P-adic Spaces of Continuous Functions I

BLAISE PASCAL

Athanasios Katsaras

P-adic Spaces of Continuous Functions I Volume 15, n^o1 (2008), p. 109-133.

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P-adic Spaces of Continuous Functions I

Athanasios Katsaras

Abstract

Properties of the so calledθo-complete topological spaces are investigated. Also, necessary and sufficient conditions are given so that the spaceC(X, E)of all contin- uous functions, from a zero-dimensional topological spaceXto a non-Archimedean locally convex spaceE, equipped with the topology of uniform convergence on the compact subsets ofX to be polarly barrelled or polarly quasi-barrelled.

Introduction

Let K be a complete non-Archimedean valued field and let C(X, E) be the space of all continuous functions from a zero-dimensional Hausdorff topological spaceXto a non-Archimedean Hausdorff locally convex space E. We will denote byC_b(X, E) (resp. by C_rc(X, E)) the space of all f ∈ C(X, E) for which f(X) is a bounded (resp. relatively compact) subset of E. The dual space of C_rc(X, E), under the topology t_u of uniform convergence, is a space M(X, E⁰) of finitely-additive E⁰-valued measures on the algebra K(X) of all clopen , i.e. both closed and open, subsets of X. Some subspaces of M(X, E⁰) turn out to be the duals of C(X, E) or of C_b(X, E)under certain locally convex topologies.

In section 2 of this paper, we give some results about the spaceM(X, E⁰).

The notion of a θ0-complete topological space was given in [2]. In section 3 we study some of the properties of θ_o-complete spaces. Among other results, we prove that a Hausdorff zero-dimensional space is θo-complete iff it is homeomorphic to a closed subspace of a product of ultrametric spaces. In section 4, we give necessary and sufficient conditions for the spaceC(X, E), equipped with the topology of uniform convergence on the compact subsets of X, to be polarly barrelled or polarly quasi-barrelled,

Keywords:Non-Archimedean fields, zero-dimensional spaces, locally convex spaces.

Math. classification:46S10, 46G10.

1. Preliminaries

Throughout this paper, K will be a complete non-Archimedean valued field, whose valuation is non-trivial. By a seminorm, on a vector space overK, we will mean a non-Archimedean seminorm. Similarly, by a locally convex space we will mean a non-Archimedean locally convex space over K (see [9]). Unless it is stated explicitly otherwise,X will be a Hausdorff zero-dimensional topological space , E a Hausdorff locally convex space and cs(E) the set of all continuous seminorms on E. The space of allK- valued linear maps onEis denoted byE^?, whileE⁰denotes the topological dual of E. A seminorm p, on a vector space G overK, is called polar if p = sup{|f|: f ∈ G^?,|f| ≤ p}. A locally convex space G is called polar if its topology is generated by a family of polar seminorms. A subset A of G is called absolutely convex if λx+µy ∈ A whenever x, y ∈ A and λ, µ ∈ K, with |λ|,|µ| ≤ 1. We will denote by β_oX the Banaschewski compactification of X (see [3]) and by υoX the N-repletion of X, where N is the set of natural numbers. We will letC(X, E) denote the space of all continuous E-valued functions on X and Cb(X, E) (resp. Crc(X, E)) the space of allf ∈C(X, E)for whichf(X)is a bounded (resp. relatively compact) subset of E. In case E =K, we will simply writeC(X), Cb(X) and C_rc(X) respectively. For A ⊂ X, we denote by χ_A the K-valued characteristic function of A. Also, for X ⊂Y ⊂βoX, we denote by B¯^Y the closure of B in Y. If f ∈ E^X, p a seminorm on E and A ⊂ X, we define

kfk_p= sup

x∈X

p(f(x)), kfk_A,p = sup

x∈A

p(f(x)).

The strict topologyβo onC_b(X, E)(see [4]) is the locally convex topology generated by the seminorms f 7→ khfk_p, where p ∈ cs(E) and h is in the space Bo(X) of all boundedK-valued functions onX which vanish at infinity, i.e. for every > 0 there exists a compact subset Y of X such that |h(x)|< if∈/ Y.

Let Ω = Ω(X) be the family of all compact subsets of β_oX \X. For H ∈ Ω, let C_H be the space of all h ∈ C_rc(X) for which the continu- ous extension h^βo to all of β_oX vanishes on H. For p ∈ cs(E), let β_H,p be the locally convex topology on Cb(X, E) generated by the seminorms f 7→ khfk_p, h ∈ C_H. For H ∈ Ω, β_H is the locally convex topology on Cb(X, E) generated by the seminorms f 7→ khfk_p, h ∈ CH, p ∈ cs(E).

The inductive limit of the topologies β_H, H ∈ Ω, is the topology β. Re- placingΩby the familyΩ₁ of allK-zero subsets ofβ_oX, which are disjoint from X, we get the topology β1. Recall that a K-zero subset of βoX is a set of the form {x ∈ β_oX : g(x) = 0}, for some g ∈ C(β_oX). We get the topologies βu and β_u⁰ replacingΩ by the familyΩu of all Q∈Ω with the following property: There exists a clopen partition (A_i)i∈I ofX such that Q is disjoint from each A_i^βoX. Nowβ_u is the inductive limit of the topologies βQ, Q∈Ωu. The inductive limit of the topologies βH,p, as H ranges over Ω_u, is denoted byβ_u,p, while β_u⁰ is the projective limit of the topologiesβu,p, p∈cs(E). For the definition of the topologyβeonCb(X) we refer to [7].

Let now K(X) be the algebra of all clopen subsets of X. We denote by M(X, E⁰) the space of all finitely-additive E⁰-additive measures m on K(X) for which the set m(K(X))is an equicontinuous subset of E⁰. For each such m, there exists a p ∈ cs(E) such that kmk_p = mp(X) < ∞, where, for A∈K(X),

m_p(A) = sup{|m(B)s|/p(s) :p(s)6= 0, A⊃B ∈K(X)}.

The space of all m ∈ M(X, E⁰) for which mp(X) < ∞ is denoted by M_p(X, E⁰). Ifm∈M_p(X, E⁰), then forx∈X we define

N_m,p(x) = inf{m_p(V) :x∈V ∈K(X)}.

In case E = K, we denote by M(X) the space of all finitely-additive bounded K-valued measures onK(X). An element m of M(X) is called τ-additive if m(Vδ) → 0 for each decreasing net (Vδ) of clopen subsets of X with ^TV_δ = ∅. In this case we write V_δ ↓ ∅. We denote by M_τ(X) the space of all τ-additive members ofM(X). Analogously, we denote by Mσ(X) the space of all σ-additivem, i.e. those m withm(Vn)→0 when V_n↓ ∅. For anm∈M(X, E⁰) ands∈E, we denote by msthe element of M(X) defined by(ms)(V) =m(V)s.

Next we recall the definition of the integral of an f ∈E^X with respect to an m∈M(X, E⁰). For a non-empty clopen subset A of X, let D_A be the family of all α={A₁, A₂, . . . , A_n;x₁, x₂, . . . , x_n}, where{A₁, . . . , A_n}is a clopen partition of A and xk ∈ Ak. We make D_A into a directed set by defining α₁ ≥α₂ iff the partition ofA inα₁ is a refinement of the one in α2. For anα ={A₁, A2, . . . , An;x1, x2, . . . , xn} ∈ DA andm ∈M(X, E⁰),

we define

ωα(f, m) =

n

X

k=1

m(A_k)f(x_k).

If the limitlimω_α(f, m)exists inK, we will say thatf ism-integrable over Aand denote this limit by^R_Af dm. We define the integral over the empty set to be 0. ForA=X, we write simply^Rf dm. It is easy to see that iff is m-integrable overX, then it ism-integrable over every clopen subsetAof X and ^R_Af dm=^R χAf dm. Ifτu is the topology of uniform convergence, then everym∈M(X, E⁰) defines aτ_u-continuous linear functionalφ_m on Crc(X, E), φm(f) = ^Rf dm. Also every φ ∈ (Crc(X, E), τu)⁰ is given in this way by some m∈M(X, E⁰).

For p ∈cs(E), we denote by Mt,p(X, E⁰) the space of all m∈Mp(X, E⁰) for which mp is tight, i.e. for each >0, there exists a compact subsetY of X such that m_p(A)< if the clopen setA is disjoint fromY. Let

M_t(X, E⁰) = ^[

p∈cs(E

M_t,p(X, E⁰).

Every m ∈ Mt,p(X, E⁰) defines a β0-continuous linear functional um on C_b(X, E),

um(f) =^R f dm. The map m7→um, fromMt(X, E⁰)to(Cb(X, E), βo)⁰, is an algebraic isomorphism. For m ∈ Mτ(X) and f ∈ K^X, we will denote by (V R)^R f dm the integral of f , with respect to m, as it is defined in [9]. We will call (V R)^R f dm the(V R)-integral of f.

For all unexplained terms on locally convex spaces, we refer to [8] and [9].

2. Some results on M(X, E⁰)

Theorem 2.1. Let m ∈ M(X, E⁰) be such that ms ∈ M_τ(X), for all s∈E, and let p∈cs(E) with kmk_p<∞. Then :

(1) m_p(V) = sup_x∈V N_m,p(x) for every V ∈K(X).

(2) The set

supp(m) =^\{V ∈K(X) :m_p(V^c) = 0}

is the smallest of all closed support sets form.

(3) supp(m) ={x:N_m,p(x)6= 0}.

(4) If V is a clopen set contained in the union of a family (V_i)i∈I of clopen sets, then

mp(V)≤sup{m_p(Vi) :i∈I}.

Proof: (1). Ifx∈V, thenN_m,p(x)≤m_p(V) and so mp(V)≥α= sup

x∈V

Nm,p(x).

On the other hand, letmp(V)> d. There exists a clopen setW, contained in V, and s∈E with|m(W)s|/p(s)> d. Letµ=ms∈M_τ(X). Then

|µ|(W) = sup

x∈W

Nµ(x).

Let x ∈ W be such that Nµ(x) > d·p(s). Now Nm,p(x) ≥ d. In fact, assume the contrary and let Z be a clopen neighborhood ofx contained in W and such thatmp(Z)< d. Now

N_µ(x)≤ |µ|(Z) = sup{|m(Y)s|:Z ⊃Y ∈K(X)} ≤p(s)·m_p(Z)≤d·p(s).

This contradiction proves (1).

(2).

X\supp(m) =^[{W ∈K(X) :m_p(W) = 0}.

Let V ∈ K(X) be disjoint from supp(m). For each x ∈ V, there exists W ∈K(X), with x ∈W and mp(W) = 0 and soNm,p(x) = 0. It follows that

mp(V) = sup

x∈V

Nm,p(x) = 0,

which proves that supp(m) is a support set for m. On the other hand, let Y be a closed support set form. There exists a decreasing net (V_δ)of clopen sets with Y =^TV_δ. Let W ∈K(X) be disjoint from Y. For each clopen set V contained inW and each s∈E, we have V ∩Vδ ↓ ∅ and so lim_δ(ms)(V ∩V_δ) = 0. Since V_δ^c is disjoint from Y, we have m(V_δ^c) = 0 and so m(V) =m(Vδ∩V), which implies that m(V)s= 0, for alls∈E, i.e. m(V) = 0, and hence m_p(W) = 0. Therefore supp(m) ⊂W^c. Taking V_δ^c in place of W, we get thatsupp(m)⊂^TV_δ=Y, which proves (2).

(3) Let G=x:Nm,p(x)6= 0}. If V ∈K(X) is disjoint fromG, then m_p(V) = sup

x∈V

N_m,p(x) = 0,

and so supp(m) ⊂ V^c, which implies that supp(m) ⊂ G. On the other hand, let x /∈ supp(m). There exists a clopen neighborhood W of x dis- joint from supp(m). Since supp(m) is a support set for m, we have that m_p(W) = 0and thus N_m,p = 0 on W, which proves that x /∈G. Thus G is contained in supp(m) and (3) follows.

(4). Let m_p(V) > α > 0. There exists a clopen set A contained in V and s ∈ E such that |m(A)s|/p(s) > α. If µ = ms ∈ M_τ(X), then

|µ|(V) ≥ |m(A)s|> α·p(s). In view of [9, p. 250] there exists an i such that m_p(V_i)≥ |µ|(V_i)/p(s)> α, which clearly completes the proof.

Theorem 2.2. Let m ∈ M(X, E⁰) be such that ms ∈ Mσ(X) for alll s∈E (this in particular holds if m∈M_σ(X, E⁰)). Let p∈cs(E) be such thatmp(X)<∞. If a clopen set V is contained in the union of a sequence (Vn) of clopen sets, then mp(V)≤sup_nmp(Vn).

Proof : We show first that, forµ∈M_σ(X), then there exists annwith

|µ|(V) ≤ |µ|(V_n). In fact, this is clearly true if |µ|(V) = 0. Assume that

|µ|(V) >0 and let W_n = ^SN₁ V_k. Since W_n^c ∩V ↓ ∅, there exists n such that |µ|(V ∩W_n^c)<|µ|(V). SinceV ⊂(V ^TW_n^c)^SWn, it follows that

|µ|(V)≤ |µ|(W_n) = max

1≤k≤n|µ|(V_k),

and the claim follows for µ. Suppose now that mp(V) > r > 0. There exists a clopen subset W of V and s∈ E such that |m(W)s| > r·p(s).

Let µ = ms. Then µ ∈ Mσ(X) and |µ|(V) ≥ |m(W)s| > r·p(s). By the first part of the proof, there exists an n such that |µ|(V_n) > r·p(s).

Hence, there exists a clopen subset D of Vn such that |µ(D)| > r·p(s).

Now |m|_p(V_n)≥ |m(D)s|/p(s)> r, which completes the proof.

For X ⊂ Y ⊂β_oX, and m ∈M(X), we denote bym^Y the element of M(Y) defined by m^Y(V) = m(V ∩X). We denote by m^υo and m^βo the m^Y forY =υoX and Y =βoX, respectively.

We have the following easily established

Theorem 2.3. Let m ∈ M(X, E⁰) be such that ms ∈ M_τ(X) for all s∈E. Then :

(1) supp(m^βo) =supp(m)^βoX. (2) supp(m) =supp(m^βo)∩X.

(3) If m has compact support, then supp(m) =supp(m^βo).

Theorem 2.4. Let m ∈ Mp(X, E⁰) and µ = m^βo. The following are equivalent:

(1) supp(µ)⊂υoX.

(2) If V_n ↓ ∅, then there exists an n_o such that m(V_n) = 0 for every n≥no.

(3) If V_n ↓ ∅, then there exists an n such that m(V) = 0 for every clopen set V contained in Vn.

(4) For every Z ∈Ω1 there exists a clopen subset A on βoX disjoint from Z and such thatsupp(µ)⊂A.

(5) If Vn↓ ∅, then there exists an n such thatmp(Vn) = 0.

Proof: (1)⇒(2). If Vn↓ ∅, then the set ^TVn βoX

is disjoint from υoX and sosupp(µ)⊂^S_nV_n^cβoX. In view of the compactness ofsupp(µ), there exists an n_o withsupp(µ)⊂V_n^c_o^βoX. If now n≥n_o, then m(V_n) = 0.

(2)⇒ (3). Let Vn ↓ ∅ and suppose that, for each n, there exists a clopen subset Aof V_nsuch that m(A)6= 0.

Claim. For each n, there exists k > n and a clopen set B with Vk ⊂ B ⊂ V_n and m(B) 6= 0. Indeed there exists a clopen subset A of V_n such that m(A) 6= 0. For each k, let Bk = Vk∩A, Dk = Vk\Bk. Then D_k ↓ ∅. By our hypothesis, there existsk > n such that m(D_k) = 0. Let B =A∪D_k. Then V_k ⊂B ⊂V_n. Since A and D_k are disjoint, we have that m(B) = m(A) 6= 0 and the claim follows. By induction, we choose n₁ = 1 < n₂ < . . . and clopen sets B_k such that V_nk+1 ⊂ B_k ⊂V_nk and m(B_k) 6= 0. Since B_k ↓ ∅ and m(B_k) 6= 0 for every k, we arrived at a contradiction.

(3)⇒(4). LetZ ∈Ω1. There exists a decreasing sequence (Vn) of clopen sets with Z = ^TV_n^βoX. By our hypothesis, there exists an n suich that m(V) = 0 for each clopen subset V of V_n. Now it suffices to take A = V_n^cβoX.

(4)⇒(1). Letz∈β_oX\υ_oX. There exists a decreasing sequence(V_n)of clopen sets with z ∈Z = ^TVn

βoX

. Clearly Z ∈ Ω1. Thus, there exists a

clopen subset A ofβoX disjoint fromZ and containing supp(µ). Hencez is not in supp(µ).

(3)⇒(5). It is trivial.

(5) ⇒ (1). Let z ∈ β_oX\υ_oX. There exists a decreasing sequence (V_n) of clopen sets with z∈ Z =^TVn

βoX

. Let n be such that mp(Vn) = 0. If G = Vn

βoX

, then µp(G) = 0 and so supp(µ) ⊂ βoX\G, which implies that z /∈supp(µ). This completes the proof.

Theorem 2.5. For an m∈M_p(X, E⁰), the following are equivalent : (1) m has a compact support, i.e. m∈M_c(X, E⁰).

(2) supp(m^βo)⊂X.

(3) If V_δ ↓ ∅, then there exists a δ_o such that m(V_δ) = 0 for all δ≥δo.,

(4) If Vδ ↓ ∅, then there exists a δ such m(V) = 0 for each clopen subset V of V_δ.

(5) If H ∈ Ω, then there exists a clopen subset A of βoX, disjoint from H and containing supp(m^βo).

(6) If V_δ↓ ∅, then there exists a δ such thatm_p(V_δ) = 0.

Proof : In view of Theorem 2.3, (1) implies (2).

(2) ⇒ (3). Let V_δ ↓ ∅. By the compactness of supp(m^β_o), there exists δ_o such that supp(m^βo)⊂V_δ^c_o and so m(V_δ) = 0 forδ≥δo.

(3) ⇒ (4). Let V_δ ↓ ∅and suppose that, for each δ, there exists a clopen subset V of V_δ withm(V)6= 0.

Claim: For each δ there exist γ ≥ δ and a clopen set A such that Vγ ⊂ A ⊂V_δ and m(A)6= 0. In fact, there exists a clopen subset G of V_δ with m(G) 6= 0. For each γ, letZγ =Vγ∩G, Wγ =Vγ\Zγ. ThenWγ ↓ ∅. By our hypothesis, there existsγ ≥δ withm(V_γ) = 0. LetA=G∪W_γ. Since the sets G and W_γ are disjoint, we have that m(A) = m(G) 6= 0. Since Vγ ⊂A⊂V_δ, the claim follows.

Let now F be the family of all clopen subsets A of X with the following property: There are γ, δ, withγ ≥δ, Vγ⊂A⊂Vδ and m(A)6= 0. Since F ↓ ∅, we got a contradiction.

(4) ⇒ (5). If H ∈ Ω, then there exists a decreasing net (Vδ) of clopen

subsets of X with ^TVδ βoX

= H. Since Vδ ↓ ∅, there exists δ such that m(V) = 0 for each clopen subset V of V_δ. Now it suffices to take A = V_δ^cβoX.

(5)⇒(1). Letz∈βoX\X. By (5), there exists a clopen subsetAof βoX containing supp(m^βo) and not containing z.

(4)⇒(6). It is trivial.

(6)⇒ (2). Letz ∈βoX\X. There exists a decreasing net (Vδ) of clopen sets with {z}=^TV_δ^βoX. Letδ be such thatmp(V_δ) = 0. Ifµ=m^βo, then µp(V_δ^βoX) =mp(V_δ) = 0 and so supp(µ) is disjoint from the closure ofV_δ in β_oX, which implies thatz /∈supp(µ).

This completes the proof.

3. θ_o-Complete Spaces

Recall that θoX is the set of all z∈βoX with the following property: For each clopen partition(V_i)ofX there existsisuch thatz∈V_i^βoX (see [2]).

By [2, Lemma 4.1] we have X ⊂ θoX ⊂ υoX. For each clopen partition α = (V_i)i∈I ofX, let

W_α = ^[

i∈I

V_i×V_i.

Then the family of all Wα, α a clopen partition of X, is a base for a uniformityU_c=U_c^X, compatible with the topology ofX, and(θ_oX,U_c^θoX) coincides with the completion of(X,U_c). We will say thatXisθ_o-complete iff X=θoX. As it is shown in [2], ifY is a θo-complete andf :X→Y is a continuous function, then f has a continuous extensionf^θo :θoX →Y. A subset A of X is called bounding if every f ∈C(X) is bounded on A.

Note that several authors use the term bounded set instead of bounding.

But in this paper we will use the term bounding to distinguish from the notion of a bounded set in a topological vector space. A set A ⊂ X is bounding iff A^υoX is compact. In this case (as it is shown in [2, Theorem 4.6]) we have that A^θoX =A^υoX = A^βoX. Clearly a continuous image of a bounding set is bounding. Let us say that a family F of subsets ofX is finite on a subset A of X if the family {f ∈ F :F∩A6=∅} is finite. We have the following easily established

Lemma 3.1. For a subset A of X, the following are equivalent :

(1) A is bounding.

(2) Every continuous real-valued function on X is bounded on A.

(3) Every locally finite family of open subsets of X is finite on A.

(4) Every locally finite family of clopen subsets ofX is finite on A.

By [1, Theorem 4.6] every ultraparacompact space (and hence every ultrametrizable space) is θo-complete.

Theorem 3.2. Every complete Hausdorff locally convex space E is θ_o- complete.

Proof: Let U be the usual uniformity onE, i.e. the uniformity having as a base the family of all sets of the form

Wp, ={(x, y) :p(x−y)≤}, p∈cs(E), >0.

Given W_p,, we consider the clopen partition α = (V_i)i∈I of E generated by the equivalence relation x ∼y iff p(x−y) ≤. Then Wp, =Wα and so U is coarser that U_c. Since (E,U) is complete and U_c is compatible with the topology of E, it follows that(E,U_c) is complete and the result follows.

Corollary 3.3. A subset B, of a complete Hausdorff locally convex space E, is bounding iff it is totally bounded.

Proof: If B is bounding, then B =B^θoE is compact and hence totally bounded, which implies that B is totally bounded. Conversely, if B is totally bounded, thenB is totally bounded. ThusBis compact and hence B is bounding.

Theorem 3.4. IfGis a locally convex space (not necessarily Hausdorff ), then every bounding subset A of G is totally bounded.

Proof: Assume first thatGis Hausdorff. LetGˆ be the completion ofG.

The closureBofAinGˆis bounding and henceBis totally bounded, which implies that A is totally bounded. If G is not Hausdorff, we consider the quotient space F =G/{0} and letu:G→F be the quotient map. Since u is continuous, the set u(A) is bounding, and hence totally bounded, in F. Let now V be a convex neighborhood of zero in G. Then, u(V)

is a neighborhood of zero in F. Let S be finite subset of A such that u(A)⊂u(S) +u(V). But then

A⊂S+V +{0} ⊂S+V +V =S+V, which proves that A is totally bounded.

Theorem 3.5. We have:

(1) Closed subspaces ofθ_o-complete spaces are θ_o-complete.

(2) If X=^QX_i,with X_i 6=∅ for all i, then X isθ_o-complete iff each Xi is θo-complete .

(3) If(Yi)i∈I is a family ofθo-complete subspaces ofX, thenY =^TYi

is θ_o-complete.

(4) θ_oXis the smallest of all θ_o-complete subspaces ofβ_oX which con- tain X.

Proof: (1). LetZ be a closed subspace of a θo-complete spaceX and let (x_δ) be a U_c^Z-Cauchy net in Z. Then (x_δ) is U_c^X-Cauchy and hence x_δ→x∈X. Moreover, x∈Z sinceZ is closed.

(2). Each X_i is homeomorphic to a closed subspace of X. Thus X_i is θo-complete if X is θo-complete. Conversely, suppose that each Xi is θo- complete. If (x^δ) is a U_c^X-Cauchy net, then (x^δ_i) is a U_c^Xi-Cauchy net in X_i and hence x^δ_i →x_i ∈X_i. If x = (x_i), then x^δ→ x, which proves that (X,U_c)is complete.

(3). Let X = ^QY_i and consider the mapf : Y → X, f(x)_i = x for all i. Then f :Y → f(Y) = D is a homeomorphism. Also D is a closed subspace of X. Since X is θ_o-complete, it follows that D is θ_o-complete and hence Y is θ_o-complete.

(4). Since θoX is θo-complete (by [2, Theorem 4.9]) and X ⊂ θoX ⊂ β_oX, the result follows from (3).

Theorem 3.6. For a point z∈β_oX, the following are equivalent : (1) z∈θ_oX.

(2) IfY is a Hausdorff ultraparacompact space andf :X →Y contin- uous, then f^βo(z)∈Y, where f^βo :β_oX→ β_oY is the continuous extension of f.

(3) For every ultrametric space Y and every f :X→ Y continuous , we have that f^βo(z)∈Y.

Proof: (1)⇒(2). SinceθoY =Y, the result follows from [2, Theorem 4.4].

(2)⇒(3). It is trivial.

(3)⇒ (1). Assume that z /∈θoX. Then, there exists a clopen partition (A_i) of X such thatz /∈^S_iA_i^βoX. Letf_i =χ_Ai and define

d:X×X→R, d(x, y) = sup

i

|f_i(x)−f_i(y)|.

Then d is a continuous ultrapseudometric on X. Let Y = Xd be the corresponding ultrametric space and letπ :X→Y_dbe the quotient map, x7→x˜_d= ˜x. Sinceπis continuous, there exists (by (3)) anx∈Xsuch that π^βo(z) = ˜x_d. Let(x_δ)be a net inXconverging toz. Thenx˜_δ=π^βo(x_δ)→ π^βo(z) = ˜x, and so d(x_δ, x) → 0. If x ∈ A_i, then |f_i(x_δ)−1| → 0, and so there exists δ_o such that x_δ ∈A_i when δ ≥δ_o. But then z ∈A_i^βoX, a contradiction. This completes the proof.

Theorem 3.7. LetXbe a dense subspace of a Hausdorff zero-dimensional space Y. The following are equivalent :

(1) Y ⊂ θoX (more precisely, Y is homeomorphic to a subspace of θ_oX).

(2) Each continuous function, fromX to any ultrametric spaceZ, has a continuous extension to all of Y.

Proof: (1) implies (2) by the preceding Theorem.

(2) ⇒ (1). We will prove first that, for each clopen subset V of X, we have that V^Y ∩V^cY =∅,and so V^Y is clopen in Y. Indeed, define

d:X×X →R, d(x, y) = max{|f₁(x)−f₁(y)|,|f₂(x)−f₂(y)|}, where f1 = χ_V, f2 = χ_V^c. Then d is a continuous ultrapseudometric on X. Let π:X→X_dbe the quotient map. By our hypothesis, there exists a continuous extension h : Y → Xd of π. Suppose that z ∈ V^Y ∩V^cY. There are nets (x_δ),(y_γ), in V, V^c respectively, such that x_δ → z, and yγ→z. Let d˜be the ultrametric ofXd and letδo, γo be such that

d(π(x˜ δ), h(z))<1 and d(π(x˜ γ), h(z))<1

when δ ≥δo, γ≥γo. Now

d(xδo, yγo) = ˜d(π(xδo), π(yδo))<1,

a contradiction. Thus V^Y is clopen in Y. If A=V^Y, B=V^cY, then A^{βoY \}B^βoY =V^{βoY \}V^cβoY =∅.

This, being true for each clopen subset V of X, implies that β_oX =β_oY and so X ⊂Y ⊂β_oY =β_oX. Now our hypothesis (2) and the preceding Theorem imply that Y ⊂θoX, and the result follows.

Theorem 3.8. For each continuous ultrapseudometric donX, there ex- ists a continuous ultrapseudometric d^θo on θ_oX which is an extension of d. Moreover, d^θo is the unique continuous extension of d.

Proof: Consider the ultrametric spaceX_d and letd˜be its ultrametric.

Let h be the coninuous extension of the quotient map π :X →X_d to all of θoX. Define

d^θo :θoX×θoX→R, d^θo(y, z) = ˜d(h(y), h(z)).

It is easy to see that d^θo is a continuous ultrapseudometric which is an extension of d. Finally, let % be any continuous ultrapseudometric on θ_oX, which is an extension of d, and let y, z ∈ θ_oX. There are nets (y_δ)δ∈∆,(zγ)γ∈Γ) in X which convergence to y, z, respectively. Let Φ =

∆×Γ and consider on Φ the order (δ₁, γ₁) ≥ (δ₂, γ₂) iff δ₁ ≥ δ₂ and γ1 ≥ γ2. For φ = (δ, γ) ∈ Φ, we let aφ = yδ, bφ = zγ. Then aφ → y, b_φ→z. Thus

%(y, z) = lim%(aφ, bφ) = lim ˜d(h(aφ), h(bφ)) (3.1)

= limd^θo(a_φ, b_φ) =d^θo(y, z) (3.2) and hence %=d^θo, which completes the proof.

Theorem 3.9. Let(H_n)be a sequence of equicontinuous subsets ofC(X).

If z ∈ θ_oX, then there exists x ∈ X such that f^θo(z) = f(x) for all f ∈^SHn=H.

Proof: Define

d:X² →R, d(x, y) = max

n min{1/n, sup

f∈Hn

|f(x)−f(y)|}.

Then d is a continuous ultrapseudometric on X. Take Y = X_d and let π : X → Y be the quotient map. Then π^βo(z) = u ∈ Y. Choose x ∈

X with π(x) = u, and let (x_δ) be a net in X converging to z in βoX.

Now f(xδ) → f^βo(z) for all f ∈ H. Since π(xδ) → π(x), we have that d(x_δ, x) → 0, and so |f(x_δ)−f(x)| → 0 for all f ∈H. Thus, for f ∈H, we have f(x) = limf(x_δ) =f^βo(z), and the result follows.

Theorem 3.10. If H⊂C(X) is equicontinuous, then the family H^θo ={f^θo :f ∈H}

is equicontinuous on θ_oX. Moreover, if H is pointwise bounded, then the same holds for H^θo

Proof: Define

d:X² →R, d(x, y) = min{1, sup

f∈H

|f(x)−f(y)|}.

Let π^θo :θ_oX →X_d be the continuous extension of the quotient mapπ : X →X_d. Let z∈θoX and >0. There exists x∈X such that π^θo(z) = π(x). Let(xδ)be a net inX converging toz. Thenπ(xδ)→π^θo(z) =π(x) and so d(x_δ, x)→0. Thus, forf ∈H, we havef^θo(z) = limf(x_δ) =f(x).

The set W ={y ∈X :d(x, y)≤} isd-clopen (hence clopen ) inX and so W^θoX =V is clopen in θ_oX. Since x_δ ∈W eventually, it follows that z ∈V. Now, for f ∈H and a∈V, we have that|f^θo(a)−f^θo(z)| ≤. In fact, there exists a net (y_γ)inW converging toa. Thus

|f^θo(a)−f^θo(z)|=|f(x)−f^θo(a)|= lim

γ |f(x)−f(yγ)| ≤. This proves that H^θo is equicontinuous on θoX. The last assertion follows from the preceding Theorem.

Theorem 3.11. U_c =U_c^X is the uniformity U generated by the family of all continuous ultrapseudometrics on X.

Proof: Let (Ai) be a clopen partition of X and let W = ^SAi×Ai. Define

d(x, y) = sup

i

|f_i(x)−f_i(y)|,

where f_i=χ_Ai. Then dis a continuous ultrapseudometric onX. Since W ={(x, y) :d(x, y)<1/2},

it follows that U_c is coarser than U. Conversely, let d be a continuous ultrapseudometric on X, >0 and D={(x, y) :d(x, y)≤}.If α is the

clopen partition of X corresponding to the equivalence relationx ∼y iff d(x, y)≤, then D=W_α and the result follows.

Theorem 3.12. Let (Y_i, f_i)i∈I be the family of all pairs (Y, f), where Y is an ultrametric space and f :X→Y a continuous map. Then

θ_oX= ^\

i∈I

(f_i^βo)⁻¹(Y_i).

Proof: It follows from Theorem 3.6.

Theorem 3.13. A Hausdorff zero-dimensional spaceX isθo-complete iff it is homeomorphic to a closed subspace of a product of ultrametic spaces.

Proof: Every ultrametric space is θ_o-complete. Thus the sufficiency follows from Theorem 3.5. Conversely, assume that X isθo-complete and let (Y_i, f_i)i∈I be as in the preceding Theorem. Then X = ^T_iZ_i, Z_i = (f_i^βo)⁻¹(Yi). Let Y =^QYi with its product topology. The map u :X → Y, u(x)_i = f_i(x), is one-to-one. Indeed, let x 6= y and choose a clopen neighborhood V of xnot containing y. Letf =χV and

d:X×Y →R, d(a, b) =|f(a)−f(b)|.

The quotient map π : X → Xd is continuous and π(x) 6= π(y), which implies that u(x)6= u(y). Clearly u is continuous. Also u⁻¹ :u(X) → X is continuous. Indeed, let V be a clopen subset of X containing xo and consider the pseudometric d(x, y) =|χ_V(x)−χ_V(y)|. Let π :X→X_d be the quotient map. There exists a i ∈ I such that Y_i = X_d and f_i = π.

Then

f_i(V) =π(V) ={π(x) : ˜d(π(x)−π(x_o))<1}.

The set π(V) is open in Yi =Xd. Let πi :Y → Yi be the ith-projection map and G =π⁻¹_i (π(V)). If x ∈X is such that u(x) ∈G, then fi(x) = u(x)_i ∈π(V) and so d(x, x_o)<1, which implies that x∈V sincex_o∈V. This proves that u :X → u(X) is a homeomorphism. Finally, u(X) is a closed subspace ofY. In fact, let(x_δ)be a net in X withu(x_δ)→y∈Y. Then f_i(x_δ)→y_i for alli. Going to a subnet if necessary, we may assume that x_δ → z ∈ β_oX. Now f_i(x_δ) → f_i^βo(z) in β_oY_i. But then f_i^βo(z) = yi ∈ Yi, for all i, and hence z ∈ θoX = X, by the preceding Theorem.

Thus y_i = f_i(z), for all i, and hence y = u(z). This proves that X is homeomorphic to a closed subspace ofY and the result follows.

Corollary 3.14. Every Hausdorff ultraparacompact space is homeomor- phic to a closed subspace of a product of ultrametric spaces.

Theorem 3.15. For a subset A of X, the following are equivalent : (1) A is bounding.

(2) A is U_c-totally bounded.

(3) For each continuous ultrapseudometric d on X, A is d-totally bounded.

Proof: In view of Theorem 3.11, (2) is equivalent to (3). Also, by [2, Theorem 4.6], (1) implies (2).

(2)⇒(1). Letf ∈C(X),

A₁ ={x:|f(x)| ≤1}, A_n+1 ={x:n <|f(x)| ≤n+ 1}

forn≥1. Then(An)is a clopen partition ofX. LetW =^S_nAn×An. By our hypothesis, there are x₁, . . . , x_N in A such that A ⊂ ^SN₁ W[x_k]. For each 1≤ k≤N, there exists n_k such that x_k ∈Ank. Then A⊂ ^SN₁ Ank

and so

kfk_A≤ max

1≤k≤Nnk, which proves that A is bounding.

4. Polarly Barrelled Spaces of Continuous Functions Definition 4.1. A Hausdorff locally convex space E is called :

(1) polarly barrelled if every bounded subset ofE_σ⁰ = (E⁰, σ(E⁰, E))is equiconinuous.

(2) polarly quasi-barrelled if every strongly bounded subset of E⁰ is equicontinuous.

We will denote byC_c(X, E)the spaceC(X, E)equipped with the topol- ogy of uniform convergence on compact subsets of X. By Mc(X, E⁰) we will denote the space of all m ∈ M(X, E⁰) with compact support. The dual space of Cc(X, E) coincides withMc(X, E⁰).

Recall that a zero-dimensional Hausdorff topological space X is called a µo-space (see [2]) if every bounding subset ofX is relatively compact. We

denote by µoX the smallest of all µo-subspaces of βoX which contain X.

Then X ⊂ µ_oX ⊂ θ_oX and, for each bounding subset A of X, the set A^βoX is contained in µoX (see [2]). Moreover, if Y is another Hausdorff zero-dimensional space and f : X → Y, then f^βo(µoX) ⊂ µoY and so there exists a continuous extensionf^µo :µ_oX →µ_oY of f.

Theorem 4.2. Assume that E⁰ 6={0} and let G=Cc(X, E). Then G is polarly barrelled iff X is a µ_o-space and E polarly barrelled.

Proof: Assume thatGis polarly barrelled.

I. E is polarly barrelled. Indeed, let Φ be a w^?-bounded subset of E⁰ and let x∈X. For u∈E⁰, let

u_x:G→K, u_x(f) =u(f(x)).

Let H={u_x :u∈Φ}. Forf ∈C(X, E), we have sup

u∈Φ

|u_x(f)|= sup

u∈Φ

|u(f(x))|<∞

and so H is a w^?-bounded subset of G⁰. By our hypothesis, there exists p∈cs(E) and Y a compact subset of X such that

{f ∈G:kfk_Y,p≤1} ⊂H^o.

But then {s∈E :p(s)≤1} ⊂Φ^o and soΦ is equicontinuous.

II. X is a µo-space. In fact, let A be a bounding subset of X and let x⁰ ∈E⁰, x⁰6= 0. Definep on E by p(x) =|x⁰(s)|.Then p∈cs(E). The set

D={f ∈G:kfk_A,p ≤1}

is a polar barrel in G and so D is a neighborhood of zero in G. Let Y a compact subset of X and q∈cs(E) be such that

{f ∈G:kfk_Y,p≤1} ⊂D.

But then A⊂Y and so A is compact.

Conversely, suppose that E is polarly barrelled and X a µ_o-space. Let H be a w^?-bounded subset of the dual spaceMc(X, E⁰)of G. Lets∈E and

D={ms:m∈H} ⊂M(X).

For h∈C_rc(X), we have that sup

m∈H

|< ms, h >|= sup

m∈H

|< m, hs >|<∞.

Thus, consideringM(X)as the dual of the Banach spaceF = (Crc(X), τu), Disw^?-bounded ofF⁰ and sosup_m∈Hkmsk=d_s<∞.Hence,|m(V)s| ≤ ds for all V ∈K(X). It follows that the set

M = ^[

m∈H

m(K(X))

is a w^?-bounded subset of E⁰. Since E is polarly barrelled, there exists p∈cs(E) such that|u(s)| ≤1 for allu∈M and all s∈E withp(s)≤1.

Hence sup_m∈Hkmk_p < ∞. We may choose p so that kmk_p ≤ 1 for all m∈H. Let

Z =S(H) = ^[

m∈H

supp(m).

Then Z is bounding. In fact, assume that Z is not bounding. Then, by [6, Proposition 6.6], there exists a sequence (m_n) inH and f ∈ C(X, E) such that < mn, f >=λⁿ, for alln, where |λ|>1, which contradicts the fact that H isw^?-bounded. By our hypothesis now, Z is compact. Since

{f ∈G:kfk_Z,p≤1} ⊂H^o, the result follows.

Corollary 4.3. Cc(X) is polarly barrelled iff X is a µo-space.

Let nowG, Ebe Hausdorff locally convex spaces. We denote byL_s(G, E) the space L(G, E) of all continuous linear maps, from G toE, equipped with the topology of simple convergence.

Theorem 4.4. Assume that E is polar and let G be polarly barrelled. If E is aµo-space (e.g. when E is metrizable or complete), then Ls(G, E) is a µ_o-space.

Proof: LetΦbe a bounding subset ofLs(G, E). Forx∈G, the set Φ(x) ={φ(x) :φ∈Φ}

is a bounding subset of E and hence its closure Mx in E is compact.

Φ is a topological subspace of E^G and it is contained in the compact set M =^Q_x∈GMx. Since the closure ofΦinE^Gis compact, it suffices to show that this closure is contained inL(G, E). To this end, we prove first that, given a polar neighborhood W of zero in E, there exists a neighborhood

U of zero in Gsuch that φ(U)⊂W for allφ∈Φ. In fact, for φ∈Φ, let φ⁰ be the adjoint map. Let

Z = ^[

φ∈Φ

φ⁰(H),

whereH is the polar ofW inE⁰. Ifx∈G, thenΦ(x)is a bounded subset of E and hence Φ(x) ⊂ αW, for some α ∈ K. If now φ∈ Φ and u ∈ H, then

|< φ⁰(u), x >| = |< u, φ(x >| ≤ |α|,

which proves that Z is a w^?-bounded subset of G⁰. As G is polarly bar- relled, the polar U = Z^o, of Z in G, is a neighborhood of zero and φ(U)⊂H^o =W, for allφ∈Φ, which proves our claim. Let now φ∈E^G be in the closure of Φ. Thenφis linear. There exists a net (φ_δ)inΦcon- verging to φ in E^G. Ifx ∈U, then φ(x) = limφ_δ(x) ∈W, which proves that φis continuous. Hence the result follows.

Corollary 4.5. If E is polarly barrelled, then the weak dualE_σ⁰ of E is a µ_o-space.

Theorem 4.6. Suppose that E is polar and G polarly barrelled. For f ∈ C(X, E), let f^µo : µ_oX → Eˆ be its continuous extension. If T : G → Cc(X, E) is a continuous linear map, then the map

T˜:G→C_c(µ_oX,E),ˆ s7→(T s)^µo, is continuous

Proof: Note thatEˆ is θ_o-complete and hence a µ_o-space. Let φ:X →L_s(G, E), < φ(x), s >= (T s)(x).

Then φis continuous. Since L_s(G,E)ˆ is aµ_o-space, there exists a contin- uous extension

φ^µo :µoX →Ls(G,E).ˆ

Let nowAbe a compact subset ofµoXandpa polar continuous seminorm on E. We denote also by pthe continuous extension ofp to all of E. Letˆ

V ={g∈C(µoX,E) :ˆ kgk_A,p ≤1}.

The set Φ =φ^µo(A) is compact in L_s(G,E).ˆ As in the proof of Theorem 4.4, there exists a neighborhood U of zero in Gsuch that

ψ(U)⊂W ={s∈Eˆ :p(s)≤1},

for all ψ∈Φ.Now, for y∈A ands∈U, we have p(( ˜T s)(y)) =p(< φ^µo(y), s >)≤1

and so T s˜ ∈V. This proves that T˜ is continuous and the result follows.

Theorem 4.7. Assume that E is polar and polarly barrelled and let τ_o be the locally convex topology on C(X, E) generated by the seminorms f 7→ kf^µok_A,p, where A ranges over the family of all compact subsets of µoX and p∈cs(E). Then :

(1) (C(X, E), τo) is polarly barrelled andτo is finer thanτb (and hence finer than τ_c).

(2) If τ is any polarly barrelled topology on C(X, E) which is finer than τ_c, then τ is finer than τ_o. Hence τ_o is the polarly barrelled topology associated with each of the topologies τb and τc.

Proof: (1). Since E is polarly barrelled, the same is true for E. Theˆ space

F =C_c(µ_oX,E)ˆ is polarly barrelled and the map S: (C(X, E), τ_o)→F, f 7→f^µo,

is a linear homeomorphism. Thus τo is polarly barrelled. Also, since for each bounding subsetB ofX, its closureB^µoX is compact, it follows that τ_o is finer thanτ_b.

(2). Let τ be a polarly barrelled topology on C(X, E), which is finer than τ_c, and letG= (C(X, E), τ). The identity map

T :G→C_c(X, E) is continuous and hence the map

T˜:G→Cc(µoX,E),ˆ f 7→f^µo,

is continuous. This proves that τo is coarser than τ and the Theorem follows.

Theorem 4.8. Suppose thatE is polar. ThenG= (C(X, E), τ_b)is polarly barrelled iff E is polarly barrelled and, for each compact subsetA of µ_oX, there exists a bounding subset B of X such thatA⊂B^µoX.

Proof: Assume that G is polarly barrelled. It is easy to see thatE is polarly barrelled. In view of the preceding Theorem, τ_b = τ_o. Thus, for each compact subset AofµoX and each non-zerop∈cs(E), there exist a bounding subset B ofX andq ∈cs(E) such that

{f ∈C(X, E) :kfk_B,q ≤1} ⊂ {f :kf^µok_A,p≤1}.

It follows easily that A ⊂ B^µoX. Conversely, suppose that the condition is satisfied. The condition clearly implies that τo is coarser than τ_b and hence τ_b =τ_o, which implies that Gis polarly barrelled by the preceding Theorem.

Let us say that a family F of subsets of a a setZ is finite on a subset F ofZ if the family of all members of F which meet F is finite.

Definition 4.9. A subset D, of a topological space Z, is said to be w- bounded if every family F of open subsets of Z, which is finite on each compact subset of Z, is also finite on D. If this happens for families of clopen sets, then D is said to be w_o-bounded. We say that Z is a w- space (resp. a wo-space ) if everyw-bounded (resp.wo-bounded) subset is relatively compact.

Lemma 4.10. A subset D, of a zero-dimensional topological space Z, is w-bounded iff it is wo-bounded.

Proof: Assume thatDis notw-bounded. Then, there exists an infinite sequence (x_n) of distinct elements of Dand a sequence (V_n) of open sets such that x_n ∈ V_n and (V_n) is finite on each compact subset of X. By [5, Lemma 2.5], there exists a subsequence (xn_k) and pairwise disjoint clopen sets W_k with x_nk ∈ W_k. We may choose W_k ⊂V_nk. Now (W_k) is clearly finite on each compact subset of X, which implies that D is not w_o-bounded. Hence the Lemma follows.

We easily get the following

Lemma 4.11. Every w_o-bounded subset of X is bounding.

Theorem 4.12. Assume that E⁰ 6= {0}. Then G = C_c(X, E) is polarly quasi-barrelled iff E is polarly quasi-barrelled and X a wo-space.

Proof: Suppose that E is polarly quasi barrelled and X a w_o-space.

LetH be a strongly bounded subset of the dual spaceMc(X, E)of G. We

show first that there exists p ∈ cs(E) such that sup_m∈Hkmk_p < ∞. In fact, let B be a bounded subset ofE and consider the set

D={ms:m∈H, s∈B}.

If h∈C_rc(X),then the set{hs:s∈B}is a bounded subset of Gand so sup

m∈H

Z

hs dm

= sup

m∈H

Z

h d(ms)

<∞.

ConsideringDa a subset of the dual of the Banach spaceF = (Crc(X), τu), we see that D is a w^?-bounded subset of F⁰ and hence equicontinuous.

Thus

d= sup

m∈H,s∈B

kmsk<∞.

Let

Φ = ^[

m∈H

m(K(X)).

Then for A ∈ K(X), s ∈ B, m ∈ H, we have |m(A)s| ≤ kmsk ≤ d.

Hence Φ is a strongly bounded subset of E⁰. By our hypothesis, Φ is an equicontinuous subset of E⁰. Thus, there exists p ∈ cs(E) such that

|m(A)s| ≤ 1 for all m ∈H and all s ∈E with p(s) ≤1. It follows from this that sup_m∈Hkmk_p = r < ∞. We may choose p so that r ≤ 1. Let now

Y =S(H) = ^[

m∈H

supp(m).

ThenY iswo-bounded. Assume the contrary. Then, there exists a sequence (V_n)of distinct clopen subsets ofX, such thatV_n∩Y 6=∅for allnand(V_n) is finite on each compact subset of X. . For each n there exists m_n ∈ H with Vn∩supp(mn)6=∅. Then (mn)p(Vn)>0. There are a clopen subset W_n of V_n and s_n∈E, with p(s_n)≤1, such thatm(W_n)s_n=γ_n 6= 0. Let

|λ|>1 and take

M ={γ_n⁻¹λⁿχ_Wns_n:n∈N}.

Since (W_n) is finite on each compact subset of X, it follows that M is a bounded subset of G and so M is absorbed by H^o. Let λo 6= 0 be such that M ⊂λ_oH^o. But then

1≥ |λ⁻¹_o γ_n⁻¹λⁿmn(Wn)sn|=|λ⁻¹_o λⁿ|

for alln, which is a contradiction. SoY iswo-bounded and hence compact by our hypothesis. Moreover

{f ∈G:kfk_Y,p≤1} ⊂H^o.

Indeed, let kfk_Y,p ≤1. The set V ={x :p(f(x))>1} is disjoint fromY and hence mp(V) = 0 for allm∈H. Thus, form∈H, we have

Z

V

f dm

≤ kfk_p·m_p(V) = 0 and so

Z

f dm =

Z

V^c

f dm

≤mp(V^c)≤1.

Conversely, suppose thatGis polarly quasi-barrelled. LetΦbe a strongly bounded subset of E⁰ and let x ∈ X. For u ∈ E⁰, define ux on G by u_x(f) = u(f(x)). Thenu_x ∈G⁰. The setH ={u_x :u ∈ Φ} is a strongly bounded subset of G⁰. Indeed, letD be a bounded subset ofG. Since the set {f(x) :f ∈D} is a bounded subset of E, we have that

sup

f∈D,u∈Φ

|u_x(f)|= sup

f∈D,u∈Φ

|u(f(x))|<∞.

By our hypothesis,His an equicontinuous subset ofG⁰. Thus, there exists a compact subset Y of X and p∈cs(E) such that

{f ∈G:kfk_Y,p≤1}.

But then {s∈E :p(s) ≤1} ⊂ Φ^o and so Φ is an equicontinuous subset of E⁰, which proves that E is polarly quasi-barrelled. Finally, let A be a wo-bounded subset of X and choose a non-zero element x⁰ of E⁰. Let p(s) =|x⁰(s)|and consider the set

Z ={f ∈G:kfk_A,p ≤1}.

Then Z is a polar set. We will show that Z is bornivorous. So, suppose that there exists a bounded subset M of G which is not absorbed by Z.

Then, there exists a sequence (fn) inM,kf_nk_A,p > n. Let Vn={x:p(fn(x))> n}.

ThenV_nintersectsA. SinceAisw_o-bounded, there exists a compact subset Y of X such that (Vn) is not finite on Y, which is a contradiction since sup_f∈Mkfk_Y,p < ∞. This contradiction shows that Z absorbs bounded