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STRICTLY WEBBED SPACES AND REGULARITY PROPERTIES OF INDUCTIVE LIMITS
ARMANDO GARCÍA-MARTÍNEZ (Received 9 January 2001)
Abstract.Sequentially complete, locally complete, locally Baire, and bornivorously web- bed are equivalent for strictly webbed spaces. For inductive limits of strictly webbed spaces these properties are equivalent. Moreover, they imply regularity.
2000 Mathematics Subject Classification. 46A13, 46A17.
1. Introduction. Throughout this noteEis a locally convex space andE1⊂E2⊂ ···
is a sequence of Hausdorff locally convex spaces with continuous identity maps id :(En, τn)→(En+1, τn+1),n∈Nwhereτnis the topology ofEn. Their locally convex inductive limit is denoted by indEn. AwebWin a locally convex spaceEis a countable family of absolutely convex subsets ofE, arranged inlayers. The first layer of the web consists of a sequence(Ap:p=1,2, . . .)whose union absorbs each point of E. For each setApof the first layer there is a sequence(Apq:q=1,2, . . .)of sets, called the sequence determined byAp, such that
Apq+Apq⊂Ap for eachq, Apq:q=1,2, . . .
absorbs each point ofAp. (1.1) Further layers are made up in a corresponding way so that each set of the kth layer is indexed by a finite row ofkintegers and at each step the above mentioned two conditions are satisfied. Suppose that we choose a setAp from the first layer, then a setApqof the sequence determined byApand so on. The resulting sequence S=(Ap, Apq, Apqr, . . .)is called a strand. Whenever we are dealing with only one strand we can simplify the notation by writingW1=Ap,W2=Apq, and so forth, thusS=(Wk) is a strand where for eachk,Wkis a set of thekth layer.
LetS =(Wk)be a strand. Consider xk∈Wkand the series∞
k=1xk. The space E iswebbed if the series∞
k=1xkis convergent for any choice ofxk∈Wk;E isstrictly webbed if∞
k=n+1xkconverges to some element inWnfor everyn∈Nand for any choice ofxk∈Wk; E isbornivorously webbed if it is strictly webbed and for every bounded setA⊂E, there exist a strand(Wk)k and a sequence(αk)k⊂Csuch that A⊂αkWk, for everyk∈N[2,6,7,9].
Adisk A⊂E is an absolutely convex, bounded and closed set. LetEA denote the linear span of A endowed with the normed topology generated by the Minkowski functionalρofA. This topology is finer than the topology inherited fromE.If(EA, ρA) is a Banach (Baire) space,Ais aBanach(Baire)disk. A locally convex space islocally
complete(locally Baire) if every bounded subset is contained in a Banach (Baire) disk.
Eis aquasi-locally complete spaceif for each bounded subsetBin(E, τ)there exists a weaker locally convex topologyς=ς(B)onEand a Banach diskAin(E, ς)such that B⊂A[8]. Note that locally complete implies quasi-locally complete.
Esatisfies theMackey convergence conditionif for every null sequence(xn)n⊂E, there exists a diskAsuch that(xn)nis aρA-null sequence. Finally,Esatisfiesproperty Kif each null sequence has a series convergent subsequence.
2. Bornivorously webbed
Lemma2.1. Let(E, τ)be a bornivorously webbed space. Then for every bounded set A⊂Ethere exists a Fréchet space(F , γ)such thatAis contained and bounded inF.
Proof. LetA⊂E be a bounded set. Then there exist a strand(Wk)k⊂W and a sequence(αk)k⊂Csuch thatA⊂αkWk, for everyk∈N. ConsiderEWk=span(Wk)and F=
k∈NEWk. Let{F∩(1/k)Wk:k∈N}be a fundamental system of neighborhoods of zero inF. This topology is metrizable and finer thanτ. We will see that it is complete.
Let(xk)k⊂F be a Cauchy sequence, and take(yk)k⊂(xk)ksuch that(yk+1−yk)∈ Wk/k. Then∞
k=1(yk+1−yk)→τ u, for some uinE.∞
k=p+1(yk+1−yk)∈Wp/p, for everyp∈N, so∞
k=1(yk+1−yk)∈F. Hence∞
k=1(yk+1−yk)→F uand ifx=u+y1, we haveyk F
→xandxk F
→x.
IfF=E, and{(1/k)Wk:k∈N}is a fundamental system of neighborhoods, thenE with the topologyγgenerated by this family is a Fréchet space. This topology is finer than the original one.
Theorem2.2. Let(E, τ)be a locally convex space. IfEis strictly webbed, then the following properties are equivalent:
(a)Eis sequentially complete.
(b)Eis locally complete.
(c)Eis locally Baire.
(d)Eis bornivorously webbed.
Proof. (a)⇒(b)⇒(c). The proof is obvious. (c)⇒(d). LetAbe a bounded subset of EandB⊂Ebe a Baire disk such thatAis contained and bounded inB. By [6, Theo- rem 5.6.3] for id :EB→E, there exists a strand(Wk)k such that id−1(Wk)∈N0(EB).
Hence, for everyk∈Nthere existsαk∈Csuch thatA⊂αkid−1(Wk)⊂EB andA⊂ αkWk⊂E.
(d)⇒(a). The argument of the proof is taken from [1, Theorem 1]: let(xn)n be a Cauchy sequence inE, andBn=clEco
{xn:m≥n},n∈N. The setB1is bounded in E which is bornivorously webbed, hence there exists a strand (Wk) in E and a sequence(αk)k⊂Csuch thatB1⊂αkWkfor eachk∈N. Denote byγthe topology on Egenerated by the subbasis{Wk:k∈N}and, for brevity, byFthe space(E, γ).
The setB1⊂Eis closed inE, and by the preceding lemma, it is closed in the locally convex spaceF. SinceB1is convex, it is also weakly closed inF.
By lemma,F is a Fréchet space. Hence the canonical imbeddingF→F, whereFis the second dual ofFequipped with the strong topology, is a topological isomorphism
intoF. SinceF is complete, it is closed inF and each functional from the strong dualFofF can be continuously extended toF. Thus theσ (F , F)-closed setB1is alsoσ (F, F)-closed inF.
Further, since B1 is bounded inF, it is equicontinuous inF. Hence by Alaoglu theorem, the setB1is relativelyσ (F, F)-compact. This, together with theσ (F, F)- closedness, implies thatB1isσ (F, F)-compact inF.
Similarly, all setsBn,n∈N, areσ (F, F)-compact. Every finite intersection {Bn: 1≤n≤m} =Bm,m∈N, is nonempty. Hence there existsx0∈
{Bn:n∈N}⊂B1⊂E.
This implies the existence of an upper triangular matrixΛ=(λnm)with all entries λnm≥0, only finite number of nonzeros in each row, and the sum of all entries in each row is equal to 1, such that the sequence{yn=∞
m=nλnmxm}nconverges tox0in the topologyγ. Then the continuity of the identity mapF →Eimplies the convergence yk→x0inE.
Take a balanced, convex, zero neighborhoodVinE. Then there existp, q∈Nsuch thatyn−x0∈Vforn≥pandxm−xn∈V form≥n≥q. Then forn≥max(p, q), we have
x0−xn= x0−yn
+ yn−xn
= x0−yn
+
∞ m=n
λnm
xm−xn
∈V+V . (2.1)
This impliesxn→x0in the spaceE.
Since propertyK implies locally Baire (see [4, Theorem 2]), this theorem proves that for strictly webbed spaces, propertyKimplies local completeness. This answers Gilsdorf’s question 3.2 in [4] in a negative way. Moreover, these different additional properties for strictly webbed spaces, which appear in [1,4,5], are proved to be all equivalent.
3. Inductive limits. Let(En, τn)nbe an inductive sequence of locally convex spaces, and let(E, τ)=ind(En, τn)be its inductive limit. The space(E, τ)isregularif for each bounded subsetBin(E, τ), there existsn=n(B)∈Nsuch thatBis contained and bounded in(En, τn).(E, τ)issequentially retractiveif for each convergent sequence (xk)kin(E, τ)there existsn=n((xk)k)∈Nsuch that the sequence converges to the same limit in(En, τn). Equivalently, each null sequence in(E, τ)is a null sequence in some(En, τn).
Sequentially retractive inductive limits were introduced and studied by Floret [3,7].
He proved that sequential retractivity implies regularity. In order to get more infor- mation about the relation between regularity and sequential retractivity, we will prove the following proposition.
Proposition3.1. Let(E, τ)=ind(En,τn)be a regular inductive limit. IfEsatisfies the Mackey convergence condition, then it is sequentially retractive.
Proof. Let(xk)kbe a null sequence inE. Since the Mackey convergence condition holds, there exists a bounded diskB⊂Esuch that(xk)kis aρB-null sequence. NowE is regular, soBis contained and bounded in someEn. So, the topologyρBinEB⊂En, is finer than that inherited fromEn. Hence(xk)kis anEn-null sequence.
In the next propositions, we present other relations between these properties and regularity for strictly webbed spaces.
Following Floret [3] and the proof of Theorem 1 in [1], if(E, τ)=ind(En, τn)is an inductive limit of an inductive sequence of bornivorously webbed spaces note that we have:
Esequentially retractive it follows thatEis regular and implies that ifxk τ
→x0, then there existsn0∈Nand a sequence{yk:yk∈conv{xm}∞m=k}∞k=1such thatyk
τn0
→x0. Theorem3.2. Let(E, τ)=ind(En, τn)be the inductive limit of an inductive sequence of strictly webbed locally convex spaces. Consider the conditions:
(a) Esatisfies propertyK (b) Eis locally Baire
(c) Eis bornivorously webbed (d) Eis sequentially complete (e) Eis locally complete
(f) Eis quasi-locally complete (g) Eis regular.
Then(a)⇒(b)(c)(d)(e)and(e)⇒(f)⇒(g).
Proof. (a)⇒(b). [4, Theorem 2].
(b)(c)(d)(e). By Theorem 2.2, since the inductive limit of strictly webbed spaces is strictly webbed.
(e)⇒(f). It is clear.
(f)⇒(g). [8, Theorem 1].
Proposition 3.3. Let (E, τ)=ind(En, τn) be the inductive limit of an inductive sequence of strictly webbed locally convex spaces such that every(En, τn)satisfies prop- ertyK. IfEis sequentially retractive thenEsatisfies propertyK.
Proof. Let(xm)m be a null sequence inE. Then there existn∈N, withxm En
→ 0 and a subsequence (xmk)k ⊂(xm)m such that ∞
k=1xmk En
→ x. Therefore ∞
k=1
xmk E
→x.
Note that combining the results of this section, and under the hypothesis of Proposition 3.3 we have (a)⇒(b)(c)(d)(e)⇒(f)⇒(g). Moreover if E satisfies the Mackey convergence condition they are all equivalent.
Acknowledgement. I would like to thank Dr. C. Bosch for his valuable sugges- tions.
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Armando García-Martínez: Instituto de Matemáticas, U.N.A.M. Area de la Investi- gación Científica, Circuito Exterior, Ciudad Universitaria, México, D.F.04510, Mexico
E-mail address:[email protected]
