~ e t isn the .upa.ce of Boclr.ner ir~tegrablef~rn,ctions. hIath. Ann., 241 (1979), 35-41 [Bl] F. Bombal, O n PI su.bspaces of Orlicz vector-valr~cdf u n c t i o n spaces. Afath. Proc. Ca~iiLr.Phil. Soc., 101 (19S7'), 107-112. [B2] F. Bolnbal, O n embedding PI as a com.plemented s~u,bspace of Orlicz vcctorvalued firnction spaces. Rev. hlat. U~iiv.Complllto~iscde Madrid, 1 (19SS), 13-17,
[B3] F . BoinI,al, O n P e l c z y n ~ h i 'property ~ (I") in vector seql~en,cespaces. Coll. Math., 39 (198S), 141-11s. [B4] F. Bornbnl, 071. (V*) sets and Pelczyn,sl:i$ property ( I f * ) . Glasgow Mat,h. .I., 32 (1990), 109-120. [B5] F. Bo~lll,wl, Sobre a1gun.a~propiedn,de.u rle E.rpu,cio.u de B a n a c h . To appear in Rev. Acatl. Ci. Madrid. [B6] F. Bonil~irl,Distinguisfr.ed s,ubset.u and com.plem.e7~ted copies of c , in vector seqlrence spaces. I!:xtra.ct.a. Matli., 5 , 11. I (1090), 4-6. [BD] J. Bourgnili a.nd J. Diestel, Limited o p e m t o r . ~and strict cosinyrrlarzt?j. h;I;~th. Nachr., 119 (1984), 55-58.
[Dl J . Dicstel, Scqr~encesand s e r i e ~i n Ban.acf1. spaces, Graduate Texts ill Matll., No. 92, Sl)r.iiiger, 1984. [DR] L. Drcw~iowski,o n Ban,ach space-u w i t h the Gelf(1,7~d-Pf~illl.p~ property. h.Iat,h. Z . , 193 (19S6), 405-411. [ E l ] G. E~nmanucle,O n t h e B a n a c l ~.upnce.u laiih. ifre property (V*) of Pelczgf~~ukz. A1111aliMat. Plira e Applicata, (19SS), 17'1-181.
[E?]G.Emmanuclr, On camplemenled copica ofc. in L:., 1 5 p < w. P r c . A. M. S., (1SSS), 785-786. [LT]J. Lindenstlauss and L. Tzafriri, CIannisal Daraach Spaces. Springer-Vmlag, Berlin. Vol. I, 1977;Vol. 11, 1979, /MI 3. Mendma, Complemented capiea o j E%m L p ( p ;E ) * Preprint.
[P] A. Pelczjnski, On Banach space3 on which
e w e q tanconditionally convergent opemtor i- wweakly compact. Bull. Aciirl. Pol. Sci., t O (19621, 641-648.
[R] R. C.Rosier, Dual spncea of certain atcfor sequence gpnceir. Pacific J. of Math., 46,NO. 2, (1973),487-501.
[SL]T. Sd~lumprecht, Limiiierde Mengen in Ilanachriumen. Maximiliaris-UnivetsitSt, M i n d ~ e n 1988. , IV] M. Valdivin, Topica in Locally
Comes space^.
Stutlies No, G7. North-Holland Fuh.
Tllesis. Ludwig
North-Holland Mathematics
Co.,Amsterdam, 1082.
Progress in Functional Analysis K.D. Bierstedt, J. Bonet, J. Horvath & M. Maestre (Eds.) O 1992 Elsevier Science Publishers B.V. All rights reserved.
Weak Topologies on Bounded Sets of a Banach Space. Associated Function Spaces
I)eparta~iicnto dc Ar~iilisisMatcrnlitico Facultad de Matemiticas 1Jniversidad Complutcnsc de Madrid I
I his paper is divided into tllrcc s e c l i o ~ ~ Section s. 1 is a n cxhaustivc study concerning
the lxi~icipalspaccs of wcakly co~ltirlllousfunctions on 13anach spaccs. S i l ~ c et h e s t r ~ ~ c t u r e of t,llesc function spaccs is closely related wit,li properties of different weak topologies, thc bounded-weak, t11c cornpact-wcak and tllc l~our~ded-weak* topologies arc introtluced. Section 2 shows tlle wcakly continuous and wcakly diirerentiable function spaces in relation with tlre extension of \Veierstrass' tlieorcin for illfinite d i ~ n c ~ l s i o n Banach al spaccs. Sectio~i3 tleals wit11 interpolation of bounded seciucrices by weakly continuous a ~ l d weakly diffcrcntiable functions. Many of t l ~ ercpresenta.tio11 results obtai~lcdin t l ~ eprevious sections arc used. Now we start wit11 some notations and tlcli~litior~s: Let 1 5 and
I: I)c real or c o ~ r ~ p l e13a1lacll x spaces; we consider ttic following linear
spaccs: - L(n12;I:),
12
E N , all cont.inuous 12-linear mappings from
En l o F .
- L,("l 0, there are 41, . . . ,+, in E' and 6 > 0 such S for all z = 1 , . . . , r z tlicn Ilf(x) - f(y)ll < E .
weakly contir~uousif for cach z E A and c that i f y E A,I $,(x
-
y)
I
0 such that if 2 , y E A, I d i ( x - 1)) I < 6 for all i = 1 , . . . ,II t l i c ~11 ~J(x) - f (y)II < E . \Vc denote as CWbu(E;1 7 ) the space of all J from E to F which are weakly unifor~nly continuous when restricted to bourldcd scts. Every J E Cwb,(E; F ) is compact, in the sense that it maps l ~ o u n d e dscts into relati\.cly cori~pactscts. bVc endow Cwbu(E; F) with t l ~ clocally convex topology
T,,
of ~lniformconvergence on bounded subsets of E.
Cwb,(E;F ) is ~ b complete. Boundcd sets for E" coir~cidefor t h e norm, thc weak* and tlic bw' topologies, arid that (13rw.)' = E'. Since Efw. is also the topological direct limit, lim,,,(B~,weak*) only i f for all
11,
precisely when
f
JIE
(B: is the 11-ball i l l El'), a fnnc1,ion f belongs to C(Erw.; F ) if and (l?:,wcak*) -t F is continuous. Then we sec that this I~appcns
IRK:
E C W b u ( EF; ) . Then we have the following topologically i s o n ~ o r p l ~ i c
representation of Cwb,(E; F):
w l ~ c r cC(Erw.; F ) is the space of all cor~ti~ir~orls functions from Er,. t,o F , endowed with the topology of uniform convergence on norr~l-boundedscts of E". It is easy to see that every relatively pseutlocompact ant1 closed s ~ ~ b s iri e t Etw. is compacl. Thus, wc have t h e following theorem:
312
J.G. Llavona
1.14. Theorem [17]. Let E , F be Banach spaces. barrelled.
A continuous linear mapping A : E
Then the space C W b u ( EF; ) is
F is compact if and only if A is weakly (uniformly) continuous on bounded sets. R u t , a non-linear mapping A : E -+ F may be +
compact, without being weakly uniformly continuous on bounded sets. For example, the mapping A : e2 --+ R given by A ( x ) = (n E N) for t h e usual basis { e n ) of
CTzl x:
e2, so t h a t
is trivially compact although A ( e n ) = 1
A
6CWbu(e2).
Valdivia [20] shows that a Ranach space E is reflexive, if and only if every weakly continuous function J : E + R is bounded on balls in E . Since every function in
Cwbu(E)is bounded on balls in E l we liave that Cwb,(E) = C w b ( E )if and only if E is reflexive. If E is a non-reflexive separable Ranach space, l h e James theorem [13] states tliat there exists 4 E E' which does not attain its norm. T h e function g : B1 -t R, g(x) =
[I1411- 4 ( x ) ] - ' , is
weakly continuous on I l l , and it is not hounded. Since E is
separable, it is W C G and therefore weakly normal. Thus, according to Tietze's tlieolcn~, there exists a weakly continuous J on El wliicll extends g. T h e function J cannot belong t o Cwbu(E).
2. On the Completion of the Space of Continuous Polynomials of Finite Type. This section foc~isesaround uniform approximation 011 bounded, and on cornpact sets, 1137
polynomials of finite type. As a natural approximation problem, weakly continuo~rs
and weakly differei~tiablefrlriction spaces arc i ~ ~ t r o d u c c d .
(a) Continuous case. Let E , F b e Danach spaces and P c ( " E ;I:) the completion of P f ( " E ;I:) wit11 respect t o tlie norm induced by P ( " E ; F ) ; P C ( " E ;F ) is in gc~leralstrictly contained in ?("I?; F ) .
2.1. Proposition [I]. Let X be a compact Ilausdorff space and C ( X ) the space of all scalar confinuovs Junctions on X . I J X is dispersrd (ccel-y closed subset o J X contains an isolated point) arzd E = C(,Y) with the s ~ iaor~rs, ~ p therl Jor every n E N, P ( " E ) = PC(",!?). 111parlicular PC("co)= P ( n c o ) . We will dellole by 'PQ("G;I:) tlie space 'P("G;F )
n
Co(13;F ) for Q, = w, wG, 2usc and
tubu. By definition F W b ( " EF; ) = 'PWbU("E; F ) wheu 13 is reflexive. B u t , a polynomial wliicl~is weakly continuous on balls is in fact weakly uniformly contiliuorls on balls [4].
313
Weak topologies on bounded sets of a Banach space
One consequence of t h e above result is the following theorem.
2.2. Theorem [4]. I f E ' 1~asllle approxinration property then P C ( " E ; F ) = Pwbu("E; F). 2.3. Remark.
1. P,,,("E; F ) = P,(" E ; F ) C PwbU("E;F) 2. If E' is separable, PwSc("E;F ) = PwbU("E; F). 3. If p(x) = C;zP=, xi, then p E P('e2) \ P,,c(2el),
and therefore p @ PWb,('ez); the
sanle p E P w s C ( ' ~\~Pu,bu('el). ) 4. Every polynomial p(x) = Cr=P=l Anxi, with X = (A,)
E
el,
belongs to PWbu('e,),
1
conditional basis o f e l ; i n this case the associated homomorl~hismR i s onto I, and o ~ ~ s .section S : E, + C w b u ( l l )oJ R. there exists also a c o n l i n ~ ~ linear
Weak topologies on bounded sets of a Banach space
317
+
4 . Let (an),cN be the sequence i71 Cl defined by an = en en+^ for each n E N . In this case the homomorl~llismR associated to (a,) is not onto e,.
(b) Interpolation of Schauder bases by C$,,(E)-functions. Let 1:' be a real Banach space, ( e n ) a normalized Schauder basis in E , with (en) -+ 0 weakly. For each f E C,"bU(E),( f ( e n ) )c R is a convergent sequence. Our interest now is to study when t h e associated liomornorpliism
is onto c and when R admits a continuous linear section.
3.7. Theorem [14]. 1 . If E is superreflexive, there exists nz E N such that R : P,bu(mE) + co admits a corttinuous linear sectioil. 2. If E is reflexive and (e,,) is syminetric, then I< : C,"b,(E) -t c is onto and a d n ~ i l s a conlinuous linear section. 3. If E = c0 then R : C,!,bu(~O) -+ c is onto. 3.8. Remark. Problem: Is there a Barlac11 space E such t h a t C S , ( E ) is reflexive? Silicc for each m E N, C,"b,(E) induces on P,bu(mE) the norm t,opology, if C s u ( E ) is reflexive then P,bu(mE) will bc reflexive for all m E N. 111 particular, for in = 1 ,
P W b u ( ' E )= E' would b e reflexive. Therefore we can restrict our attention to reflexive Uanach spaces. From 3.7.1 and 3.7.2 it can easily b e verified that if E is superreflexive wit11 basis or E is reflexive with sy~nriictrichasis, then C z , ( E ) is not reflexive. If T i is t h e Tsirelson space, 7'' is reflexive and has uncondit,ional basis. We don't know if
C,"b,,(T1) is reflexive.
REFERENCES
R.M.Compact
polynor~zials and compact diJerentia6le mappings, in: Sdm. Picrrc Lelong, Lccture Not,cs in hlatli. 524, Springer-Vcr1a.g (197G) 213-222.
[l] Aron,
[2] A r o ~ i ,It.M., Uiestcl, J. and liajappa, A.Iilcll In, -, cu
wit11 In = ( m l , . . . , ? n ~ ) , mk ', t, E N'*. Namely, if f , g E H 2 ( D " ) wit11 f = El,,. f l , l ~ n l , 9 = ~ , g l , z e , , Lthen , (21) implies inlillcdiately that ri,hnl'(f,g) i5 cclual to tllca f i l \ t aritllnletic irieari of f g E H'(D"). By [Z], Tllcorein XVII.1.23 and (15) ; ~ I I O I Y ~ wcs (20). We considcr now bilinear complex partial differelitial operatols wit11 l)oly~~oiiiii~l coefficients,
Factorization of multilinear operators
325
where n,7i E NA', z = (*I,. . . , z M ) E C" and a n , h is a polyiio~llialof Ad conlplcx variables. We write an,& = an,n,pzP
C P
E C is for all n, ?different i from 0 only for a finite number of multi-indices where EN ~ . It is elementary to see that P E L(2Ak(D"), H 1 ( D M ) ) if N = ( N I , . . . ,AT,,) E PdA' where N, is the highest order of the z:th partial derivative occurring in (22); notc that multiplication by znl is for all rn E Nh' a bounded operator in H 1 ( D n l ) . The Hausdorff-Young inequality works also in the case of M variables so that \vc liilve a continuous canonical inclusion H ' ( D M ) -+ A?(DM). Since tlic identity 1napl3ing A r ( D M ) -+ A f - l - c ) M e ( ~ Mis) continl~ousfor all E > 0 by (IS), we sec that P E L(2Ak(D"), Af-l-E)h,a(DA')). \Ire show that P has even a tensor product representation. So let f = C , f,,e,, and g = CTn g,,,e,, belong to A;. Thcn we have
where a,,,, E R is such that aV1em+n - a ,,,,, e ,,,. Note t,hat
The expression (23) is the Taylor scrim of the filnction P( f ,g ) E _ E , Ale(D"); hence, (23) converges for all f , g E A ~ ( D " ' ) . This implies that the tcnsor prot1uc.t represent ntion
converges in ( ( A $ ( D ~ ) ) '@ ( A ~ ( D A ' ) ) ' ) 6 1 ; ' A ~ - l - EA~1 ).A , e ( D
326
J. Taskinen
5 . MAIN RESULT.
Using the notation and preliminary facts presented in the previous section xvc prow Theorem 7. Let P be the symmetric bilinear complex partial differential operator with polynomial coefficients
wl~erethe notation is as in ( 2 2 ) , f , g E A L ( D M ) and N = ( N 1 , .. . , N A l ) E N" is s~lcll that each N , , 1 5 i 5 M , is the highest order of i:th partial derivative occur~rirlgin (2%). For all E > 0 there exist bounded linear operators A E L(A%(D"), ~ ' ( 0 ":ili~d )) B E L ( H ' ( D ~ ) , A $ - ~ _ , )sucll ~ ~that ~(D for~all) )f,!, E A % ( D M )
Since P is symmetric, we call assunie a n,,,,p = a ,,,, for all 11, ii and p i11 ( 2 6 ) . Proof. In view of Lenlma 2.2 of [TI it is enough to construct bounded lincar operatols A' E L ( ( H 2 ( D M ) ) (' ,A $ ( D " ~ )and ) ' ) B as above such that
(A' 8 A' @ B ) ( * z ) = P
('8)
holds in ( ( A % ( D M ) )12 ' ( A & ( D ' f ) ) ' ) & p A ~ - 1 1 E ) M 8 (see D A( 4 2 0) ;) ,(25) and [TI, Sectioil 2, for the notation and definitions connected with (28). Let us denote by I i the n u ~ n b e rof terms in (26). We define B by
Be,,, =
if
i n
# ( 1 < ' 3 q ( ~ ) )0+, .~.,. , 0 ) for all
( ~ ( k ) l ) l t E e k , if
in
= ( 1 < ' 3 q ( ~ ) + ~ ,.0, ,,.0 ) ,
+
where r/ is as in ( 1 7 ) and I 0, s > 0, then all complex geodesics 4: D + 'D extend to continuous~nctions4: D -+ D.
0 and 5 1 so that
If p, = 0 and la, 1 < 1, then analyticity of $ j ( ( ) forces cuj = Pj and two terms cancel in the expression for $j(C). Then from 4j(O) = $j(O) = xj, we conclude that cj = tc, and t = 1. So i$j = G j in this situation. In the remaining cases, 4, ( 0 ) = $, ( 0 ) = x j yields
and hence that
aj
= tP. Then $,(s) = 4,(s) = yj and some cancellation of common terms shows
( s - a5)(1- cup) = t ( s - P)(1 - Ps).
Combining this with c u j = tP yields
356
S. Dineen, R.M. Timoney
Hence t = 1 or t = I/31-2. In the second case,
Since laj[ 5 1 and
1/31 5 1, we must have laj[ = 1/31 = 1 SO that t
=
1. Hence aj
= /3
and
d ~( j0 = dj(C). We can summarise our results for e p as follows.
Corollary 5.8 Let R, denote the unit ball of ep, 1 5 p < m. Then
(i)Any two distinct points in B, can be joined by a unique normalized complex geodesic. (ii)All complex geodesics in B, are continuous. (iii) A map 4: D -+ Bp is a complex geodesic if and only i f it is a non-constant map of the form given in Proposition.5.5 PROOF: For all 1 5 p < m, existence follows from Theorem 2.5 (or from [l I]). Uniqueness for p > 1 follows from Theorem 3.2 and uniqueness for p = 1 has just been established in Lemma 5.7. (ii) follows from Theorem 4.4. For p = 1, this has already been noted in Examples 4.5. It is straightforward that Theorem 4.4 applies to @'for 1 p < m because &' is uniformly convex in the real sense (see [8]). (iii) follows from Proposition 5.5 and (i). We suspect that a more general version of this result holds for LP@) in place of C\ but we have not managed to prove LP-versions of Propositions 5.5 or 5.7. Example5.9 Let X =
(ll~ll;,
epl
$,
CPZ
=
{x
= (y, 2)
: y E lp1,z E
lP2)
normed by II.rII =
111.
+
ll41;~) -
One can check using Proposition 5.3 that for 1 5 p, < m, 1 5 r < oo all nonconstant maps 4: D -+ Bs of the following form are complex geodesics.
where Iai,I 5 1, Ir;l < 1, 171 < 1, Pij is Oor 1, and the following relations hold
Complex geodesics on convex domains
where
The proof of this involves observing first that for x = (y, z) E A' with llxll = 1,
with Nv/IIyII and Nz/llzIl given as in the proof of Proposition 5.4 for@'. take p(C) = 11 - ?(I2 and
To apply Proposition 5.3,
where r-P,
e;j = ri
( C ; ~ ( " ' - ~ C ~(i ~
= 1,2; 1 5 j
< 00).
We suspect that all complex geodesics in Bx are of this form. Other examples of complex geodesics in spaces which are direct sums of more than two summands of CP-type can also be exhibited.
Remark 5.10 The case p = oo is excluded in all of the previous calculations because it is Even for the unit ball (and I,"). well known that almost everything is different for B,,2 = ( ( ~ 122) , E C2 : max, 1z,1 < 1) (polydisc) of Ey, many of the differences are apparent. The only points of BB,,2 that are complex extreme points are those where lzll = 1221 = 1 and therefore the result of Vesentini [33] cited at the beginning of Section 3 shows that there are many complex geodesics joining 0 to 2 = (zl, z2) if lzll $ 1 ~ 2 1 . In fact, if lz21 < 1211, the normalized complex geodesics joining 0 to (21,22) are
where g is any analytic function on D with y(0) = 0, g(lzl()= z2 and sup p : Aluu~~cy's jnctorization theorem ([19], see also [lo], 18.9.) states, in particular, that for T E C(E, L,(q)) the operator
is continuous if and only if T factors through a multiplication operator L,(v) see [lo], 18.9. for more information about tliis result. 3.3 Some remarks about A,
011
G8E
c I,,
@
-+
Lp(v);
E:
( 1 ) The theorem implies (talie p = q ) that the defi~~ition of A, on G @E does not depend on the measures /I wit11 the property t l ~ n tG is isometrically ernbedded illto L,(p); in other words: G Bap E is well-defined.
( 2 ) x = A , on L1 @ E - but this does no longer hold on G @ E: Just take a noncomplemented reflexive subspace G C L1[0,1], define E := G' and look at the restriction map (L1 @a, G")' = C(L1, G) --+ C(G,G ) = ( G 8 , G')'
A. Defant, K Floret
374
which is not surjective; the Hahn-Banach theorem gives n
(3)
# A1 on G @ G'
e2gApC2 = C2 holds isomorphically for all 1 5 p < co.
This result is due to Rosentlial and Szarclc [22]. We find it informative to give a proof of this using a tensornorrn close to A,: First note that d i , = A, on L, @ E if (and only if) E is isometric to a subspace of some L,(v) (this is also a result of Kwapien's, see [lo], 25.10.). Since d;, is left-injective ([lo], 20.4.) and P2 is isometric to a subspace of some L,(p) (via Gauss functions; [lo], 8.7.) it follows that
Moreover, di, is equivalerit to A2 on P2 @ P2 for 1 < p < co (see [lo], 26.6.) and, clearly, C2@A,12 = C2 holds isometrically. 3.4. Take 1 5 p
< q < 2 and 1:4
A
LPb,)
the (isometric) Ldvy embedding (where p, is the q-stable Ldvy measure; see [lo], section 24). It was observed by Bcnnctt [2] (see also Rosenthal-Szarek [22] and [lo], 26.3.) that
is not continuous. This and I' @ I' give the
2 follows by dualization.
Note that using tlie Gauss ~ncasureand tlie embedding
for arbitrary s < co the arguments in the last proof give corollary 1 in 4.2. for p, q < co. But for q = co (and p = 1) tlre result is much stronger: by theorem 4.3. it implies Grothendieck's inequality. 5. C o m p l e x i f i c a t i o ~ lof O p e r a t o r s
5.1. As an application of the results presented, tlie con~plexificationof operators will be treated. Recall from 1.3. that = L:
L:
and
sC= S @ idq
whenever S : L% + L r is a real-linear operator. Since R;is isometric to a subspace of some L, (via Gauss functions), theorem 3.2. yields that for every S E L ( L F , L F ) and l l q l p l m llid@ s : @A, L% a;8,; L,RII = Ilsll.
m;
-
@A;L: (isometrically) this implics IISGII= llSll - an observation Since R;BA, Lf = due to Figiel-Iwaniecz-Pbtczylis1;i [12]. 111 the "bad" case 1 5 p 5 q 5 m one obtains directly from theorem 4.3. 5 LX3,,""'(idq)1lSl1.
IlsCll
and the constant is best possible. For the worst case q = co and y = 1 the considerations in 4.3. on the finite dimensional Grotlie~ldicckconstant yield
-
a.
It might be difficult Krivine [15] calculated (actually for this prrrpose) that 111g(2)= to find the exact val~ieof Ljj",'3,,""'(zdR;) in general. For given (p, q) with 1 5 p 5 q 5 m
380
A. Detant, K. Floret
it is obvious that k,,, := LEiCsUr(idR;)5 LEi,(idR;) holds for the best "cornplexification constant" k,, , hence [20], 22.1.5, impl~est h e estimate
Since the operator ideal CE-', is surjective and k,,
= k,,, ,, by duality, one o l ~ t a i n st h e
Theorem: For 1 5 p 5 2
c ~ n d k 2 , = k,,
2
Jz
= --s
4
1, 11072
References
[I] Beckner, W.: Irlequalztzes
LIL
F o u r ~ e rAnalysis; Annals of Math. 102(1975)159-182.
[2] Bennett, G.: S c h u r m u l t i p l ~ e r s ;Duke J.Math. 44(1977)603-639. [3] Bergh, ,I.-J.Lofstrom: Interpolation spaces; Grundl.math.Wiss. 223, Springer, 1976. [1] Bourgain, J.: Sornc rernarks oil U a n a c h spaces Lrl whzch rr~arttr~galedzfference sequences are unconditional; Ark.Mat. 22(1983)163-168.
1 .51 . Burkholder, D.L.: A g e o i n e t ~ i c a lcondition that implies the e.zis1er~c.eof certain singular ~ n t e g r a l sof B a r ~ a c hspace ualuetl f ~ ~ n c t r o n sProc. ; Conf. IIarnioriic A n a l y i ~ sin Honour of Antony Zygmund, Chicago (1981)270-286. [6] Carl, B.-A.Defant T e n s o r products and Grothendleck type l n e q u a l l t ~ e sof operntorn ziz L,-spaces; to appear in Trans.Amer.Math.Soc. g of'finitt [7] Carl, B.-A.Defant: i l n inequality between the p-and ( p , l ) - , s u r i l n l i ~ ~r10rrr1 rank oprrators from C ( l i ) - s p a c e s : to appear in Israel J.1lath. r~; Olcl~libul~g. 1986. [8] Defant, A,: Pr,odukte (Ion T r r ~ s o r n o r n ~ rIIahilitationsschrift,
[9] Defa~lt,A,-Ii.Floiet. .-tsptct5 of the rrretrxc t l ~ ~ o rofy ferlsor. p t o d u c f s ar1C1 operator zdeals; Note di h l a t e m a t ~ c a8(1988)181-281 1101 Defarlt, A,-K.Floret: T e n s o r n o r m s and operator ideals; to appear Math. Studies/Notas de hlatematica.
i l l No1
th-Holland
[ l l ] Defa~it,M.: Or1 t h e vector. valued Hilbert t m n s f o r ~ n ;hIath.Nac111.. 141 ( 1!189)231-2G5.
h 'orrlt [12] Figiel, 'I'-'T.Iwaniec - A . Prlczyriski: C " o r r ~ p ~ ~ rt ~ no yr ~ nof' spaces; Studia Math. 79(1984)227-274.
LP-
O ~ P I Y L ~ OI17~ S
[13] Garcia-C'uerva, J.-J.L.Rubio de Francia: Weighted rlorm ineqnalities a n d inelated topics; Math. StudiesJNotas d e Maternitica 104, North-Hollantl, 1985.
38 1
Continuity of tensor product operators
[14] Grothendieck, A.: Re'sume' de la the'orie me'trique des produits tensoriels topologiques; Bol.Soc.Mat.Sio Paulo S(1956)l-79. [15] Krivine, J.L.: Sur la complexification des ope'rateurs de L" 284(1977)377-379.
duns L'; C.R.A.S.Paris
[16] Kwapien, S.: O n operators factoring through Lp-space; Bull.Soc.Math.France MCnioire 31-32(1972)215-225. 1171 Kwapien, S.: Isomorphic characterization of inner product spaces b y orthogonal series with vector valued coeficients; Studia Math. 44(1972)583-595. [18] Marcinkiewicz, J.-A.Zygmund: Quelques ine'galite's pour les ope'rations line'aires; Fund.Math. 32(1939)115-121. [19] hlanrey, B.: Tlkor6mes de factorisation pour les ope'rateurs line'aires ci valeurs dans les espaces LP; Asthrisclue 11, 1974. [20] Pietsch, A.: Operator ideals; Deutschcr Verlag der Wiss., 1978 and North-IIolland, 1980. 1211 Pisier, G.: Factorization of linear operators and geometry of Banach spaces; CBMS Regional Conf. Series 60, A~ner.Math.Soc.,1986. [22] Roscnthal, 1I.P.-S.J. Szarek: On tensor products of operators from LP t o Lq; in: Longhorn Notes (Austin), Funct.Anal.Sem., 1987-89. [23] Virot, B.: Extensions vectoricllcs d'ope'rateurs line'aires born& sur LP; C.R.A.S.Paris 293(1981)413-415.
A. Defant, K. Floret Fachbcrcich Mathematik der U~liversitat D-2900 Oldenburg Germany
Ilirs t l ~ t 3. ball ~ ~ . o l w for r ~ arbilrary y t:o1111)1t!)r: SIIX(.YS ill vvcry co111pac1( . U I I V ~ ' Xbe1 I\ c C' 10r wl~ichall f ( l i )arc disks for f E (c2)' r~ecessarily11- a ccnlcr of s y ~ ~ ~ r r l e l r y . ~
Cornpact t o ~ ~ v csets x li for whicl~all f ( l i )a t : disks will be said t o satisfy Lllc Yoat it was shown t h a t they need not have a centcr of syrnnictry scc also s c c t ~ o n3 1 ) so Lllal 11y Ll1c above r r ~ u l t st l l e ~ cis i l l fart also in rornplrx ~ ~ C Ca ScliKerc~~ctb e t w e r ~tllc 2 ball property and the 3-ball property I l o w c v c ~ the ,
I H O ~ I P ~I I~~ ~I YV ~J Recently
(['I,
S
~
Yost property implies some other interesting consequences for I i which we are goillg tir describe in this paper. They concern strong regularity properties o l t h e topological alitl colivex structure; details are provided in section 3 after some prepuatiorls i r i sectiorl 2. In section 4 we sketch an example of a set with the Yost propcrty but without points of symmetry, and finally in section 5 we indicate some open problems.
2 Preliminaries I,ef, I i C c2be a cornpact, convex and nonvoid w t . W h a t are the interior properties of I i which are special for the K with the Yost property? In order t o treat this problem it, is desirable to have the followi~lgdescription of li by means of lower dimensiorlal objects:
Let x : C' -+ C be the pvwjection ( = , t o ) H
2.1 Lemma:
2.
( i ) Let lhc jamrly ( I i z ) r E n ( h -be) defined b y I(, := {wI ( r ,t o ) E I < } . T h e n ( 1 ) li,,+,l-,)w > f l i , ( 1 - l ) l i w jor. z,ru E a ( K ) u n d 0 d 1 6 1.
+
( 2 ) :H
/c
0 then
YO
IS
lipper s e m r r o i i l ~ n r ~ o i r r. . ~ , r . rj a K,,
ra
co~rltrznt~lr r r
ILIL
ol1r11
.-I
I
is I i , for :CIOYC t o zo.
(ii) Converclely, lcl ah a n d ( ~ , ) , , be , , ironuord compact conver s u b s c t ~of c wlth ( I ) . ( I ) , a n d (4). T l r c i ~I i := { ( z ,w ) l z-E x h , w E I(,} dcjille5 a cornpucl roliucr aulrscl o j c 2 I U I ~ I A~ ( 1 =0 ~h u n d I\', = li, jar all :E r h . 'l'he e l e r n e ~ ~ t a rproof y is omitted. Now let K bc the collection of nonvoitl compact convex subsets of C , provided wit11 tlie Hxusdorff 11ietl.i~:
(where U(O,E) tlenotes the closed d i ~ kwith center 0 and radius
Let I i bc a.9 uboue. T h e n z
2.2 Lemma: ~(li).
I-+
E).
li, is c o n t i n u o u s o n llie i i i l c ~ ~ o or .j
Proof: Let zu be in the interior of K ( I ~i )~ i dE > 0. We have to show tliat t l l e ~ el h a neighl~ourl~ood I / of 20 such that I\', c h', D(0, E ) , liz, c li, D ( 0 , : ) for 2 E Li By ? . l ( i ) ( 2 ) we have l i : C li., U(0, E ) for z close to z o For the proof of the othel inclusion wr first show that z H l i , is lower scmicantinuouv a t zu, i.c. l i , n O # fl for z ill a iieigl~bourhoodof 30 if Ii,, n 0 # 0 a l ~ d0 is open.
+
+
+
U be given and choose wo E liz,n 0, rl > 0 with U ( w u 1, 1 ) c O. Sinrt> O such that the following holds: evrry 2 with 1: - zol 6 b hau .L ( 1 - 1 ) T wit11 5 E ~ ( l iant1 ) t E [l 11 (here H tlenotes arly 11~111lber tiucll that U(0. R ) bounds all h',). We claim that li, meets U for tllese z . Let such
a11
E r ( I i ) O there is a 6 representation 2 = t:o 3"
+
A,
In fact, if 6 E Ii; is arbitrary we know from 2 . l ( i ) ( l )tllat w := ltoo so that, since - wOl 6 11 - tI(I111~l li^ol) c 71, 111 E U fl I i - .
IW
+
+ (1
-
t ) ~ ^E o
li,,
385
Compact mnvex sets with the Yost property
To conclude thc proof let wl, . . . , ru, bc an e/2-net in li,. By t h e preceding step there is a neighbourhood Uof 20 such that K, n D ( w , , E / ~ ) 0 for p = 1,. . . , r and z E U. But this yields i m m e d i ~ t e l yK , C I(, $ D(O,E) for these z .
+
Note:
T h e proof shows even more: one haa c o n t i ~ ~ u i tayt every zo such that
l'liis is satisfied. e.g., for all zo if a ( / < ) is a square but only for Lhe ~ ( l iil l )the case of a disk.
ill
t h e interior of
-
d stressc(1 that rnntinl~itycan not br: g ~ l a r a n t e ~adt all zu in general: CarlIt s l ~ o ~ l lbc sider, for e x a ~ n p l eany T : {;I 1-1 = 1) K which is uppcr semicontinuous but not c o n t i n ~ l o ~and ~ s put I 8. Then them is an 7 > 0 such lhat s u p { R e ~ u I u ~l E i , ) > 6 i j l z - 11 < 7.
Proof: At first we will c o n r r r ~ t r a t eon tlic r close to 1 with jzl = 1 which wc will writ(% as c" with small real 0. T h c following elementary fact will bc of ilnportance: if 0 a11t1 Oo are given such that 0 < 1001 0.01 and 101 (= I sin81 Id - 0,I a 181 10 - 0,l) 6 2le0l 10 - e,I.
0.01, 10 - 001 (
IQOl,
then I cos 0
-
cos Oul
Now suppose that R e lie,% is strictly bounded by some a0 E [ - I , + I ] , i.e. Re u, < au for w t l i e n s , .By 'L.l(i)('L) t h e Rame is then true for tlle R e I i , with Iz - eleol 6 26, say, whcrc 6 > 0 and w. 1. o. g. 26 < IOol. If thcn 0 satisfies lei'-eiB.I 6 3~ for a n E < 613 the11 10 001 < GE < Ieol so that cos 0 6 cos Oo 1210olc; more generally Kc z 6 cou 0, 121U01c whenever Izl 1 a11d 1 - e l k I < BE (note that the m a x i m ~ l mof the functio~iz HL I O I I { z l IZ 6 1, Iz eiB.I < 3 ~ is) acliieved itt a point of t h e form elB).
-
+
-
+
-
< 1 be given
Let any ,- with lzl
;1
Case 1:
- eiBO1
and
E
6 613.
a 3i.
'l't~en,for w t I i L , wc know that Ilr w < nuSO that I l e ( 2 in partict~lar,cne0 E(ao 12(Q01)$ 2 + e l < , .
+
+
+ E W ) < HeetBu+ 1 2 ~ 1 0 +~ 1Eno;
:1 - L"O( > 3 ~ . \Vt: hair(. lcrol 121Uol 6 2. 1Ie11ct.1- - r A B o ~ ( o u 121001)l > c, itlld t111isalso : EI\.=tlocs not c o ~ ~ t ar 'iO~O l c ( n u I2lOUl). Case 2:
+
+
+
+
+
+
ill
t l ~ i scay,.
S u n ~ n i i r ~N[I, g we have showl~so far that
B such t h a t there exists a t u t I\'l wit11 and choose a 0 with Po > n o > p . Wc claim that sup{He wluj E I i 2 } LIU for all z with Izl = 1 i t ~ i t l1s - 11 small. To prove this we have to make uge of We now turn t o the proof of 3.1. I"ix any po >
>
Rr w =
('1
There is a constant h.I > 0 with ttlr following property: Wherlever Ou m d : givrr~S L I ~ I tIl ~ a t o, < i < 1~~ 6 0.01 one has nil Jr' (sine. 2 (cos oU)(1 +c/jU) - AICI sill uOl; 11crcr n l , n t 2 ,r d e ~ i o t ethe r(aa~11um11ersS L I C I I L I I ~ I , Ac = D ( m l i 1 n 2 , I.).
+
arts
-
+
i (*),
t o u whenever 1 ~ ' '- 1 1 6 6. 1'111 ij := \VC will show that silp Rc Suppose that this werc not truc. Wc then could, by ( I ) , find a Ou # 0 wltll ItAB0 - 11 < IS A I I ( ~ I t l ~ a tr h 5 :(nu 121001) $ Ac for sufficiently small E . Uut on t h e othcr hand. LY ('21, ( [ ( I t E , ~ U ) C O-Y ~~ hOf I s 1 i l O ~ 1 1m l ) ' (si1i6'~ ~r12)'< r A ,
+
+
+
i.e. riBO
+ ( / j O c o s UU
Sincc A,
-
-
M sill OU)c t A,.
n (ciao + f?) i~ convcx this irnplirs (/I0 cos Oo
-
A1 sir1 On)& < ( 0 0
+ li!10u1)E7
and this l c d s aftcr sornc c l c r n c ~ ~ t a rcalculatior~ y to a contradiction to our choice of UO.
387
Compact mnvex sets m'lh the Yost property
It rc~nainsto co~lsitlerull 2 in D ( 0 . 1 ) with :1 - 11 slnall (arld llot orlly those with IzI = 1 ) and to prove (2). To fill t h e first gap let z E U ( 0 , I ) with Iz - 11 6 7 := ij/2. We may write r = I-.Izo (1 - [zI)O, where 20 := is +close to 1 . By t h e preceding 1-1 invcvtigations we have Re li, ao. Thus, with an arbitrary r7, E Iio alld any wo E Ii,, with He t u ao, w := lzlwo (1 - Izl)G E li, ( 2 . l ( i ) ( l ) )and Re w (1 - rl)ao - 11 or Re w no - q if a 0 0 or Q 0 respectively. Since both ( 1 - q ) a o - and au - are larger than by t h e choice of q this leads to sup Re li, ,&.
+
>
>
+
>
+
>
>
>
Finally we prove ( 2 ) . Let c and Oo be given aa in thitr rtssertion and write A. as D(ri11 irn2, r ) . Tllcn, since sup{& w lw E K l } ,& and D ( 0 , l - e ) c A, c D ( O , l + E ) it follows that
+
(3)
>
Irllll, lmal
< E , 1. E [ I - E , 1 t E
] , ~t I1.
>It
EBO.
Choose. an M I > 0 with
for 0
< c < (11 < 0.01.
'I'he iollowing steps are easy consequences of (3):
(I'or a suitable
M 2 and
c, t
M
abovc) ;
(for a suitable
M3 and
c, t
ay
above) ;
(for a suitable
AI.,
E,
and
t
as above),
and this ~ i e l d a( 2 ) ii we put t = sill 00 (note tbat, by (3).
( E )2 ~
(-)2).
3.2 T h e o r e m : Let li be a compact convtlz set with the Yost p r o p e r t y . Then, wrth lhe notation ojsection 2, z H lir is continuow on A(/ b } , and by applying 3.1 for the family ( e ' P I i , ) I , I ~we l conclude that t h c same is true for all halfspaces.
Now let E > 0 be arbitrary. Sirlce upper semi-continuity always holds (2.1(1)(%))wt. only have to find a 6 > 0 surh that D(0, E ) Ii, > Iil for Iz - 11 6 6. To this end choosc: an open set 0 > KI and open halfspaces H I , . . . , H, such t h a t t h e following holds: H, n A', # 0, and whenever y, E H J n O are given then U(O,&) (4) C O { Y I , . . . , Y 3~ 1} i l . (Choose e.g. strongly exposed points 21,. . . , z, E K 1 with Iil = co (21,. . . , z,}, T > U and exposing normalized functionals f l , . . . , f, such that s, := sup Re j,(A') = Re j , ( z , ) , diarn {zlz E l i l , Y, - T ( Re f,(z) ( s,} 6 E; then 0 := 111' D(O,T/'L)' iuld HJ := {zlRr j , ( z ) > Y, - T I ? ) have the required properties.)
+
+
+
By the preceding c~nsiderations and ? . l ( i ) ( 2 ) we find a 6 > 0 such that, for 11 6 6, Ii, C 0 and I\', f l H, # 0 for all j. ( 4 ) immediately implies that the11 lil C I
From
1
+ ~ a ( 0+) i 6
1
+
.5'ri,"'+.
=
we now sty that, :
Ell1
3.5 Proposition: u(U)= ,tu cos(0 - Uu)
E
+ ~ ( c ~ (-0 613) ) + ~6 1 + ~ c r ( 0+ ) $6 2 1
7.
'llitrr u1.c pu L 0 , Uo E j o r U E [O,:'TI.
+ r =: D,,h,,(U)
[ O , ' L r ] , und
I.
E
R such Lhut
Proof: 1,et 3 c C [ 0 , 2 ~ be ] the c,ollcction or all /jM,oo,,. It is easy to see t h a t 3 is closed. ; r r ~ t l 11enc.c it suffices t o show tllat n can bc arbitrarily well approximated by thc elcmcnts ol 7 . '1'0 l l ~ i n erld. Ict 6 > 0 I)? give^^. Wc cl~oosc EU > 0 as i l l 3.4 ; ~ n d fix any L l l c * disk A, = I ) ( I I L ( E ) , I . ( E )IIPS ) b e t w r c ~.St,~ and ,qz-+, a ~ it ~ d follows [lorn r l r ~ ~ ~ c ~c ~o n~ st iadrcyr d ~ o ~l l~~sa t l ~ c n with , I I I ( E ) = Geton, I &n(U)- 2 ~ $ 6
r € ] O , i u ] . Hy 3. 1 T(E) 1,
+ El
.= (7.(c)
C U S ( ~-
-
OD)
l ) / ~ .
6 1
+ e n ( 0 ) + 2 ~ 6 .llrnrc
+
110
- @,,,,8,,,))
6 26, wl~ere/ d o .=
-
v 7
I hc cxxlsequcl1ces of 11r0110sitio113 . 5 are c-ollectt~li l l
3.6 Theorem: T , P ~ Ii be a con1l)act col~vexsubset of C' wit11 thc Yost prop(-rty nucl~ tllat 7r(I
+
+ &, arld
+
lie(t) := r ( t )
+ m(1)cnsB.
a t lrast for crveli s n ~ a l l e rllnrllj
the iwsociated sct I U such thal t h r Ii E KV,p,whicli are I)oundl:d i r ~norm 11scmc, say, can I>e 71-irpproxi~riatcdby a x r t i l l L S . ~(or, l . w l ~ a t~ I ~ I O LLOI ~I(~ R t l ~ esanlr, t h r tliarneter of the range! or tllr function r H rnitlpoint of j l , , ( l i ) is always small).
+
A sec.ond problem ~ l ~ o ~b(* i l ~~lclllionecl. d Since by .I. 1 tlic. Yost property docs not illiply sy1r1metl.yonr wo~lldl i k ~to Iiave a11ntlditio~~al u ~ ~ a t , ~ ~sufficient rill" condition which could c*wily IN. for~r~ulatcd i r ~t ( . r i ~ ~ ofs t t ~ c . c.ollvcx ge!olll(-t.~.yof li.
392
E. Behrends
References [1] E. hl. AI,FSI;:NA N D E. G . EFFIIOS.S t r ~ ~ c t a r111e rcnl Banach spacts. Pnl~tI c ~ t r c l II. A n n . of M a t h . 96 (1972), 98 - 173.
I'[
E. B E I I R E N IP) ~o ~. ~ z of t s symmetry of coil11fr s p f s I T Z the triro-d17r~enororrcLIcomplcs ( I ~ o u i ~ f e r e x n n z pto l e D. Yost's problrrn. (Preprint.).
q~om
[3] A . L I M A .I1rtc7.sccf1onp~opertztsof balls and subspaces zn 13anach spaces. T r a ~ i s . Amcr. M a t h . Soc. 227 (1977), 1 - 62. [4] A . LIMA. Conaplt~.rBnnnch spaces whose d7~alsa7.e I,' Matli. 24, 1976, 5:) 72.
spacts. Isldcl Jouillal of
Progress in Functional Analysis K.D. Bierstedt. J. Bonet, J. Horvath & M. Maestre (Eds.) O 1992 Elsevier Science Publishers B.V. All rights reserved.
S O M E K E M A K K S ON A L T M r T CLASS O F A P P R O X l M A T l O N I D E A L S
Fernando CobosX
N
and
*
Thomas ~ u h n
' ~ e ~ a r t a m e n L od e Matemhticas, Facultad de Ciencias, Universidad A u t h n o m a d e M a d r i d , 2 8 0 4 9 M a d r i d . S p a i n . HH
Sektion Mathematik, UniversiLat
I,eipzig, 7 0 1 0 L e i p z i g .
Germany. D e d i c a t e d Lo P r o f e s s o r M . V a l d i v i a o n h i s 6 0 L h b i r t h d a y
Abstract We g i v e s o m e r e s u l t s that s h o w t h e d i f f e r e n c e b e t w e e n approxination ideals
A,,,and
c1;lssicsI 1.oreritz i d e a l s Z L l t q .
Lel FI be a H i l b e r t s p a c e and let '1':H l i n e a r o p e r a t o r . D e n o t e by o p e r a ~ o r sucli that
I TI
=
['1'*'1']
IT/'=T"'~. Then
Consequc,ntly the spectr-unl of
112
1.I-I
f I be a c o m p a c t t h e u n i q u e posit ive
i s c o m p a c t and p o s i t i v e .
1'1') i s n o n - n e g a t i v e , anti a l l iis
non-zero spectral values a r e eigenvalues, having no accumulat i o n point e x c e p t p o s s i b l y z e r o . W e c a n o r d e r t h e e i g e n v a l u e s w i t h r e s p e c t to d e c r e a s i n g v a l u e s , r c p e a t i n g e a c h o n e a s many t i m e s a s ~ t s( a l g e b r a i c ) m u l t i p l i c i t y i n d i c a t e s ,
TI
IT1 lids o n l y
f l r ~ l t e rlunlber rJ of e l g c n v a l u e s , t h e n w e
c o n ~ p l e t e t h e seclucncc b y s e ~ ltn g X (
I'rl)
=
0 for
11
> N.
F. Cobos, T. Kuhn
The n - t h
s i n g u l a r number
In t h e t h e o r y of r o l e i s played bility
T i s g i v e n by
of
o p e r a t o r s on a H i l b e r t
space,
a major summa-
by o p e r a t o r s p a c e s d e f i n e d by m e a n s o f
c o n d i t i o n s on t h e s i n g u l a r n u m b e r s .
So L o r e n t z o p e r a t o r
spaces a r e defined a s follows
Mere 0 < p
find
0,
< E
(f)
2
+
E
[14].
inequality
Let c i n
1
~
X , % re-
~
3
40 1
Some remarks on a limit class of approximation ideals
Hence, using
( I ) , ( I I ) a n d ( r I I ) , we g e t
E :!2n ( J ( f , g ) )
5
=
5
IIJ(f,g) - J(fO,gO)IIZ
lIZ
I I ~ ( f - f ~ , g+ )J ( f 0 9 g - g O )
c 3 ~ [ I I f - f O l I X 1 1 ~ 1 1 +Y
5 c3M[(E
(f)
+
€)llglly + c l ( E
2" 5 c3M[(~
lIXII~-~ollyl
IIf0-fff 2
(f)
+ e ) lIglly +
c1(211f
( f ) + E + I l f l l y ) ( ~" ( g ) 2
lly
+ E)(E
2
( g ) + E)]
+€)I .
2
P a s s i n g t o t h e l i m j t when
E
+
0 w e o b t a i n ( x ) w i t h c=2Mc c 1 3'
Now we h a v e
where
M1
is a
Lhe p r o o f .
//
constant
independent o f
f
and
g.
This conlplctes
F. Cobos, T. Kuhn T h e o r e m 3 i m p r o v e s [l],
Thrn.3.2, w h e r e t h e s a m e c o n c l u s i o n
w a s o b t a i n e d a s s u m i n g that the g r o u p Z i s normed and that O < q 5 1 . As a d i r e c t c o n s e q u e n c e of T h e o r e m 3 w e h a v e that i d e a l s a r e t e n s o r s t a b l e for any 0 < q
I
X2(T)
I > ... > 0 ,
counting multiplicities; s e e (C71, 3.2.20). The formula ( X , , ( T ) )E I ,.,
x , , ( T ) )E I,.
(n"r
means t h a t
The following criterion is d u e t o G. Pisier ( [ I l l , Theorem 2.9) f o r p = 0 and t o S. Geiss (121, Prop. 3 . 5 ) for 0 < p < 1/2; s e e a l s o (101, Theorem 5.5).
Theorem. Let 0 < p < 1/2
and
1
< I-
1 such that for every I , , , m i n zd the c o i ~ d i t i o n
iinplies
1~ n ( n ) Q ~ ~ " ( i zo) ( a ) ~ ~ ~ ' ~ ( i r <x )c'.l ' -
nES
For thc proof cf. [P-W],Prol~osition1.1.
2 P r o o f of t h e 1mai11 r e s u l t
First wc, ~ e c a l la wrll kllown constr~~ction essr~ltinllydue to Fouriiicr [F]. 2.1 L e m m a . Let (rrlk) C Zd satisfy
426
A. Pelczyriski, M. Wojciechowski
T h e n for every positive integer N and for arbitrary complex n u m b e r s c l , cz, . . . , c~ there exists a trigonometric polynomial F : T d + C s u c h t h a t
(2.2)
~ ( r r z k= )
ck for k = 1 , 2 , . . . , N ,
(2.3)
I~(t)l'
e
N
x / c k J 2for t E T ~ ,
k= 1
x N
if
P(m) # 0
t h e n rn =
71;iizk for s o m e k= 1 71,712,. . . , T N s u c h t h a t 7k E ( 0 , -1, + 1 ) for k = 1 , 2 , . . . , N = 1. and if ko = max{k: 71; # 0 ) t h e n
(2.4)
Outline of the corlstruction. -1/2
N
If E k = ] ckI2 = 0 we put F = 0. Otherwise put 71 = ck ~ ~ ~ ( : ~=l ' ) 1 for I; = 1 , 2 , . . . , N and defirie functiorls Fo, F l , . . . ,F,, and G o ,G I , .. . , G,, by the recursive forrnulae Fo = 0; Go = 1 Fk = Fk-1 yk exp(i(., I , L ~ ) ) G I ; - ~
(
+
G I , = Gk-1
-
yk cxp(-i(.,
Then
+ IGkI2 = ( 1 + N Therefore IFNI2 < n ( 1 + l y k 1 2 )
+ [GI;->1' )
IFk12
k= 1
? ~ z ~ ) ) F k( -k ~= 1 , 2 , . . . , N ) . ( k = 1 , 2 , . . . ,N ) .
5 e . Finally we put
Then F satisfies (2.2) - (2.4). For details see [F].
111the next proposition we overcorne essential technical difficulties 2.1 Proposition. Let S satisfy
c Z"
be a n arbitrary smoothness and lct a sequence lim iilf k
l
