ON WEIGHTED SPACES WITHOUT A FUNDAMENTAL SEQUENCE OF BOUNDED SETS

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ON WEIGHTED SPACES WITHOUT A FUNDAMENTAL SEQUENCE OF BOUNDED SETS

J. O. OLALERU

Received 5 February 2001 and in revised form 21 September 2001

The problem of countably quasi-barrelledness of weighted spaces of continuous functions, of which there are no results in the general setting of weighted spaces, is tackled in this paper. This leads to the study of quasi-barrelledness of weighted spaces in which, unlike that of Ernst and Schnettler (1986), though with a similar approach, we drop the assump- tion that the weighted space has a fundamental sequence of bounded sets. The study of countably quasi-barrelledness of weighted spaces naturally leads to definite results on the weighted (DF)-spaces for those weighted spaces with a fundamental sequence of bounded sets.

2000 Mathematics Subject Classification: 46A08, 46E30.

1. Introduction and notations. The countably barrelledness, countably quasi-bar- relledness, barrelledness, and quasi-barrelleness of the space C(X) of continuous functions on a completely regular Hausdorff spaceXequipped with the compact open topology (c-op), including when it is a (DF)-space and (gDF)-space, is well known (cf.

[7,13]). In the more general setting of weighted spaces, Ernst and Schnettler [3] studied the (gDF) and the quasi-barrelledness of weighted spaces by constructing a Nachbin family onX which is based on the assumption that the weighted space has a fun- damental sequence of bounded sets. A weighted space need not have a fundamental system of bounded sets. There are classical examples of such spaces, for example, letX be a noncompact locally compact and σ-compact space, and letC(X)be the space of all continuous real-valued functions onXequipped with the compact-open topology.C(X)is a weighted space [14], metrizable and not normable [11, Observa- tion 10.1.25] and hence it does not have a fundamental sequence of bounded sets [6].

See also [10, page 9] for another example. Following the same approach, the count- ably barrelledness and barrelledness of weighted spaces was studied in [10] without assuming that the weighted space has a fundamental sequence of bounded sets. This paper is a follow up of [10] although we have made it independent of it. We characterise countably quasi-barrelled (Section 3) and quasi-barrelled (Section 4) weighted spaces by a constructed Nachbin family onXwithout assuming that the spaces have a fun- damental sequence of bounded sets. This approach makes the study of (DF)-weighted spaces easy as it can be seen inSection 5. InSection 6, we show that every countably quasi-barrelled weighted space satisfies the countable neighborhood property (cnp).

As an application, we recover the known results for the countably quasi-barrelled- ness and quasi-barrelledness ofC(X)equipped with compact open topology (Corollary 3.10) and at the same time get new results for the space of bounded continuous

functions onXequipped with the strict topology (cf. Corollaries3.13and4.3). Also [1, Theorem 1.8] on the (DF)-weighted inductive limits was recovered as a corollary to Proposition 4.5.

We adopt the notation and definitions of weighted spaces (cf. [1,3,12]).

Let X denote, unless otherwise indicated, a completely regular Hausdorff space andE, a normed linear space.F(X,E)is the vector space of all mappings fromXtoE, C(X,E)is the vector subspace of all continuous mappings fromXtoE.B(X,E)is the vector subspace of all mappings fromXintoEsuch thatf (X)is a bounded subset ofE, andB0(X,E)is the vector subspace ofB(X,E)consisting of those bounded mappings f fromXintoEthat vanish at infinity, that is, thosef∈B(X,E)such that, for any >0, there is a compact subsetK⊂Xsuch thatf (x) ≤for everyx∈Xoutside of K. The vector subspaceC(X,E)∩B(X,E)is denoted byC_b(X,E), andC(X,E)∩B0(X,E) is denoted byC0(X,E). The setC_c(X,E)denotes the subset ofC(X,E)consisting of those functions that are identically zero inEoutside of some compact subset ofX. If each function is complex valued, we write the corresponding spaces omittingE.

A nonnegative real-valued upper semicontinuous function (usc) on X is called a weight. IfU and V are two sets of weights onX, we writeU ≤V whenever, given u∈U, there isu,v∈Vsuch thatu≤v(pointwise onX). We writeU∼Vif and only ifU≤VandV≤U. A set of weightsV onXis said to be directed upwards if forv1, v2inVand everyλ >0, there isv∈V such thatλvi≤v,i=1,2. If a set of weights VonXis directed upwards andV >0, that is, if, given anyx∈X, there isv∈Vwith v(x) >0, thenVwill be referred to as a Nachbin family onX.

LetVbe a Nachbin family onXsuch that, for everyx∈X, there is anf∈FV (X,E) such thatf (x)≠0, then we define the following weighted spaces:

FV (X,E)=f∈F(X,E):(vf )(X)is bounded inE∀v∈V,

FV0(X,E)=f∈F(X,E):vf vanishes at infinity∀v∈V. (1.1)

We considerFV (X,E)endowed with the locally convex topologywvdefined by the family of seminorms

pv(f )=sup

x∈Xv(x)f (x), v∈V , (1.2) with a basis of neighborhoods of the origin of the form

Vv=

f∈FV (X,E):pv(f )≤1, v∈V

. (1.3)

ClearlyFV0(X,E)is a closed subspace ofFV (X,E).

Examples of those spaces are found in [13]. Furthermore, we will denoteCV (X,E) and CV₀(X,E), respectively, as FV (X,E)∩C(X,E) and FV₀(X,E)∩C(X,E) each equipped with the topology induced byw_v.

We recall that a subsetM⊂FV (X,E)isfullif M=

f∈FV (X,E):f ≤µ

, µ(x)=sup

f∈M

f (x)∀x∈X. (1.4)

In the sequel, we puta+∞ = ∞forn∈R⁺∪∞, 1/∞ =0, 1/0= ∞.

2. Preliminary lemmas. The following definitions are recalled for reference. For the definitions of other terminologies mentioned below, see [4].

Definition2.1. A subset of a locally convex space is called abarrelif it is abso- lutely convex, absorbent, and closed.

Definition2.2. A locally convex space E is calledbornivorous ifB absorbs all bounded subsets ofE.

Definition2.3. A locally convex space is calledbarrelled (quasi-barrelled)if every barrel (bornivorous barrel) is a neighborhood of zero.

The following lemma, which was proved by Bonet [1, Lemma 1.2] forCV (X,E), is also true forFV (X,E).

Lemma2.4. LetVbe a Nachbin family onXfor the weighted spaceFV (X,E)such thatv,w∈V, then,V_v≤V_w if and only ifw(x)≤v(x)for everyx∈X.

Following the argument of the proof of [14, Theorem 3.1], if X is a locally com- pact space orV≤B0(X), thenLemma 2.4holds forFV0(X,E) and thus we have the following result.

Lemma2.5. LetVbe a Nachbin family onXfor the weighted spaceFV0(X,E)such thatV≤B0(X)orXis locally compact. Ifv,w∈V, thenVv≤Vwif and only ifw(x)≤ v(x)for everyx∈X.

The following lemma, which was proved forCV (X)in [3, Lemma 1.3], is also true for FV (X,E), and of course true for any of the weighted spacesFV0(X,E)andCV0(X,E).

Lemma2.6. Let{Mi, i∈N}be a sequence of full subsets of the spaceFV (X,E)and µi(x)=sup_f∈Mif (x). Then M=

Mi is a full subset of FV (X,E)andM= {f ∈ FV (X,E):f (x) ≤inf_iµ_i(x)}.

The following lemmas are very important for our work.

Lemma2.7. LetVbe a Nachbin family onXandsan arbitrary function fromXto [0,∞]such that1/s∈B0(X)and the setM= {f∈FV (X,E):f ≤s}is absorbent in FV (X,E). ThenM is bornivorous. Suppose thatinff∈Mf (x)is usc onX, then there is a smallest weightv_s>1/sandM=V_vs.

Proof. In view of [3, Lemma 1.5], it is sufficient to show thatM is bornivorous.

This is immediate sincef·1/sis bounded onX.

Similarly, in view ofLemma 2.5, we have the following lemma.

Lemma2.8. LetVbe a Nachbin family onXsuch thatXis locally compact orV≤ B0(X). Ifs is an arbitrary function fromX to[0,∞]such that1/s∈B0(X)and the setM= {f ∈FV0(X,E):f ≤s}is absorbent in FV0(X,E), thenM is bornivorous.

Suppose thatinff∈Mf (x)is usc onX, then there is a smallest weightvs>1/sand M=Vvs.

3. Countably quasi-barrelledness of weighted spaces

Definition3.1. LetVbe a Nachbin family onX, we define a new Nachbin family V^∗onXby adjoining the smallest weights (including all their multiples) greater than the suprema of all countable weights inV(with the condition that the suprema of all such weights vanish at infinity) toV.

Lemma3.2. LetVbe a Nachbin family onXand let(v_n)_nbe the set of all sequences (v_n)inVsuch thatsup_nv_n∈B0(X)for each(v_n)∈V. Letv_s(n)be the smallest weight greater thansup_nvnfor each(vn)∈V. ThenV^∗is the system of all positive multiples ofvs(n)as(vn)runs through all sequences inV.

Remark3.3. Suppose that U is a bornivorous barrel inFV (X,E)such thatU= {U_n, n∈N}, where eachU_nis a closed and absolutely convex neighborhood of zero inFV (X,E). Then for eachUn, there is avninVsuch thatVvn∈Unand thus

Vvn⊆ Un. Vvn= {f ∈FV (X,E):f (x) ≤1/vn(x)for allx∈X}. Clearly,

{Vvn, vn∈ V} is absorbent. ByLemma 2.6, V_vn= {f ∈FV (X,E):f (x) ≤inf_n(1/v_n(x))}. If we set infn(1/vn)=s(n)and assume that the conditions in Lemma 2.7 are all satisfied, then there is a smallest weightvs(n)greater than 1/s(n)and

Vvn=Vv_s(n). Furthermore, ifV∼V^∗,v_s(n)∈V^∗ thusV_vs and henceUis a neighborhood of zero inFV (X,E)=FV^∗(X,E)and thusFV (X,E)is countably quasi-barrelled. Conversely, assume thatFV (X,E)is countably quasi-barrelled.

We show thatV ∼V^∗ under the same conditions in the remark above. First, it is clear thatV≤V^∗ from the construction ofV^∗. Also, sinceFV (X)is assumed to be countably barrelled, thenU=U_n is a W_v neighborhood of zero. Hence there is a v∈V such thatVv⊆

{Un, n∈N}. But sinceVv_s(n)=

Vvn⊆

Unandvs(n) is the smallest weight greater than 1/s(n)and in view of the fact that U is arbitrary, we can choosevsuch thatV_v⊆V_vs(n)and thus, byLemma 2.7,v_s(n)≤v. Hence,V^∗≤V.

ThereforeV∼V^∗, thus we have the following theorem.

Theorem3.4. LetV be a Nachbin family onXsuch thatsup_nvn∈B0(X)for each (v_n)∈Vand for every subsetMinFV (X,E),inf_f∈Mf (x)is usc onX, thenFV (X,E) is countably quasi-barrelled if and only ifV∼V^∗.

Similarly, in view ofLemma 2.8, we have the following result.

Theorem3.5. LetV be a Nachbin family onXsuch thatsup_nv_n∈B0(X)for each (vn)∈V. Assume thatEis a normed linear space and eitherV≤B0(X)orXis locally compact. If for every subsetMinFV0(X,E),inff∈Mf (x)is usc onX, thenFV0(X,E) is countably quasi-barrelled if and only ifV∼V^∗.

Consequently, the following results follow from Theorems3.4and3.5, respectively.

Proposition3.6. LetV be a Nachbin family onXsuch thatsup_nv_n∈B0(X)for each(vn)∈V, thenCV (X,E)is countably quasi-barrelled if and only ifV∼V^∗.

Proposition3.7. LetV be a Nachbin family onXsuch thatsup_nv_n∈B0(X)for each(v_n)∈V. Assume that eitherV≤B0(X)orXis locally compact, thenCV0(X,E) is countably quasi-barrelled if and only ifV∼V^∗.

The following definition is needed at this point.

Definition 3.8. A subsetB of a topological spaceX is said to be F(X)-quasi- compact if every functionfinF(X,E)bounded onBis inB0(X,E).

Clearly, every F(X)-quasicompact set is F(X)-pseudocompact and the converse need not be true.

Remark3.9. It should be observed from the construction ofV^∗ that V≤V^∗ is always valid. Suppose that

V=χc(X)=

λχK:λ≥0 andK⊂X, Kis compact

, (3.1)

whereχKis the characteristic function ofK. It is well known thatCV0(X,E)=(C(X,E), c-op). We show when V^∗ ≤V. This is true when every sup_vn∈Vv_n which vanishes at infinity onX is inV, that is, sup_vn∈Vv_n is usc and also has a compact support.

Let vn ∈ V and assume that the compact support of vn is Kn for each n. Then sup_vn∈Vv_n is bounded, and by assumption vanishes at infinity, on K_n. Thus, in view ofProposition 3.7, we have the following result.

Corollary3.10. LetX be a completely regular Hausdorff space. Then(C(X,E), c-op)is countably quasi-barrelled if and only if everyF(X)-quasicompact subset ofX, which is a countable union of compact subsets ofX, is compact.

The following example shows that anF(X)-pseudocompact set need not beF(X)- quasicompact. LetWbe the space of ordinals less than the first uncountable ordinals, and letTbe the Thychonov plank. ThenT=(_∞

n=1(W×n)).TisF(X)-pseudocompact but not compact and henceC(T )is not countably barrelled [4, Example 4, page 142].

Alsoχ_Tis bounded onTbut is clearly not inB0(T ). HenceTis notF(X)-quasicompact.

However, we are yet to construct an example of a countably quasi-barrelled space, which is not countably barrelled, using the above result.

Remark 3.11. If X is a locally compact space and if E is a Banach space and χ_c(X)≤V, it is well known thatCV0(X,E)is complete and since a complete countably quasi-barrelled space is countably barrelled, then in view ofProposition 3.7, we have conditions under whichV^∗can also characterize countably barrelled weighted spaces as shown in the next result.

Proposition3.12. LetV be a Nachbin family on a locally compact spaceXsuch thatsup_nv_n∈B0(X)for each (v_n)∈V whereE is a Banach space andχ_c(X)≤V, then the following are equivalent:

(i) CV0(X,E)is countably quasi-barrelled;

(ii) CV0(X,E)is countably barrelled;

(iii) V∼V^∗.

(C(X,E), c-op)is countably barrelled if and only if everyF(X)-pseudocompact sub- set ofX, which is a countable union of compact subsets ofX, is compact [10, Corol- lary 2.6]. Since(C(X,E),c-op)is complete, and a complete countably quasi-barrelled space is countably barrelled, then as a consequence ofProposition 3.12, we have the following result.

Corollary3.13. LetXbe a locally compact space andEa Banach space. Then the following are equivalent:

(i) (C(X,E), c-op)is countably quasi-barrelled;

(ii) (C(X,E), c-op)is countably barrelled;

(iii) everyF(X)-pseudocompact subset ofX, which is a countable union of compact subsets ofX, is compact;

(iv) everyF(X)-quasicompact subset ofX, which is a countable union of compact subsets ofX, is compact;

(v) V∼V^∗.

Consequently,Corollary 3.13gives a condition as to when anF(X)-pseudocompact set inXisF(X)-quasicompact.

Corollary 3.14. Let X be a locally compact space and E a Banach space. If (C(X,E), c-op)is countably quasi-barrelled, then everyF(X)-pseudocompact isF(X)- quasi-compact.

Remark 3.15. We now consider the countably quasi-barrelledness of Cb(X,E) equipped with the strict topology (β). SinceV∼B0(X)is the defining Nachbin family onXforβ, in view ofProposition 3.7, we show whenV^∗≤V. It is sufficient to show when sup_vn∈Vvnis usc. This is the case when the countable union of closed sets inX is closed. Thus we have the following result.

Corollary3.16. LetXbe a completely regular Hausdorff space. Then(C_b(X,E),β) is countably quasi-barrelled if and only if every countable union of closed sets in X is closed.

IfEis complete andXis a locally compact space, it is well known that(Cb(X,E),β) is complete. The fact that (Cb(X,E),β) is countably barrelled if and only if every countable union of compact sets inXis relatively compact [10, Corollary 2.8] coupled with the fact that a complete countably quasi-barrelled space is countably barrelled, gives the following result as a consequence ofCorollary 3.16.

Corollary3.17. LetXbe a locally compact space andEa Banach space. Then the following are equivalent:

(i) (Cb(X,E),β)is countably quasi-barrelled;

(ii) (Cb(X,E),β)is countably barrelled;

(iii) every countable union of closed sets inXis closed;

(iv) every countable union of compact sets inXis relatively compact.

Remark3.18. Thus ifXis a metrizable locally compact space andEis a Banach space, then(Cb(X,E),β)is countably quasi-barrelled and also countably barrelled.

The following example, taken from [2], which we use to show that a countably quasi-barrelled space need not be countably barrelled, was communicated to us by Ian Tweddle. The ordinaryl^∞(S)(S=positive integers) is just the bounded (continuous) functions on the discrete spaceNof natural numbers. According to Collins, its dual under the strict topology isl¹. The closed unit ball of l¹is separable and bounded for the pairing ofl^∞andl¹, and ifl^∞is countably barrelled under the strict topology, the closed unit ball ofl¹is equicontinuous and so, being closed, it is weakly compact

for the weak topology defined byl^∞, which is false, sincel¹is not reflexive under the weak topology. Sol^∞is not countably barrelled but it is countably quasi-barrelled by Corollary 3.16.

4. Quasi-barrelledness of weighted spaces. In view of the fact that a separable or metrizable countably quasi-barrelled space is quasi-barrelled, the quasi-barrelledness ofCV (X,E)andCV0(X,E)follow from Propositions 3.6and 3.7, respectively. Thus we have the following results.

Proposition4.1. LetV be a Nachbin family onXsuch thatsup_nvn∈B0(X)for each(v_n)∈V. IfCV (X,E)is metrizable or separable, thenCV (X,E)is quasi-barrelled if and only ifV∼V^∗.

Proposition4.2. LetV be a Nachbin family onXsuch thatsup_nv_n∈B0(X)for each(vn)∈V. Assume that eitherV≤B0(X)orXis locally compact. IfCV0(X,E)is separable or metrizable, thenCV0(X,E)is quasi-barrelled if and only ifV∼V^∗.

Conditions under which weighted spaces are metrizable or separable are well known. For example, see [8,9].

Specifically, we have the following result.

Corollary4.3. LetXbe separably submetrizable, then the following are equiva- lent:

(i) (C_b(X,E),β)is countably quasi-barrelled;

(ii) every countable union of closed sets inXis closed;

(iii) (Cb(X,E),β)is quasi-barrelled.

Proof. The proof is easy in view ofCorollary 3.16and the fact that ifXis separably submetrizable, then(Cb(X,E),β)is separable, see [9].

For the definition of separably submetrizable, see [9] or [10].

Remark 4.4. There are other conditions for which a weighted countably quasi- barrelled space is quasi-barrelled. For example if the separability or metrizability con- dition onCV (X,E) and CV0(X,E) in Propositions4.1 and 4.2is replaced with the condition that the weighted spaces have a bounded absorbing sequence of metriz- able subsets (see [4, Corollary 9, page 137]). Of particular importance of this result is its application to the study of countably quasi-barrelled inductive limits of weighted spaces, which leads to results of special interest.

Let ᐂ =(v_n,n= 1,2,...) be a decreasing sequence of strictly positive continu- ous weights onX.Vn is a system of weights(avn, a >0)for eachnand we write C(vn)(X,E)andC(vn)0(X,E)instead ofC(Vn)(X,E)andC(Vn)0(X,E), respectively.

Denote the weighted inductive limit as

ᐂ^C(X,E)=indCv_n(X,E), n=1,2,..., ᐂ0C(X,E)=ind

C vn

0(X,E), n=1,2,...

. (4.1)

Define the following system of weights associated toᐂ as ¯V = {v¯:X→R: ¯v is a weight onXsuch that ¯v/vnis bounded from above onXfor eachn}.

The weights of the formv(t)=inf(a(n)vn(t):n=1,2,...)constitute a directed fundamental family of members of ¯V. IfX is locally compact thenᐂ0C(X,E)is a dense topological subspace ofCV¯0(X,E), and ifEis complete thenCV¯0(X,E)is the completion ofᐂ0C(X,E). SimilarlyCV (X,E)¯ =ᐂC(X,E)algebraically. Those are well known, for example see [11].

We show the conditions under which ¯V^∗∼V. It is sufficient to show when ¯¯ V^∗≤V.¯ Let ¯vp=inf_n∈Nα^pnvn∈V¯and assume sup_p∈Nv¯p∈B0(X)

supp∈Nv¯p=sup

p∈Ninf

n∈Nα^pnvn=inf

n∈Nsup

p∈Nα^pnvn. (4.2) Set sup_p∈Nα^pn=λ_n. Since sup_p∈Nv¯_p∈B0(X),λ_nis finite and hence

supp∈Nv¯p=inf

n∈Nλnvn∈V .¯ (4.3)

If for every ¯v∈V¯there is a continuous weight ˜v∈V¯such that ¯v≤v, then it is clear˜ that ¯V^∗≤V¯. SinceCV (X,E)¯ andCV¯0(X,E)each have a bounded absorbing sequence of metrizable subsetsC(vn)(X,E)andC(vn)0(X,E), respectively. In view ofRemark 4.4, Propositions4.1and4.2, respectively, give the following.

Proposition 4.5. LetX be a completely regular Hausdorff space andᐂ^=(vn) a sequence of continuous weights onXsuch that(vn)∈B0(X). If for everyv¯∈V, there¯ is a continuous weightv˜∈V¯such thatv¯≤v, then˜ CV (X,E)¯ is quasi-barrelled.

Proposition 4.6. LetX be a locally compact space andᐂ=(vn)a sequence of continuous weights onXsuch that(vn)∈B0(X). If for everyv¯∈V, there is a continuous¯ weightv˜∈V¯such thatv¯≤v, then˜ CV¯0(X,E)is quasi-barrelled.

5. (DF)-weighted spaces. A countably quasi-barrelled space with a fundamental sequence of bounded sets is a (DF)-space. Thus all our results on countably quasi- barrelled weighted spaces and quasi-barrelled weighted spaces are also true for (DF)- weighted spaces provided that those spaces have a fundamental sequence of bounded sets. Thus for example, according toCorollary 3.16, ifXis a completely regular Haus- dorff space, then(C_b(X,E),β)is a (DF)-space if and only if every countable union of closed sets inX is closed. IfX=R,(Cb(X,E),β)is then not a (DF)-space and thus a (gDF)-(C_b(X,E),β)need not be a (DF)-space [5, pages 266, 269]. Of special interest is Proposition 4.5. SinceCV (X,E)¯ has a fundamental sequence of bounded sets, then, under the assumptions in the proposition,CV (X,E)¯ is also a (DF)-space. This, in the light of application, is exactly [1, Theorem 1.8] and of course the main result in that paper with a more technical proof. Thus the results, which are true for quasi-barrelled CV (X,E), are also true for countably quasi-barrelled or (DF)-C¯ V (X,E).¯

6. Countable neighborhood property (cnp). A spaceEis said to satisfy the cnp if, given any sequence(Un, n=1,2,...)of zero neighborhoods inE, there isa(n) >0 such thatU=

a(n)Un:n=1,2,... is a neighborhood of zero (see [11, Definition 8.3.4]). In view of Propositions3.6and3.7, a close observation of the construction of V^∗ inRemark 3.3gives, respectively, the following results which are generalizations of [11, Proposition 8.3.5].

Proposition 6.1. LetV be a Nachbin family on a completely regular Hausdorff spaceXsuch thatsup_nvn∈B0(X)for each(vn)∈V, then a countably quasi-barrelled CV (X,E)satisfies cnp.

Proposition6.2. LetV be a Nachbin family onXsuch thatsup_nv_n∈B0(X)for each(vn)∈V. If eitherV ≤B0(X)or X is locally compact, then a countably quasi- barrelledCV0(X,E)satisfies cnp.

Acknowledgments. I am grateful to Abdul Salam ICTP for hospitality and Ian Tweddle whose careful reading, suggestions, and corrections of our previous work led to the writing of this paper. I am also grateful to the referee for his helpful comments and suggestions.

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J. O. Olaleru: Mathematics Department, University of Lagos, Yaba, Lagos, Nigeria E-mail address:[email protected]