WEBBED AND

VOL. 14 NO. (1991) 17-26

RESEARCH PAPERS

MACKEY CONVERGENCE AND

QUASI-SEQUENTIALLY

WEBBED SPACES

THOMASE.GILSDORF

Department

of Mathematics/Computer

Systems

University of Wisconsin-River Falls

RiverFalls, Wisconsin 54022 (Received April 5, 1990)

ABSTRACT;. Theproblemof characterizing those locally convexspacessatisfying theMackey convergence condition is still open. Recently in [4], a partial description was given using compatiblewebs. Inthispaper,those resultsareextended by using quasi-sequentially webbed spaces (seeDefinition1).

In

particular,it isshown that strictly barrelledspacessatisfy theMackey convergenceconditionandthat they areproperlycontainedinthesetof quasi-sequentially webbed spaces.

A

relatedproblemisthatofcharacterizing thoselocallyconvexspacessatisfying the so- called fastconvergence condition.

A

partial solution to this problem isobtained. Several

examplesaregiven.

KEY

WORDS

AND

PHRASES: Webbed space, quasi-sequentially webbed space, Mackey convergence,fastconvergence,localcompleteness,inductivelimit.

1.980

MATHEMATICS

SUBJECT CLASSIFICATION CODE

(1985Revision):

Primary 46A05;Secondary46A30.

1. Introduction and Definitions.

In [7],28.3, Krthe pointedoutthatacharacterizationoflocally convexspacessatisfying the Mackeyconvergencecondition(seedefinitionbelow)didnotexist. Thisproblemisstillopen. In [4],apartialsolution isgiven, usingspaceswithwebstructures. Inthispaper,thoseresultsare extended. Also,the relatedproblemofdescribingthoselocally convexspacessatisfyingthefast convergencecondition(def’med below)isbriefly examined. Severalexamplesare givenalongthe way.

Throughoutthispaper, Ewill denoteaHausdorfflocallyconvexspace. IfAisasubset of

E

whichis absolutelyconvex, we willcallAadisk, and we letE

A

denote the linearspan ofA, endowedwiththetopology generated bythe Minkowskifunctional ofA. When

A

isbounded, EA^isanormedspace,andthenormed topology^isfinerthanthetopologyinherited fromE. If E

A

^{isaBanachspace,wecallAaBanach}

disk. A

locally convexspaceislocally_ complete if each closed, boundeddisk is aBanachdisk. If(x

n)

^{is asequenceinEwhichconvergestox in}

the normed space EA for some bounded diskA, we say that (x

n)

^isMackey (or locally convergent tox. Incase x 0,wesay(x

n)

^isMackeynull. Becausealltopologiesinvolved hereare translation invariant, we willalwaysusenullsequences. Also, it is obvious thatevery locallynullsequenceis anullsequenceforthe original topologyonE. Wheneachnullsequence islocally null,wesaythatEsadsfiesthe Mackeyconvergence

condition.

^If

(Xn)

^{is aMackeynull} sequenceand thecorrespondingdiskAis a Banach disk,

(Xn)

^isfastconvergenAto0. Ifevery nullsequenceisfastconvergentto0, thenEsatisfiesthe fastconvergencecondition.

Wewillalso need the following information^onwebs. Detailed discussions ofwebs maybe foundin [3], [10], and [11]. A web on a locallyconvex spacewill be denoted by ^/. A sequence

{Wml,m2 mk:

^{ke N}} of members from aweb

’

^{such that}

^Wml

^mk+l

Wml

mk^iscalled astrand. Forconvenience, we willdenoteastrandby

(Wk).

Also, weassumethroughoutthispaper, thatforastrand

(Wk)

^of

Wk+l

^c

- ^Wk.

IfWisawebon alocally convexspace E,wesay that^,@iscompatible

with

^Eifgivenany

zeroneighborhoodUinE,andany strand(W

k)

^from

,,

^thereexists keNsuchthatW_k^cU.

A compatible web ,#iscompleting ^ifforeach strand

(Wk)

^{from 74/and for each series}

webbed.

^{If is}completing and foreachstrand(W

k)

and each series

klXk

^{with xk eWk,we}

have

_r=k+[ x

^{e Wk_1,}

then

W

is and wesaythatEisstrictly webbed,

2. Ouasi-Seo_uentiallv^WebbedSpacesandMackey

Convergence

Oneobviousproperty of compatible websisthatthe members ofanystrandcan be made smallenoughtofitintoanyzeroneighborhood. If each nullsequencehas the property thatits membersare eventuallycontained in a f’mite collectionof strands, thenitappearsthatthesequence isconvergingwith respecttoafinertopology. This istheidea behindsequentiallywebbedspaces and quasi-sequentially webbedspaces.

Definition

^{1; Let Ebea}locally convexspacewithacompatible web,@. ThenEissequentially webbediffor each null sequence

(Xn)

there exists a finite collectionofstrands

{(Wk(1)), (Wk(2)) (Wk(m))

^from

’

such that for each ke NthereexistsN_ke Nsuchthat

m xne

U Wk(i)

i=l

for each n> N_k. Ifwehavethe weakercondition,that foreachn>

Nk

m xne

U Vk(i),

i=l

where V_k is theclosed, convexbalanced hull ofW_k, then wesay Eisq.uasi-sequentially webbed.

Remark._.__.’

Every

sequentially webbedspaceisquasi-sequentiallywebbed. Anexampleofa quasi- sequentially webbedspacewhich is notsequentiallywebbedisgiven below.

Example 1" LetE

tl/2,

withthenon-locally convex metrizabletopology generatedbythe decreasingsequence

{Wn:

neN}. Itiseasy^toshowthat 7’ {W

n"

^ne

N}

isacompatible web for

(E,).

Let r denote thetopologyonEinducedby thenormedtopology

on/1. ,

is thenacompatible web forrl^also. Nowlet^7,#

{W’n"

ne N},where

W’n

^{isthe /l_closure}

of the convexbalanced hullofWn,for each neN. Since

’

forms a baseof closedzero neighborhoods forrl, (E,rl) isquasi-sequentiallywebbedunder^7.g Onthe otherhand,we may pick_xne

W’n\W

n for each n,sothat

Xn

^{0, but isnot}contained in the(only)^strand

(Wn).

Hence, Ewiththeweb7,isnotsequentially webbed.

Inthefollowing proposition, let

{En:n

eN} be a collectionoflocallyconvexspaceswithE_n

En+

^{for eachn, and}each injectionid:

EnEn+

continuous. Then we writeE

indnlim

^En torepresentthe inductive limit ofthespaces En. Aninductve limitE

indnlim ^E

n issequentially retractive if foreach convergentsequence ⁱⁿ Ethere is some neN such that the sequence convergestothesame limit inE_n.

Theorem1:

(a)

Every

metrizablelocallyconvexspaceisquasi-sequentially^webbed.

(b) LetE

indnlim

^Enbe sequentiallyretractive, witheachE_nclosedinE. If each E_nis quasi-sequentially webbed,thensoisE.

(c)

Every

strict(LF)-space^isquasi-sequentially webbed.

(d)

Every

subspaceof aquasi-sequentially webbed space^isquasi-sequentially webbed.

Proof:

(a) Let

’ ^{Un:

^neN}beabase of zeroneighborhoods ofthe metrizablespace E consisting of absolutelyconvex,closedsetsUn,with

Un+l

^{c 1/2U}n. Clearly, isa compatible web forEandEisquasi-sequentially webbedunder

(b) Let E

indnlim

^Ensatisfyingthehypothesis. Foreachn,let

(n)

denotethe webon E_{n. Define}the web

,, {Wm mk}

^k,m

...

^{mk eN} onEasfollows:}

Let

{Wm

_1"m eN}consist of the collection

{w(n)/l"/’1

^{,neN}wherethe}subscripts areput inone-onecorrespondence.

Let

{Wm

1,m2:ml,m2 N} bethecollection

{w(n)l,/2:/1,/2,

^{n N},and soon. Itiseasyto}

verifythat

’

^{is aweb onE,andWiscompatiblewithE}by [4], Proposition 9.

Nowlet_xm^---)o inE. Then_xm oinE_nfor some neN. Hence,there are!strands

(Wk(n’l)) (Wk(n,/))

in

74(n)

_{suchthatforeachke}

N,

there is

Nk

^eNsuchthat

x

me ^{U Vk(n,i)}

i=l

for each m _>N_k,where

Vk(n’i)

^isthe

_En-closure

^ofW

k(n’i)

^{for t’andkeN. Each}

Vk(n,i)

^{isclosed inE}_{n which in}^{turnisclosedinE;hence,eachV}

k(n’i)

is closed inE.

Moreover, bythe constructionof 7’onE,each strand of

74/(n)

is a strandof7,#. Hence, Eis quasi-sequentiallywebbed.

(c)

Every

strict(LF)-spacesatisfied theassumptionin(b).

(d) Let

E

be aquasi-sequentiallywebbedspaceandletFbe asubspace^ofE. If^,#isthewebon E,define_/’=the web

"

^onFby

{Wml

mk F"k,m mkeN}

Itis routinetoshow that4/’isa compatible webonF. Next,ifxn^---)o inF,x_n^---)o inE,so therearestrands

(W

K(1))

^(W

K(m))

^c

suchthatforeach ke

N,

thereis

Nke

^{N sothat}

m Xn e

U Vk(i),

i=l

whereV_kisthe E-closure ofthe convex,balancedhull of

w(i)

_k

Hence,

x

ne

^(U

mvk(i)

) F i=l

U^m

(Vk(i)

^F)

i=l

U^m

Vk(i)’,

i=l

where

Vk

^(i)’isthe F-closure of theconvex,balancedhullof

Wk

^{(i) F.}

ThisshowsthatFis

quasi-sequentially

^webbed.

Remark:

^{If all thespaces E}_n^{in(b)are infact}sequentiallywebbed, then the assumption^thateach

E

_n isclosedinE maybedropped,since in this case,Eissequentiallywebbed by [4],

Proposition^9.

Theproofof ournextresult is similartothe one for[4],Theorem12;itsproofislefttothe reader.

Theorem

2:

Every

quasi-sequentiallywebbedspacesatisfiestheMackeyconvergencecondition.

RCmk:

The motivation fordefining quasi-sequentiallywebbedspacesis in thetwocorollaries below;theyallow ustoenlargethe list oflocallyconvexspaceswhichsatisfytheMackey convergencecondition. First, we need thefollowingdefinition,givenrecently byValdivia 12]in connection withthe closedgraph theorem. Inthiscontext,aweb is ordered ifgiven arbitrary positive integersk,_{ml mk}and_{nl nk}such thatmi-<ni for k,then

Wml

^mkc

Wnl

nk

Definition2:

A

locallyconvexspace Eisstrictlybarrelledifgivenanyordered andabsolutely convex web^74/=

{Wml _mk"

^{k,m m}k eN} onE,thereisasequence(m_n ofpositive integerssuch that W

ml

mn

^{isa zero}neighborhoodinE,for each neN.

Remark: Strictly ban’elled spacesare studied in detail in Section6of 12]. Wenotehere that strictly ban’elledspacesincludeunordered Baire-like spaces properly. (See 12]again).

Corollary_ 1"

Every

strictly barrelledlocallyconvexspacesatisfiestheMackey convergence condition.

Proof: Let Ebestrictly barrelledand let74’ by any orderedandabsolutelyconvexweb onE.

Define the web^7, by

7,0"= {2^-k

Wm

1,m2 ^{mk:k, ml,m2 mke} N}

ThenW’isanotherordered, absolutelyconvex web onE,which satisfied the condition

Wk+l

^c

- ^Wk

foreachstrand(Wk)of74 BecauseEisstrictlybarelled, thereisastrand (Wk) of

’

^such

that Wk is a zeroneighborhoodofEfor each k

:

N. Thus,Eisquasi-sequentiallywebbed.

Corollary2: IfEisa Bairespacewith acompatibleweb, thenEsatisfiestheMackey convergence condition.

Proof:

By

lemma2 page 158 of 11],ifEisa Bairespacewithacompatibleweb^7,0’thenthere is a strand

(Wk)

of 4 such that Wk isa zeroneighborhoodⁱⁿEforeach keN. ThismakesE quasi-sequentially webbed.

Wewill now obtain apartialconversetoTheorem2. Wenotethat in[4],Theorem18it is shown that ifEislocallycomplete, strictly webbed, andsatisfiesthe Mackey convergence condition, thenEissequentially^webbed. Wewillgeneralizethis. First, we needtointroducethe following:

Definition 3: A locallyconvexspaceEislocallyBaire if for eachboundedsubsetAofEthere exists aboundeddisk B^c

A

such that

EB

^{is a Bairespace.}

Rem..ark: Every

locallycomplete spaceislocallyBaire. Moreover,in[2],page3-4, example 6,an example ofanormedBairespacewhichisnotcompleteisgiven; thisrepresents alocallyBaire spacewhich is notlocally complete. Also,anystrict(LF)-spacerepresentsanexample of alocally Bairespacewhich is neither Baire nor metrizable.

Theorem

3:

Let E

be webbedandlocallyBaire. IfEsatisfies theMackey convergencecondition thenEis

quasi-sequentially

webbed.

Proof: Letx

n-o

^inE. Using Kothe [7], 28.3,there is asequence

(rn)C(o

^,oo)such that r

noo

^as

n--,,*,and

rnxn---o

^{inE. Let}

^A {rnxn:n

eN}. ThenAisboundedsothereexists abounded diskB

A

suchthat

EB

^{isaBairespace.} The injection

id:EBE

is continuousso ithas aclosed graph. IfEiswebbed using 74,’, then using Theorem 19, cor. page722of[10], thereisa strand (Wk)c7’and thereisasequence Ctk of numbers such that

idB B_{c Ok}

-k

foreach k eN. Hence,foreachneN

rnXn

^{etk Wkc tXk}

^Vk

whereV_k convbal

(-Wk) convbal(Wk)

so that

Ikl

< for eachn>

Nk.

rn

Thenwehave

Thus, for eachfixedkeN,wefind

Nk

^eN

ICtkl

Xn e

n ^Vk

^c

^Vk

sinceeach

Vk

^isbalanced.

Corollary_" Let EbelocallyBaireandwebbed. ^ThenEsatisfies the Mackeyconvergence conditionifand onlyifEisquasi-sequentiallywebbed.

Example2: We mayusetheabove resulttofind anexampleofaquasi-sequentially^webbedspace whichisnotstrictly barrelled. Let

E

be thestrongdual of a FrEchet-Schwartz

space.

Then

E

is complete, hence locallyBaire. Moreover, Eiswebbed by Proposition 2page157 of 11]. Infact the webWon

E

isdescribedasfollows:

Let {Un:n e N}beadecreasingbase of zero neighborhoods for the Fr6chet-Schwartzspace Fsuch thatEisthe strong dual ofF. Wedefine{Wm

1"

^ml,eN}

{Un:

neN},where

Un

^isthe

polar of

Un

^{foreachneN. Thendefine}

{Wm_1,m

2"ml,m2eN}=[ Un’neN},

and,ingeneral,

k,ml,., mkeN}

{Wm

mk _f

Un’neN},

Clearly,

’ ^{Wml

^{mk k,ml mk eN}isanorderedweb onE}consisting ofabsolutely convex,closedsets. Furthermore,Eisnotnormable,so none ofthemembers of74/canbe zero neighborhoodsinE,whichmeans thatEcannotbestrictly barrelled. Finally, by 12.5.9of[5],E satisfiestheMackey convergencecondition,sobyTheorem3,Eisquasi-sequentiallywebbed.

3. The

Fast Convergence

Condition.

In [6],asequenceis definedtobe fastconvergent in alocallyconvexspace

E

ifthereisa compact diskBinEsuchthat(Xn)is convergent in

EB.

This differsfromour definition,but this difference iseasilyrectified intheorem4 below. Also,in[6],it isshown thatalocally complete bomologicalspacesatisfiestheMackey convergencecondition ifandonlyif itsatisfiesthefast convergencecondition. Inthis section wewill showthat "bornological"maybe removed from that statementandthattheonlydifferencebetweenalocallyconvexspacesatisfying theMackey convergenceconditionand onesatisfyingthe fastconvergencecondition isthepresenceof local completeness. Finally,wewillmakesomestatementsconnectingthis section with theprevious section.

Theorem4: Let Ebe alocallyconvexspace. Thenthe followingareequivalent:

(a) Esatisfiesthefastconvergencecondition.

(b) Foreachnullsequence(Xn)inEthere is a compact diskKinEsuch thatXn--)o in

EK.

(c) Eislocally completeand satisfiesthe Mackey convergencecondition.

Proof: Toshow(a)(b),letXn--)oin

EB

^{whereBis abounded}Banach disk. Then since

EB

^{is a}

FrEchetspace, 35.7,(4)of[8] applies; namely,thereisa compact diskKsuch thatXnOⁱⁿ

EK.

Next,toshowthat (b)=,(c)notethatundertheassumptionsin(b),weneedonlyshowthatEis locally complete. Todo this,weuse5.1.11of[9],where it is shown thata locally convexspaceis locally completeiftheclosedabsolutelyconvexhullofeach nullsequenceiscompact. Let (Xn)be a nullsequenceinE. ThenXn---)oin

EK

for some compact diskKinE.

EK

^isaBanach space ([9], 3.2.5)so if

A

denotesthe

EK

^{closureofconvbal({Xn:ne N}),thenA}is compact in

EK.

Sincethe injectionid:EK---)Eiscontinuous,AiscompactinE,too. The assertion isnowobtained byshowing that theE-closureof convbal({_Xn:neN })isA. Denoteby

Ao,

theclosure of

A

inE.

SinceAisboundedin

EK,

thereisa

,

>osuch thatAc,K. kKisclosedinE,so wehave

Ao

^c

,K c

EK.

^NowtakeXo e

Ao

^{andlet(xct)beanetinAsuchthat}xct---)XointhetopologyofE.

Notethatid:

EK

^Eis continuousand that{n" K:ne N}is abase ofzeroneighborhoodsfor

EK

consisting ofsetsclosedinE,hence alsoclosedinthelinear hull ofK. Thus, by 3.2.4 of [5],

Xct

---)Xointhetopologyof

EK.

Moreoever, AisEK-closed,soXo eA,whichshowsthat

Ao

^A.

Finally,(c)=,(a)is obvious.

Corollary1:

Any

locallyconvexspacesatisfying^thefastconvergence conditionislocally complete.

Corollary_ 2:

Any

metrizable,incomplete locallyconvexspacesatisfies theMackey convergence conditionbutnotthefast convergencecondition.

oof:

Any

suchspaceisquasi-sequentially webbed byTheorem (a),butcannotbelocally completeby ], II.2.

Corollary_ 3: IfEislocally complete, thenEsatisfies thefast convergencecondition if andonlyifE satisfiestheMackey convergencecondition.

Thenext tworesults are combinations of Theorem4above and resultsfromtheprevioussection.

They givecharacterizationsoflocallyconvexspacessatisfyingthefastconvergencecondition for the case wherethespacesarewebbed.

T.h.eorem

^{5: LetEbe}awebbedlocallyconvexspace. Thenthe followingareequivalent:

(a) Esatisfiesthefastconvergencecondition.

(b) Eislocally completeandquasi-sequenti’,dlywebbed.

This is animmediateconsequenceofTheorems2,3,and4.

Forthe resultbelow,werecallthat an inductive limitEoflocally

convex spaces isgulrifeach boundedsetinE iscontained in and bounded in oneofthe constituentspaces.

Theorem

^6" Let

E

be aregularinductive limitof locally completewebbedspaces. Then the followingareequivalent:

(a) Esatisfiesthe fast convergence condition.

(b) Esatisfies theMackey convergencecondition.

(c) Eisquasi-sequentially^webbed.

Toshow (b) <=>(a),it sufficesby Corollary 3 of^{Theorem 4toshow that}Eislocally complete. Hence,letAbeabounded subset ofE,whereE

indn

^lim

En.

^{ThenAisboundedin}

Eno

^{for some}

^no

^{eN. Thus,ifBisthe}

Eno-closure

^{of convbal(A),thenBisa}bounded Banach disk in

En

_o,^hencealsoinE. Moreoever, A^c_.B,sowehaveshown thatevery bounded subset of Eiscontained in aBanachdisk.

It

followsnowby[9],5.1.6,thatEislocally complete. Finally, weobtain(a) <=> (c) bynoting thataninductive limitofwebbedspacesiswebbed([3], IV.4.6);

theassertionthenfollowsfromthe localcompletenessofEandTheorem5.

Asinthe previoussection,we givesomeexamples.

]Example^3" Itisshownin[4], Example 15,

that/lwith

itsweaktopologyisalocallycomplete, non-bornological, non-metrizable locally convex space satisfying ^the Mackey convergence condition. Thisspacealso satisfies the fastconvergencecondition since it islocallycomplete.

Example4: Inthisexample,weshow thattherearelocally completewebbedspaceswhichdonot satisfytheMackey convergencecondition,hence,notthe fastconvergencecondition either. Let(E, I1.11)beanyBanachspacehavingweaklyconvergentsequencesthat arenotnorm convergent. For instance,Ecouldbe theBanachspace

LP([0,1]),

where < p <oo. LetBdenote the closed unit ball ofE. Then the web

, {2-nB

ne N} is acompatible webonEforwhichEiswebbed.

Moreoever,weshowthatEwith itsweak topology isalsowebbedwithrespectto^7,#.

First,

’

^{iscompatiblewithtsinceeachweak zero}neighborhoodcontains some member of’.

Next,let(Xn)be anysequenceinEsuchthat Xn e 2^-n Bfor eachneN. Since

’

^{isacompleting}

web with respecttoI1.11,

Xn

is norm convergent inE,hence this series is alsoweaklyconvergent n=l

inE. Thus,7’iscompleting for

Furtheremore,it isclearthat aclosed,bounded disk inEisaBanach disk with respecttoT ifandonlyifitisaBanachdisk with respecttoT

1,

whereTandT areany topologieswhichare compatiblewithrespecttothe duality

<E,E’>.

Hence,since(E,I1.11)iscomplete, (E, )islocally complete. Therefore, by corollary ofTheorem3, (E,o) satisfies the Mackeyconvergence conditionifandonlyif(E,o’)isquasi-sequentiallywebbed. However,if(Xn)isaweaklynull sequencewhich isnotnormconvergent, then(Xn)cannotbe containedinthe(only)strand(2^-nB) of4/,since thiswouldimply^tt,c,ormconvergenceof(xn). Thus,(E,)isnotquasi-sequentially

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Author’scurrent address:

Department

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