<figure><img src="image_1.png" /></figure>

BLAISE PASCAL

ATHANASIOS KATSARAS

P-adic Spaces of Continuous Functions I

Volume 15, n° 1 (2008), p. 109-133.

<http://ambp.cedram.org/item?id=AMBP_2008__15_1_109_0>

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Publication éditée par le laboratoire de mathématiques
de l'université Blaise-Pascal, UMR 6620 du CNRS
Clermont-Ferrand — France
# P-adic Spaces of Continuous Functions I

ATHANASIOS KATSARAS

## Abstract

Properties of the so called $\theta_o$-complete topological spaces are investigated. Also, necessary and sufficient conditions are given so that the space $C(X, E)$ of all continuous functions, from a zero-dimensional topological space $X$ to a non-Archimedean locally convex space $E$, equipped with the topology of uniform convergence on the compact subsets of $X$ 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 space $X$ to a non-Archimedean Hausdorff locally convex space $E$. We will denote by $C_b(X, E)$ (resp. by $C_{rc}(X, E)$) the space of all $f \in C(X, E)$ 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 $M(X, E')$ of finitely-additive $E'$-valued measures on the algebra $K(X)$ of all clopen, i.e. both closed and open, subsets of $X$. Some subspaces of $M(X, E')$ 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 space $M(X, E')$. The notion of a $\theta_0$-complete topological space was given in [2]. In section 3 we study some of the properties of $\theta_o$-complete spaces. Among other results, we prove that a Hausdorff zero-dimensional space is $\theta_o$-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 space $C(X, E)$, equipped with the topology of uniform convergence on the compact subsets of $X$, to be polarly barrelled or polarly quasi-barrelled,

<footnote>Keywords: Non-Archimedean fields, zero-dimensional spaces, locally convex spaces.</footnote>

<footnote>Math. classification: 46S10, 46G10.</footnote>
# 1. Preliminaries

Throughout this paper, $\mathbb{K}$ will be a complete non-Archimedean valued field, whose valuation is non-trivial. By a seminorm, on a vector space over $\mathbb{K}$, we will mean a non-Archimedean seminorm. Similarly, by a locally convex space we will mean a non-Archimedean locally convex space over $\mathbb{K}$ (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 all $\mathbb{K}$-valued linear maps on $E$ is denoted by $E^*$, while $E'$ denotes the topological dual of $E$. A seminorm $p$, on a vector space $G$ over $\mathbb{K}$, is called polar if $p = \sup\{|f| : f \in G^*, |f| \le p\}$. 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 $\lambda x + \mu y \in A$ whenever $x, y \in A$ and $\lambda, \mu \in \mathbb{K}$, with $|\lambda|, |\mu| \le 1$. We will denote by $\beta_o X$ the Banaschewski compactification of $X$ (see [3]) and by $v_o X$ the $\mathbb{N}$-repletion of $X$, where $\mathbb{N}$ is the set of natural numbers. We will let $C(X, E)$ denote the space of all continuous $E$-valued functions on $X$ and $C_b(X, E)$ (resp. $C_{rc}(X, E)$) the space of all $f \in C(X, E)$ for which $f(X)$ is a bounded (resp. relatively compact) subset of $E$. In case $E = \mathbb{K}$, we will simply write $C(X), C_b(X)$ and $C_{rc}(X)$ respectively. For $A \subset X$, we denote by $\chi_A$ the $\mathbb{K}$-valued characteristic function of $A$. Also, for $X \subset Y \subset \beta_o X$, we denote by $\bar{B}^Y$ the closure of $B$ in $Y$. If $f \in E^X, p$ a seminorm on $E$ and $A \subset X$, we define

$$ \|f\|_p = \sup_{x \in X} p(f(x)), \quad \|f\|_{A,p} = \sup_{x \in A} p(f(x)). $$

The strict topology $\beta_o$ on $C_b(X, E)$ (see [4]) is the locally convex topology generated by the seminorms $f \mapsto \|hf\|_p$, where $p \in cs(E)$ and $h$ is in the space $B_o(X)$ of all bounded $\mathbb{K}$-valued functions on $X$ which vanish at infinity, i.e. for every $\epsilon > 0$ there exists a compact subset $Y$ of $X$ such that $|h(x)| < \epsilon$ if $\notin Y$.

Let $\Omega = \Omega(X)$ be the family of all compact subsets of $\beta_o X \setminus X$. For $H \in \Omega$, let $C_H$ be the space of all $h \in C_{rc}(X)$ for which the continuous extension $h^{\beta_o}$ to all of $\beta_o X$ vanishes on $H$. For $p \in cs(E)$, let $\beta_{H,p}$ be the locally convex topology on $C_b(X, E)$ generated by the seminorms $f \mapsto \|hf\|_p$, $h \in C_H$. For $H \in \Omega, \beta_H$ is the locally convex topology on $C_b(X, E)$ generated by the seminorms $f \mapsto \|hf\|_p$, $h \in C_H, p \in cs(E)$.
P-ADIC SPACES OF CONTINUOUS FUNCTIONS I

The inductive limit of the topologies $\beta_H, H \in \Omega$, is the topology $\beta$. Replacing $\Omega$ by the family $\Omega_1$ of all $\mathbb{K}$-zero subsets of $\beta_o X$, which are disjoint from $X$, we get the topology $\beta_1$. Recall that a $\mathbb{K}$-zero subset of $\beta_o X$ is a set of the form $\{x \in \beta_o X : g(x) = 0\}$, for some $g \in C(\beta_o X)$. We get the topologies $\beta_u$ and $\beta'_u$ replacing $\Omega$ by the family $\Omega_u$ of all $Q \in \Omega$ with the following property: There exists a clopen partition $(A_i)_{i \in I}$ of $X$ such that $Q$ is disjoint from each $\bar{A}_i^{\beta_o X}$. Now $\beta_u$ is the inductive limit of the topologies $\beta_Q$, $Q \in \Omega_u$. The inductive limit of the topologies $\beta_{H,p}$, as $H$ ranges over $\Omega_u$, is denoted by $\beta_{u,p}$, while $\beta'_u$ is the projective limit of the topologies $\beta_{u,p}$, $p \in cs(E)$. For the definition of the topology $\beta_e$ on $C_b(X)$ we refer to [7].

Let now $K(X)$ be the algebra of all clopen subsets of $X$. We denote by $M(X, E')$ the space of all finitely-additive $E'$-additive measures $m$ on $K(X)$ for which the set $m(K(X))$ is an equicontinuous subset of $E'$. For each such $m$, there exists a $p \in cs(E)$ such that $\|m\|_p = m_p(X) < \infty$, where, for $A \in K(X)$,

$$m_p(A) = \sup\{|m(B)s|/p(s) : p(s) \neq 0, A \supset B \in K(X)\}.$$

The space of all $m \in M(X, E')$ for which $m_p(X) < \infty$ is denoted by $M_p(X, E')$. If $m \in M_p(X, E')$, then for $x \in X$ we define

$$N_{m,p}(x) = \inf\{m_p(V) : x \in V \in K(X)\}.$$

In case $E = \mathbb{K}$, we denote by $M(X)$ the space of all finitely-additive bounded $\mathbb{K}$-valued measures on $K(X)$. An element $m$ of $M(X)$ is called $\tau$-additive if $m(V_\delta) \to 0$ for each decreasing net $(V_\delta)$ of clopen subsets of $X$ with $\bigcap V_\delta = \emptyset$. In this case we write $V_\delta \downarrow \emptyset$. We denote by $M_\tau(X)$ the space of all $\tau$-additive members of $M(X)$. Analogously, we denote by $M_\sigma(X)$ the space of all $\sigma$-additive $m$, i.e. those $m$ with $m(V_n) \to 0$ when $V_n \downarrow \emptyset$. For an $m \in M(X, E')$ and $s \in E$, we denote by $ms$ the element of $M(X)$ defined by $(ms)(V) = m(V)s$.

Next we recall the definition of the integral of an $f \in E^X$ with respect to an $m \in M(X, E')$. For a non-empty clopen subset $A$ of $X$, let $D_A$ be the family of all $\alpha = \{A_1, A_2, \dots, A_n; x_1, x_2, \dots, x_n\}$, where $\{A_1, \dots, A_n\}$ is a clopen partition of $A$ and $x_k \in A_k$. We make $D_A$ into a directed set by defining $\alpha_1 \ge \alpha_2$ iff the partition of $A$ in $\alpha_1$ is a refinement of the one in $\alpha_2$. For an $\alpha = \{A_1, A_2, \dots, A_n; x_1, x_2, \dots, x_n\} \in D_A$ and $m \in M(X, E')$,
we define

$$ \omega_{\alpha}(f, m) = \sum_{k=1}^{n} m(A_k) f(x_k). $$

If the limit $\lim \omega_{\alpha}(f, m)$ exists in $\mathbb{K}$, we will say that $f$ is $m$-integrable over $A$ and denote this limit by $\int_A f dm$. We define the integral over the empty set to be 0. For $A = X$, we write simply $\int f dm$. It is easy to see that if $f$ is $m$-integrable over $X$, then it is $m$-integrable over every clopen subset $A$ of $X$ and $\int_A f dm = \int \chi_A f dm$. If $\tau_u$ is the topology of uniform convergence, then every $m \in M(X, E')$ defines a $\tau_u$-continuous linear functional $\phi_m$ on $C_{rc}(X, E)$, $\phi_m(f) = \int f dm$. Also every $\phi \in (C_{rc}(X, E), \tau_u)'$ is given in this way by some $m \in M(X, E')$.

For $p \in cs(E)$, we denote by $M_{t,p}(X, E')$ the space of all $m \in M_p(X, E')$ for which $m_p$ is tight, i.e. for each $\epsilon > 0$, there exists a compact subset $Y$ of $X$ such that $m_p(A) < \epsilon$ if the clopen set $A$ is disjoint from $Y$. Let

$$ M_t(X, E') = \bigcup_{p \in cs(E)} M_{t,p}(X, E'). $$

Every $m \in M_{t,p}(X, E')$ defines a $\beta_0$-continuous linear functional $u_m$ on $C_b(X, E)$,

$u_m(f) = \int f dm$. The map $m \mapsto u_m$, from $M_t(X, E')$ to $(C_b(X, E), \beta_o)'$, is an algebraic isomorphism. For $m \in M_\tau(X)$ and $f \in \mathbb{K}^X$, we will denote by $(VR) \int f dm$ the integral of $f$, with respect to $m$, as it is defined in [9]. We will call $(VR) \int f dm$ the $(VR)$-integral of $f$.

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

## 2. Some results on $M(X, E')$

**Theorem 2.1.** Let $m \in M(X, E')$ be such that $ms \in M_\tau(X)$, for all $s \in E$, and let $p \in cs(E)$ with $\|m\|_p < \infty$. Then:

(1) $m_p(V) = \sup_{x \in V} N_{m,p}(x)$ for every $V \in K(X)$.

(2) The set

$$ \text{supp}(m) = \bigcap \{V \in K(X) : m_p(V^c) = 0\} $$

is the smallest of all closed support sets for $m$.
P-ADIC SPACES OF CONTINUOUS FUNCTIONS I

$$
(3) \operatorname{supp}(m) = \overline{\{x : N_{m,p}(x) \neq 0\}}.
$$

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

$$
m_p(V) \leq \sup\{m_p(V_i) : i \in I\}.
$$

Proof: (1). If $x \in V$, then $N_{m,p}(x) \le m_p(V)$ and so

$$
m_p(V) \ge \alpha = \sup_{x \in V} N_{m,p}(x).
$$

On the other hand, let $m_p(V) > d$. There exists a clopen set $W$, contained
in $V$, and $s \in E$ with $|m(W)s|/p(s) > d$. Let $\mu = ms \in M_\tau(X)$. Then

$$
|\mu|(W) = \sup_{x \in W} N_{\mu}(x).
$$

Let $x \in W$ be such that $N_\mu(x) > d \cdot p(s)$. Now $N_{m,p}(x) \ge d$. In fact, assume the contrary and let $Z$ be a clopen neighborhood of $x$ contained in $W$ and such that $m_p(Z) < d$. Now

$$
N_{\mu}(x) \leq |\mu|(Z) = \sup\{|m(Y)s| : Z \supset Y \in K(X)\} \leq p(s) \cdot m_p(Z) \leq d \cdot p(s).
$$

This contradiction proves (1).
(2).

$$
X \setminus \mathrm{supp}(m) = \bigcup \{W \in K(X) : m_p(W) = 0\}.
$$

Let $V \in K(X)$ be disjoint from $\text{supp}(m)$. For each $x \in V$, there exists $W \in K(X)$, with $x \in W$ and $m_p(W) = 0$ and so $N_{m,p}(x) = 0$. It follows that

$$
m_p(V) = \sup_{x \in V} N_{m,p}(x) = 0,
$$

which proves that *supp*(m) is a support set for *m*. On the other hand, let *Y* be a closed support set for *m*. There exists a decreasing net (*V*<sub>δ</sub>) of clopen sets with *Y* = *∩* *V*<sub>δ</sub>. Let *W* ∈ *K*(X) be disjoint from *Y*. For each clopen set *V* contained in *W* and each *s* ∈ *E*, we have *V* ∩ *V*<sub>δ</sub> ↓ ∅ and so lim<sub>δ</sub>(*ms*)(*V* ∩ *V*<sub>δ</sub>) = 0. Since *V*<sub>δ</sub><sup>c</sup> is disjoint from *Y*, we have *m*(*V*<sub>δ</sub><sup>c</sup>) = 0 and so *m*(*V*) = *m*(*V*<sub>δ</sub> ∩ *V*), which implies that *m*(*V*)*s* = 0, for all *s* ∈ *E*, i.e. *m*(*V*) = 0, and hence *m*<sub>*p*</sub>(*W*) = 0. Therefore *supp*(m) ⊂ *W*<sup>c</sup>. Taking *V*<sub>δ</sub><sup>c</sup> in place of *W*, we get that *supp*(m) ⊂ ∩ *V*<sub>δ</sub> = *Y*, which proves (2).

(3) Let $G = \overline{x : N_{m,p}(x) \neq 0}$. If $V \in K(X)$ is disjoint from $G$, then

$$
m_p(V) = \sup_{x \in V} N_{m,p}(x) = 0,
$$
and so $supp(m) \subset V^c$, which implies that $supp(m) \subset G$. On the other hand, let $x \notin supp(m)$. There exists a clopen neighborhood $W$ of $x$ disjoint from $supp(m)$. Since $supp(m)$ is a support set for $m$, we have that $m_p(W) = 0$ and thus $N_{m,p} = 0$ on $W$, which proves that $x \notin G$. Thus $G$ is contained in $supp(m)$ and (3) follows.

(4). Let $m_p(V) > \alpha > 0$. There exists a clopen set $A$ contained in $V$ and $s \in E$ such that $|m(A)s|/p(s) > \alpha$. If $\mu = ms \in M_\tau(X)$, then $|\mu|(V) \ge |m(A)s| > \alpha \cdot p(s)$. In view of [9, p. 250] there exists an $i$ such that $m_p(V_i) \ge |\mu|(V_i)/p(s) > \alpha$, which clearly completes the proof.

**Theorem 2.2.** Let $m \in M(X, E')$ be such that $ms \in M_\sigma(X)$ for all $s \in E$ (this in particular holds if $m \in M_\sigma(X, E')$). Let $p \in cs(E)$ be such that $m_p(X) < \infty$. If a clopen set $V$ is contained in the union of a sequence $(V_n)$ of clopen sets, then $m_p(V) \le \sup_n m_p(V_n)$.

*Proof:* We show first that, for $\mu \in M_\sigma(X)$, then there exists an $n$ with $|\mu|(V) \le |\mu|(V_n)$. In fact, this is clearly true if $|\mu|(V) = 0$. Assume that $|\mu|(V) > 0$ and let $W_n = \bigcup_{1}^{N} V_k$. Since $W_n^c \cap V \downarrow \emptyset$, there exists $n$ such that $|\mu|(V \cap W_n^c) < |\mu|(V)$. Since $V \subset (V \cap W_n^c) \cup W_n$, it follows that

$$|\mu|(V) \le |\mu|(W_n) = \max_{1 \le k \le n} |\mu|(V_k),$$

and the claim follows for $\mu$. Suppose now that $m_p(V) > r > 0$. There exists a clopen subset $W$ of $V$ and $s \in E$ such that $|m(W)s| > r \cdot p(s)$. Let $\mu = ms$. Then $\mu \in M_\sigma(X)$ and $|\mu|(V) \ge |m(W)s| > r \cdot p(s)$. By the first part of the proof, there exists an $n$ such that $|\mu|(V_n) > r \cdot p(s)$. Hence, there exists a clopen subset $D$ of $V_n$ such that $|\mu(D)| > r \cdot p(s)$. Now $|m|_p(V_n) \ge |m(D)s|/p(s) > r$, which completes the proof.

For $X \subset Y \subset \beta_o X$, and $m \in M(X)$, we denote by $m^Y$ the element of $M(Y)$ defined by $m^Y(V) = m(V \cap X)$. We denote by $m^{\nu_o}$ and $m^{\beta_o}$ the $m^Y$ for $Y = \nu_o X$ and $Y = \beta_o X$, respectively.

We have the following easily established

**Theorem 2.3.** Let $m \in M(X, E')$ be such that $ms \in M_\tau(X)$ for all $s \in E$. Then:

$$(1) \quad supp(m^{\beta_o}) = \overline{supp(m)}^{\beta_o X}.$$

$$(2) \quad supp(m) = supp(m^{\beta_o}) \cap X.$$
P-ADIC SPACES OF CONTINUOUS FUNCTIONS I

(3) If $m$ has compact support, then $\text{supp}(m) = \text{supp}(m^{\beta_o})$.

**Theorem 2.4.** Let $m \in M_p(X, E')$ and $\mu = m^{\beta_o}$. The following are equivalent:

(1) $\text{supp}(\mu) \subset v_o X$.

(2) If $V_n \downarrow \emptyset$, then there exists an $n_o$ such that $m(V_n) = 0$ for every $n \ge n_o$.

(3) If $V_n \downarrow \emptyset$, then there exists an $n$ such that $m(V) = 0$ for every clopen set $V$ contained in $V_n$.

(4) For every $Z \in \Omega_1$ there exists a clopen subset $A$ on $\beta_o X$ disjoint from $Z$ and such that $\text{supp}(\mu) \subset A$.

(5) If $V_n \downarrow \emptyset$, then there exists an $n$ such that $m_p(V_n) = 0$.

*Proof:* (1) ⇒ (2). If $V_n \downarrow \emptyset$, then the set $\bigcap \overline{V_n}^{\beta_o X}$ is disjoint from $v_o X$ and so $\text{supp}(\mu) \subset \bigcup_n \overline{V_n}^{\beta_o X}$. In view of the compactness of $\text{supp}(\mu)$, there exists an $n_o$ with $\text{supp}(\mu) \subset \overline{V_{n_o}^c}^{\beta_o X}$. If now $n \ge n_o$, then $m(V_n) = 0$.
(2) ⇒ (3). Let $V_n \downarrow \emptyset$ and suppose that, for each $n$, there exists a clopen subset $A$ of $V_n$ such that $m(A) \ne 0$.

**Claim.** For each $n$, there exists $k > n$ and a clopen set $B$ with $V_k \subset B \subset V_n$ and $m(B) \ne 0$. Indeed there exists a clopen subset $A$ of $V_n$ such that $m(A) \ne 0$. For each $k$, let $B_k = V_k \cap A$, $D_k = V_k \setminus B_k$. Then $D_k \downarrow \emptyset$. By our hypothesis, there exists $k > n$ such that $m(D_k) = 0$. Let $B = A \cup D_k$. Then $V_k \subset B \subset V_n$. Since $A$ and $D_k$ are disjoint, we have that $m(B) = m(A) \ne 0$ and the claim follows. By induction, we choose $n_1 = 1 < n_2 < \dots$ and clopen sets $B_k$ such that $V_{n_{k+1}} \subset B_k \subset V_{n_k}$ and $m(B_k) \ne 0$. Since $B_k \downarrow \emptyset$ and $m(B_k) \ne 0$ for every $k$, we arrived at a contradiction.

(3) ⇒ (4). Let $Z \in \Omega_1$. There exists a decreasing sequence $(V_n)$ of clopen sets with $Z = \bigcap \overline{V_n}^{\beta_o X}$. By our hypothesis, there exists an $n$ such that $m(V) = 0$ for each clopen subset $V$ of $V_n$. Now it suffices to take $A = \overline{V_n}^{\beta_o X}$.

(4) ⇒ (1). Let $z \in \beta_o X \setminus v_o X$. There exists a decreasing sequence $(V_n)$ of clopen sets with $z \in Z = \bigcap \overline{V_n}^{\beta_o X}$. Clearly $Z \in \Omega_1$. Thus, there exists a
clopen subset $A$ of $\beta_o X$ disjoint from $Z$ and containing $\text{supp}(\mu)$. Hence $z$ is not in $\text{supp}(\mu)$.

(3) $\Rightarrow$ (5). It is trivial.

(5) $\Rightarrow$ (1). Let $z \in \beta_o X \setminus \nu_o X$. There exists a decreasing sequence $(V_n)$ of clopen sets with $z \in Z = \bigcap \overline{V_n}^{\beta_o X}$. Let $n$ be such that $m_p(V_n) = 0$. If $G = \overline{V_n}^{\beta_o X}$, then $\mu_p(G) = 0$ and so $\text{supp}(\mu) \subset \beta_o X \setminus G$, which implies that $z \notin \text{supp}(\mu)$. This completes the proof.

**Theorem 2.5.** For an $m \in M_p(X, E')$, the following are equivalent :

(1) $m$ has a compact support, i.e. $m \in M_c(X, E')$.

(2) $\text{supp}(m^{\beta_o}) \subset X$.

(3) If $V_\delta \downarrow \emptyset$, then there exists a $\delta_o$ such that $m(V_\delta) = 0$ for all $\delta \ge \delta_o$,

(4) If $V_\delta \downarrow \emptyset$, then there exists a $\delta$ such that $m(V) = 0$ for each clopen subset $V$ of $V_\delta$.

(5) If $H \in \Omega$, then there exists a clopen subset $A$ of $\beta_o X$, disjoint from $H$ and containing $\text{supp}(m^{\beta_o})$.

(6) If $V_\delta \downarrow \emptyset$, then there exists a $\delta$ such that $m_p(V_\delta) = 0$.

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

(2) $\Rightarrow$ (3). Let $V_\delta \downarrow \emptyset$. By the compactness of $\text{supp}(m_o^{\beta_o})$, there exists $\delta_o$ such that $\text{supp}(m^{\beta_o}) \subset V_{\delta_o}^c$ and so $m(V_\delta) = 0$ for $\delta \ge \delta_o$.

(3) $\Rightarrow$ (4). Let $V_\delta \downarrow \emptyset$ and suppose that, for each $\delta$, there exists a clopen subset $V$ of $V_\delta$ with $m(V) \ne 0$.

**Claim:** For each $\delta$ there exist $\gamma \ge \delta$ and a clopen set $A$ such that $V_\gamma \subset A \subset V_\delta$ and $m(A) \ne 0$. In fact, there exists a clopen subset $G$ of $V_\delta$ with $m(G) \ne 0$. For each $\gamma$, let $Z_\gamma = V_\gamma \cap G$, $W_\gamma = V_\gamma \setminus Z_\gamma$. Then $W_\gamma \downarrow \emptyset$. By our hypothesis, there exists $\gamma \ge \delta$ with $m(V_\gamma) = 0$. Let $A = G \cup W_\gamma$. Since the sets $G$ and $W_\gamma$ are disjoint, we have that $m(A) = m(G) \ne 0$. Since $V_\gamma \subset A \subset V_\delta$, the claim follows.

Let now $\mathcal{F}$ be the family of all clopen subsets $A$ of $X$ with the following property: There are $\gamma, \delta$, with $\gamma \ge \delta$, $V_\gamma \subset A \subset V_\delta$ and $m(A) \ne 0$. Since $\mathcal{F} \downarrow \emptyset$, we got a contradiction.

(4) $\Rightarrow$ (5). If $H \in \Omega$, then there exists a decreasing net $(V_\delta)$ of clopen
subsets of X with $\cap \overline{V}_{\delta}^{\beta_o X} = H$. Since $V_{\delta} \downarrow \emptyset$, there exists $\delta$ such that $m(V) = 0$ for each clopen subset $V$ of $V_{\delta}$. Now it suffices to take $A = \overline{V_{\delta}^{c\beta_o X}}$.

(5) ⇒ (1). Let $z \in \beta_o X \setminus X$. By (5), there exists a clopen subset $A$ of $\beta_o X$ containing $\text{supp}(m^{\beta_o})$ and not containing $z$.

(4) ⇒ (6). It is trivial.

(6) ⇒ (2). Let $z \in \beta_o X \setminus X$. There exists a decreasing net $(V_\delta)$ of clopen sets with $\{z\} = \cap \overline{V}_\delta^{\beta_o X}$. Let $\delta$ be such that $m_p(V_\delta) = 0$. If $\mu = m^{\beta_o}$, then $\mu_p(\overline{V}_\delta^{\beta_o X}) = m_p(V_\delta) = 0$ and so $\text{supp}(\mu)$ is disjoint from the closure of $V_\delta$ in $\beta_o X$, which implies that $z \notin \text{supp}(\mu)$.

This completes the proof.

### 3. $\theta_o$-Complete Spaces

Recall that $\theta_o X$ is the set of all $z \in \beta_o X$ with the following property: For each clopen partition $(V_i)$ of $X$ there exists $i$ such that $z \in \overline{V_i}^{\beta_o X}$ (see [2]). By [2, Lemma 4.1] we have $X \subset \theta_o X \subset \nu_o X$. For each clopen partition $\alpha = (V_i)_{i \in I}$ of $X$, let

$$W_{\alpha} = \bigcup_{i \in I} V_i \times V_i.$$

Then the family of all $W_\alpha$, $\alpha$ a clopen partition of $X$, is a base for a uniformity $\mathcal{U}_c = \mathcal{U}_c^X$, compatible with the topology of $X$, and $(\theta_o X, \mathcal{U}_c^{\theta_o X})$ coincides with the completion of $(X, \mathcal{U}_c)$. We will say that $X$ is $\theta_o$-complete iff $X = \theta_o X$. As it is shown in [2], if $Y$ is a $\theta_o$-complete and $f: X \to Y$ is a continuous function, then $f$ has a continuous extension $f^{\theta_o}: \theta_o X \to Y$. A subset $A$ of $X$ is called bounding if every $f \in C(X)$ 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 $A \subset X$ is bounding iff $\overline{A}^{\nu_o X}$ is compact. In this case (as it is shown in [2, Theorem 4.6]) we have that $\overline{A}^{\theta_o X} = \overline{A}^{\nu_o X} = \overline{A}^{\beta_o X}$. Clearly a continuous image of a bounding set is bounding. Let us say that a family $\mathcal{F}$ of subsets of $X$ is finite on a subset $A$ of $X$ if the family $\{f \in \mathcal{F}: F \cap A \neq \emptyset\}$ 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 of X is finite on A.

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

**Theorem 3.2.** Every complete Hausdorff locally convex space E is $\theta_o$-complete.

*Proof:* Let $\mathcal{U}$ be the usual uniformity on E, i.e. the uniformity having as a base the family of all sets of the form

$$W_{p,\epsilon} = \{(x, y) : p(x - y) \le \epsilon\}, p \in \mathcal{C}(E), \epsilon > 0.$$

Given $W_{p,\epsilon}$, we consider the clopen partition $\alpha = (V_i)_{i \in I}$ of E generated by the equivalence relation $x \sim y$ iff $p(x - y) \le \epsilon$. Then $W_{p,\epsilon} = W_\alpha$ and so $\mathcal{U}$ is coarser that $\mathcal{U}_c$. Since $(E, \mathcal{U})$ is complete and $\mathcal{U}_c$ is compatible with the topology of E, it follows that $(E, \mathcal{U}_c)$ 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 $\bar{B} = \bar{B}^{\theta_o E}$ is compact and hence totally bounded, which implies that B is totally bounded. Conversely, if B is totally bounded, then $\bar{B}$ is totally bounded. Thus $\bar{B}$ is compact and hence B is bounding.

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

*Proof:* Assume first that G is Hausdorff. Let $\hat{G}$ be the completion of G. The closure B of A in $\hat{G}$ is bounding and hence B is totally bounded, which implies that A is totally bounded. If G is not Hausdorff, we consider the quotient space $F = G/\{0\}$ and let $u : G \to F$ 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)
P-ADIC SPACES OF CONTINUOUS FUNCTIONS I

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

$$
A \subset S + V + \overline{\{0\}} \subset S + V + V = S + V,
$$

which proves that $A$ is totally bounded.

**Theorem 3.5.** *We have:*

(1) Closed subspaces of θ<sub>o</sub>-complete spaces are θ<sub>o</sub>-complete.

(2) If $X = \prod X_i$, with $X_i \neq \emptyset$ for all $i$, then $X$ is $\theta_o$-complete iff each $X_i$ is $\theta_o$-complete.

(3) If $(Y_i)_{i \in I}$ is a family of $\theta_o$-complete subspaces of $X$, then $Y = \bigcap Y_i$ is $\theta_o$-complete.

(4) $\theta_o X$ is the smallest of all $\theta_o$-complete subspaces of $\beta_o X$ which contain $X$.

Proof: (1). Let $Z$ be a closed subspace of a $\theta_o$-complete space $X$ and let $(x_\delta)$ be a $U_c^Z$-Cauchy net in $Z$. Then $(x_\delta)$ is $U_c^X$-Cauchy and hence $x_\delta \to x \in X$. Moreover, $x \in Z$ since $Z$ is closed.

(2). Each $X_i$ is homeomorphic to a closed subspace of $X$. Thus $X_i$ is $\theta_o$-complete if $X$ is $\theta_o$-complete. Conversely, suppose that each $X_i$ is $\theta_o$-complete. If $(x^\delta)$ is a $U_c^X$-Cauchy net, then $(x_i^\delta)$ is a $U_c^{X_i}$-Cauchy net in $X_i$ and hence $x_i^\delta \to x_i \in X_i$. If $x = (x_i)$, then $x^\delta \to x$, which proves that $(X, U_c)$ is complete.

(3). Let $X = \prod Y_i$ and consider the map $f : Y \to X$, $f(x)_i = x$ for all $i$. Then $f : Y \to f(Y) = D$ is a homeomorphism. Also $D$ is a closed subspace of $X$. Since $X$ is $\theta_o$-complete, it follows that $D$ is $\theta_o$-complete and hence $Y$ is $\theta_o$-complete.

(4). Since $\theta_o X$ is $\theta_o$-complete (by [2, Theorem 4.9]) and $X \subset \theta_o X \subset \beta_o X$, the result follows from (3).

**Theorem 3.6.** For a point $z \in \beta_o X$, the following are equivalent :

(1) $z \in \theta_o X$.

(2) If $Y$ is a Hausdorff ultraparacompact space and $f: X \to Y$ continuous, then $f^{\beta_o}(z) \in Y$, where $f^{\beta_o}: \beta_o X \to \beta_o Y$ is the continuous extension of $f$.
A. KATSARAS

(3) For every ultrametric space Y and every $f: X \to Y$ continuous, we have that $f^{\beta_o}(z) \in Y$.

*Proof:* (1) ⇒ (2). Since $\theta_o Y = Y$, the result follows from [2, Theorem 4.4].

(2) ⇒ (3). It is trivial.

(3) ⇒ (1). Assume that $z \notin \theta_o X$. Then, there exists a clopen partition $(A_i)$ of X such that $z \notin \bigcup_i \overline{A_i}^{\beta_o X}$. Let $f_i = \chi_{A_i}$ and define

$$d: X \times X \to \mathbf{R}, \quad d(x, y) = \sup_i |f_i(x) - f_i(y)|.$$

Then $d$ is a continuous ultrapseudometric on $X$. Let $Y = X_d$ be the corresponding ultrametric space and let $\pi: X \to Y_d$ be the quotient map, $x \mapsto \tilde{x}_d = \tilde{x}$. Since $\pi$ is continuous, there exists (by (3)) an $x \in X$ such that $\pi^{\beta_o}(z) = \tilde{x}_d$. Let $(x_\delta)$ be a net in $X$ converging to $z$. Then $\tilde{x}_\delta = \pi^{\beta_o}(x_\delta) \to \pi^{\beta_o}(z) = \tilde{x}$, and so $d(x_\delta, x) \to 0$. If $x \in A_i$, then $|f_i(x_\delta) - 1| \to 0$, and so there exists $\delta_o$ such that $x_\delta \in A_i$ when $\delta \ge \delta_o$. But then $z \in \overline{A_i}^{\beta_o X}$, a contradiction. This completes the proof.

**Theorem 3.7.** Let $X$ be a dense subspace of a Hausdorff zero-dimensional space $Y$. The following are equivalent:

(1) $Y \subset \theta_o X$ (more precisely, $Y$ is homeomorphic to a subspace of $\theta_o X$).

(2) Each continuous function, from $X$ to any ultrametric space $Z$, 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 $\overline{V}^Y \cap \overline{V}^V = \emptyset$, and so $\overline{V}^Y$ is clopen in $Y$. Indeed, define

$$d: X \times X \to \mathbf{R}, \quad d(x, y) = \max\{|f_1(x) - f_1(y)|, |f_2(x) - f_2(y)|\},$$

where $f_1 = \chi_V$, $f_2 = \chi_{V^c}$. Then $d$ is a continuous ultrapseudometric on $X$. Let $\pi: X \to X_d$ be the quotient map. By our hypothesis, there exists a continuous extension $h: Y \to X_d$ of $\pi$. Suppose that $z \in \overline{V}^Y \cap \overline{V}^{V^c}$. There are nets $(x_\delta)$, $(y_\gamma)$, in $V$, $V^c$ respectively, such that $x_\delta \to z$, and $y_\gamma \to z$. Let $\tilde{d}$ be the ultrametric of $X_d$ and let $\delta_o, \gamma_o$ be such that

$$\tilde{d}(\pi(x_\delta), h(z)) < 1 \quad \text{and} \quad \tilde{d}(\pi(x_\gamma), h(z)) < 1$$
P-ADIC SPACES OF CONTINUOUS FUNCTIONS I

when $\delta \ge \delta_o, \gamma \ge \gamma_o$. Now

$$d(x_{\delta_o}, y_{\gamma_o}) = \tilde{d}(\pi(x_{\delta_o}), \pi(y_{\delta_o})) < 1,$$

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

$$\overline{A}^{\beta_o Y} \cap \overline{B}^{\beta_o Y} = \overline{V}^{\beta_o Y} \cap \overline{V}^{c\beta_o Y} = \emptyset.$$

This, being true for each clopen subset $V$ of $X$, implies that $\beta_o X = \beta_o Y$ and so $X \subset Y \subset \beta_o Y = \beta_o X$. Now our hypothesis (2) and the preceding Theorem imply that $Y \subset \theta_o X$, and the result follows.

**Theorem 3.8.** For each continuous ultrapseudometric $d$ on $X$, there exists a continuous ultrapseudometric $d^{\theta_o}$ on $\theta_o X$ which is an extension of $d$. Moreover, $d^{\theta_o}$ is the unique continuous extension of $d$.

*Proof:* Consider the ultrametric space $X_d$ and let $\tilde{d}$ be its ultrametric. Let $h$ be the continuous extension of the quotient map $\pi: X \to X_d$ to all of $\theta_o X$. Define

$$d^{\theta_o}: \theta_o X \times \theta_o X \to \mathbf{R}, \quad d^{\theta_o}(y, z) = \tilde{d}(h(y), h(z)).$$

It is easy to see that $d^{\theta_o}$ is a continuous ultrapseudometric which is an extension of $d$. Finally, let $\varrho$ be any continuous ultrapseudometric on $\theta_o X$, which is an extension of $d$, and let $y, z \in \theta_o X$. There are nets $(y_\delta)_{\delta \in \Delta}$, $(z_\gamma)_{\gamma \in \Gamma}$ in $X$ which convergence to $y, z$, respectively. Let $\Phi = \Delta \times \Gamma$ and consider on $\Phi$ the order $(\delta_1, \gamma_1) \ge (\delta_2, \gamma_2)$ iff $\delta_1 \ge \delta_2$ and $\gamma_1 \ge \gamma_2$. For $\phi = (\delta, \gamma) \in \Phi$, we let $a_\phi = y_\delta, b_\phi = z_\gamma$. Then $a_\phi \to y, b_\phi \to z$. Thus

$$\varrho(y, z) = \lim \varrho(a_\phi, b_\phi) = \lim \tilde{d}(h(a_\phi), h(b_\phi)) \quad (3.1)$$

$$= \lim d^{\theta_o}(a_\phi, b_\phi) = d^{\theta_o}(y, z) \quad (3.2)$$

and hence $\varrho = d^{\theta_o}$, which completes the proof.

**Theorem 3.9.** Let $(H_n)$ be a sequence of equicontinuous subsets of $C(X)$. If $z \in \theta_o X$, then there exists $x \in X$ such that $f^{\theta_o}(z) = f(x)$ for all $f \in \bigcup H_n = H$.

*Proof:* Define

$$d: X^2 \to \mathbf{R}, \quad d(x, y) = \max_n \min\{1/n, \sup_{f \in H_n} |f(x) - f(y)|\}.$$

Then $d$ is a continuous ultrapseudometric on $X$. Take $Y = X_d$ and let $\pi: X \to Y$ be the quotient map. Then $\pi^{\beta_o}(z) = u \in Y$. Choose $x \in$
X with $\pi(x) = u$, and let $(x_\delta)$ be a net in $X$ converging to $z$ in $\beta_o X$. Now $f(x_\delta) \to f^{\beta_o}(z)$ for all $f \in H$. Since $\pi(x_\delta) \to \pi(x)$, we have that $d(x_\delta, x) \to 0$, and so $|f(x_\delta) - f(x)| \to 0$ for all $f \in H$. Thus, for $f \in H$, we have $f(x) = \lim f(x_\delta) = f^{\beta_o}(z)$, and the result follows.

**Theorem 3.10.** If $H \subset C(X)$ is equicontinuous, then the family

$$H^{\theta_o} = \{f^{\theta_o} : f \in H\}$$

is equicontinuous on $\theta_o X$. Moreover, if $H$ is pointwise bounded, then the same holds for $H^{\theta_o}$

*Proof:* Define

$$d: X^2 \to \mathbb{R}, \quad d(x, y) = \min\{1, \sup_{f \in H} |f(x) - f(y)|\}.$$

Let $\pi^{\theta_o}: \theta_o X \to X_d$ be the continuous extension of the quotient map $\pi: X \to X_d$. Let $z \in \theta_o X$ and $\epsilon > 0$. There exists $x \in X$ such that $\pi^{\theta_o}(z) = \pi(x)$. Let $(x_\delta)$ be a net in $X$ converging to $z$. Then $\pi(x_\delta) \to \pi^{\theta_o}(z) = \pi(x)$ and so $d(x_\delta, x) \to 0$. Thus, for $f \in H$, we have $f^{\theta_o}(z) = \lim f(x_\delta) = f(x)$. The set $W = \{y \in X : d(x, y) \le \epsilon\}$ is $d$-clopen (hence clopen) in $X$ and so $\overline{W}^{\theta_o X} = V$ is clopen in $\theta_o X$. Since $x_\delta \in W$ eventually, it follows that $z \in V$. Now, for $f \in H$ and $a \in V$, we have that $|f^{\theta_o}(a) - f^{\theta_o}(z)| \le \epsilon$. In fact, there exists a net $(y_\gamma)$ in $W$ converging to $a$. Thus

$$|f^{\theta_o}(a) - f^{\theta_o}(z)| = |f(x) - f^{\theta_o}(a)| = \lim_{\gamma} |f(x) - f(y_\gamma)| \le \epsilon.$$

This proves that $H^{\theta_o}$ is equicontinuous on $\theta_o X$. The last assertion follows from the preceding Theorem.

**Theorem 3.11.** $U_c = U_c^X$ is the uniformity $\mathcal{U}$ generated by the family of all continuous ultrapseudometrics on $X$.

*Proof:* Let $(A_i)$ be a clopen partition of $X$ and let $W = \bigcup A_i \times A_i$. Define

$$d(x, y) = \sup_i |f_i(x) - f_i(y)|,$$

where $f_i = \chi_{A_i}$. Then $d$ is a continuous ultrapseudometric on $X$. Since

$$W = \{(x, y) : d(x, y) < 1/2\},$$

it follows that $U_c$ is coarser than $\mathcal{U}$. Conversely, let $d$ be a continuous ultrapseudometric on $X$, $\epsilon > 0$ and $D = \{(x, y) : d(x, y) \le \epsilon\}$. If $\alpha$ is the
P-ADIC SPACES OF CONTINUOUS FUNCTIONS I

clopen partition of X corresponding to the equivalence relation $x \sim y$ iff $d(x, y) \le \epsilon$, then $D = W_\alpha$ and the result follows.

**Theorem 3.12.** Let $(Y_i, f_i)_{i \in I}$ be the family of all pairs $(Y, f)$, where $Y$ is an ultrametric space and $f: X \to Y$ a continuous map. Then

$$ \theta_o X = \bigcap_{i \in I} (f_i^{\beta_o})^{-1}(Y_i). $$

*Proof:* It follows from Theorem 3.6.

**Theorem 3.13.** A Hausdorff zero-dimensional space $X$ is $\theta_o$-complete iff it is homeomorphic to a closed subspace of a product of ultrametric spaces.

*Proof:* Every ultrametric space is $\theta_o$-complete. Thus the sufficiency follows from Theorem 3.5. Conversely, assume that $X$ is $\theta_o$-complete and let $(Y_i, f_i)_{i \in I}$ be as in the preceding Theorem. Then $X = \bigcap_i Z_i$, $Z_i = (f_i^{\beta_o})^{-1}(Y_i)$. Let $Y = \prod Y_i$ with its product topology. The map $u: X \to Y$, $u(x)_i = f_i(x)$, is one-to-one. Indeed, let $x \ne y$ and choose a clopen neighborhood $V$ of $x$ not containing $y$. Let $f = \chi_V$ and

$$ d: X \times Y \to \mathbb{R}, \quad d(a, b) = |f(a) - f(b)|. $$

The quotient map $\pi: X \to X_d$ is continuous and $\pi(x) \ne \pi(y)$, which implies that $u(x) \ne u(y)$. Clearly $u$ is continuous. Also $u^{-1}: u(X) \to X$ is continuous. Indeed, let $V$ be a clopen subset of $X$ containing $x_o$ and consider the pseudometric $d(x, y) = |\chi_V(x) - \chi_V(y)|$. Let $\pi: X \to X_d$ be the quotient map. There exists a $i \in I$ such that $Y_i = X_d$ and $f_i = \pi$. Then

$$ f_i(V) = \pi(V) = \{\pi(x) : \tilde{d}(\pi(x) - \pi(x_o)) < 1\}. $$

The set $\pi(V)$ is open in $Y_i = X_d$. Let $\pi_i: Y \to Y_i$ be the $i$th-projection map and $G = \pi_i^{-1}(\pi(V))$. If $x \in X$ is such that $u(x) \in G$, then $f_i(x) = u(x)_i \in \pi(V)$ and so $d(x, x_o) < 1$, which implies that $x \in V$ since $x_o \in V$. This proves that $u: X \to u(X)$ is a homeomorphism. Finally, $u(X)$ is a closed subspace of $Y$. In fact, let $(x_\delta)$ be a net in $X$ with $u(x_\delta) \to y \in Y$. Then $f_i(x_\delta) \to y_i$ for all $i$. Going to a subnet if necessary, we may assume that $x_\delta \to z \in \beta_o X$. Now $f_i(x_\delta) \to f_i^{\beta_o}(z)$ in $\beta_o Y_i$. But then $f_i^{\beta_o}(z) = y_i \in Y_i$, for all $i$, and hence $z \in \theta_o X = X$, 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 of $Y$ and the result follows.
**Corollary 3.14.** Every Hausdorff ultraparacompact space is homeomorphic 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). Let $f \in C(X)$,

$$A_1 = \{x : |f(x)| \le 1\}, \quad A_{n+1} = \{x : n < |f(x)| \le n + 1\}$$

for $n \ge 1$. Then $(A_n)$ is a clopen partition of X. Let $W = \bigcup_n A_n \times A_n$. By our hypothesis, there are $x_1, \dots, x_N$ in A such that $A \subset \bigcup_1^N W[x_k]$. For each $1 \le k \le N$, there exists $n_k$ such that $x_k \in A_{n_k}$. Then $A \subset \bigcup_1^N A_{n_k}$ and so

$$\|f\|_A \le \max_{1 \le k \le N} n_k,$$

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 of $E'_\sigma = (E', \sigma(E', E))$ is equicontinuous.

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

We will denote by $C_c(X, E)$ the space $C(X, E)$ equipped with the topology of uniform convergence on compact subsets of X. By $M_c(X, E')$ we will denote the space of all $m \in M(X, E')$ with compact support. The dual space of $C_c(X, E)$ coincides with $M_c(X, E')$.

Recall that a zero-dimensional Hausdorff topological space X is called a $\mu_o$-space (see [2]) if every bounding subset of X is relatively compact. We
P-ADIC SPACES OF CONTINUOUS FUNCTIONS I

denote by $\mu_o X$ the smallest of all $\mu_o$-subspaces of $\beta_o X$ which contain $X$. Then $X \subset \mu_o X \subset \theta_o X$ and, for each bounding subset $A$ of $X$, the set $\bar{A}^{\beta_o X}$ is contained in $\mu_o X$ (see [2]). Moreover, if $Y$ is another Hausdorff zero-dimensional space and $f: X \to Y$, then $f^{\beta_o}(\mu_o X) \subset \mu_o Y$ and so there exists a continuous extension $f^{\mu_o}: \mu_o X \to \mu_o Y$ of $f$.

**Theorem 4.2.** Assume that $E' \neq \{0\}$ and let $G = C_c(X, E)$. Then $G$ is polarly barrelled iff $X$ is a $\mu_o$-space and $E$ polarly barrelled.

*Proof:* Assume that $G$ is polarly barrelled.

I. $E$ is polarly barrelled. Indeed, let $\Phi$ be a $w^*$-bounded subset of $E'$ and let $x \in X$. For $u \in E'$, let

$$u_x: G \to \mathbb{K}, \quad u_x(f) = u(f(x)).$$

Let $H = \{u_x: u \in \Phi\}$. For $f \in C(X, E)$, we have

$$\sup_{u \in \Phi} |u_x(f)| = \sup_{u \in \Phi} |u(f(x))| < \infty$$

and so $H$ is a $w^*$-bounded subset of $G'$. By our hypothesis, there exists $p \in \text{cs}(E)$ and $Y$ a compact subset of $X$ such that

$$\{f \in G: \|f\|_{Y,p} \leq 1\} \subset H^o.$$

But then $\{s \in E: p(s) \leq 1\} \subset \Phi^o$ and so $\Phi$ is equicontinuous.

II. $X$ is a $\mu_o$-space. In fact, let $A$ be a bounding subset of $X$ and let $x' \in E'$, $x' \neq 0$. Define $p$ on $E$ by $p(x) = |x'(s)|$. Then $p \in \text{cs}(E)$. The set

$$D = \{f \in G: \|f\|_{A,p} \leq 1\}$$

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

$$\{f \in G: \|f\|_{Y,p} \leq 1\} \subset D.$$

But then $A \subset Y$ and so $\bar{A}$ is compact.

Conversely, suppose that $E$ is polarly barrelled and $X$ a $\mu_o$-space. Let $H$ be a $w^*$-bounded subset of the dual space $M_c(X, E')$ of $G$. Let $s \in E$ and

$$D = \{ms: m \in H\} \subset M(X).$$

For $h \in C_{rc}(X)$, we have that

$$\sup_{m \in H} |<ms, h>| = \sup_{m \in H} |<m, hs>| < \infty.$$
A. KATSARAS

Thus, considering $M(X)$ as the dual of the Banach space $F = (C_{rc}(X), \tau_u)$,
$D$ is $w^*$-bounded of $F'$ and so $\sup_{m \in H} \|ms\| = d_s < \infty$. Hence, $|m(V)s| \le$
$d_s$ for all $V \in K(X)$. It follows that the set

$$
M = \bigcup_{m \in H} m(K(X))
$$

is a w\*-bounded subset of E'. Since E is polarly barrelled, there exists
$p \in cs(E)$ such that $|u(s)| \le 1$ for all $u \in M$ and all $s \in E$ with $p(s) \le 1$.
Hence $\sup_{m \in H} \|m\|_p < \infty$. We may choose $p$ so that $\|m\|_p \le 1$ for all
$m \in H$. Let

$$
Z = S(H) = \overline{\bigcup_{m \in H} \mathrm{supp}(m)}.
$$

Then $Z$ is bounding. In fact, assume that $Z$ is not bounding. Then, by
[6, Proposition 6.6], there exists a sequence $(m_n)$ in $H$ and $f \in C(X, E)$
such that $< m_n, f > = \lambda^n$, for all $n$, where $|\lambda| > 1$, which contradicts the
fact that $H$ is $w^*$-bounded. By our hypothesis now, $Z$ is compact. Since

$$
\{f \in G : \|f\|_{Z,p} \le 1\} \subset H^o,
$$

the result follows.

**Corollary 4.3.** $C_c(X)$ is polarly barrelled iff X is a $\mu_o$-space.

Let now $G, E$ be Hausdorff locally convex spaces. We denote by $L_s(G, E)$
the space $L(G, E)$ of all continuous linear maps, from $G$ to $E$, equipped
with the topology of simple convergence.

**Theorem 4.4.** *Assume that E is polar and let G be polarly barrelled. If E is a μ<sub>o</sub>-space (e.g. when E is metrizable or complete), then L<sub>s</sub>(G, E) is a μ<sub>o</sub>-space.*

*Proof:* Let $\Phi$ be a bounding subset of $L_s(G, E)$. For $x \in G$, the set

$$
\Phi(x) = \{ \phi(x) : \phi \in \Phi \}
$$

is a bounding subset of E and hence its closure $M_x$ in E is compact.
$\Phi$ is a topological subspace of $E^G$ and it is contained in the compact set
$M = \prod_{x \in G} M_x$. Since the closure of $\Phi$ in $E^G$ is compact, it suffices to show
that this closure is contained in $L(G, E)$. To this end, we prove first that,
given a polar neighborhood $W$ of zero in $E$, there exists a neighborhood
P-ADIC SPACES OF CONTINUOUS FUNCTIONS I

$U$ of zero in $G$ such that $\phi(U) \subset W$ for all $\phi \in \Phi$. In fact, for $\phi \in \Phi$, let $\phi'$ be the adjoint map. Let

$$Z = \bigcup_{\phi \in \Phi} \phi'(H),$$

where $H$ is the polar of $W$ in $E'$. If $x \in G$, then $\Phi(x)$ is a bounded subset of $E$ and hence $\Phi(x) \subset \alpha W$, for some $\alpha \in \mathbb{K}$. If now $\phi \in \Phi$ and $u \in H$, then

$$| \langle \phi'(u), x \rangle | = | \langle u, \phi(x) \rangle | \le |\alpha|,$$

which proves that $Z$ is a $w^*$-bounded subset of $G'$. As $G$ is polarly barrelled, the polar $U = Z^o$, of $Z$ in $G$, is a neighborhood of zero and $\phi(U) \subset H^o = W$, for all $\phi \in \Phi$, which proves our claim. Let now $\phi \in E^G$ be in the closure of $\Phi$. Then $\phi$ is linear. There exists a net $(\phi_\delta)$ in $\Phi$ converging to $\phi$ in $E^G$. If $x \in U$, then $\phi(x) = \lim \phi_\delta(x) \in W$, which proves that $\phi$ is continuous. Hence the result follows.

**Corollary 4.5.** If $E$ is polarly barrelled, then the weak dual $E'_\sigma$ of $E$ is a $\mu_o$-space.

**Theorem 4.6.** Suppose that $E$ is polar and $G$ polarly barrelled. For $f \in C(X, E)$, let $f^{\mu_o} : \mu_o X \to \hat{E}$ be its continuous extension. If $T : G \to C_c(X, E)$ is a continuous linear map, then the map

$$\tilde{T} : G \to C_c(\mu_o X, \hat{E}), \quad s \mapsto (Ts)^{\mu_o},$$

is continuous

*Proof:* Note that $\hat{E}$ is $\theta_o$-complete and hence a $\mu_o$-space. Let

$$\phi : X \to L_s(G, E), \quad \langle \phi(x), s \rangle = (Ts)(x).$$

Then $\phi$ is continuous. Since $L_s(G, \hat{E})$ is a $\mu_o$-space, there exists a continuous extension

$$\phi^{\mu_o} : \mu_o X \to L_s(G, \hat{E}).$$

Let now $A$ be a compact subset of $\mu_o X$ and $p$ a polar continuous seminorm on $E$. We denote also by $p$ the continuous extension of $p$ to all of $\hat{E}$. Let

$$V = \{g \in C(\mu_o X, \hat{E}) : \|g\|_{A,p} \le 1\}.$$

The set $\Phi = \phi^{\mu_o}(A)$ is compact in $L_s(G, \hat{E})$. As in the proof of Theorem 4.4, there exists a neighborhood $U$ of zero in $G$ such that

$$\psi(U) \subset W = \{s \in \hat{E} : p(s) \le 1\},$$
for all $\psi \in \Phi$. Now, for $y \in A$ and $s \in U$, we have

$$p((\tilde{T}s)(y)) = p(< \phi^{\mu_o}(y), s>) \le 1$$

and so $\tilde{T}s \in V$. This proves that $\tilde{T}$ is continuous and the result follows.

**Theorem 4.7.** Assume that $E$ is polar and polarly barrelled and let $\tau_o$ be the locally convex topology on $C(X, E)$ generated by the seminorms $f \mapsto \|f^{\mu_o}\|_{A,p}$, where $A$ ranges over the family of all compact subsets of $\mu_o X$ and $p \in \text{cs}(E)$. Then:

(1) $(C(X, E), \tau_o)$ is polarly barrelled and $\tau_o$ is finer than $\tau_b$ (and hence finer than $\tau_c$).

(2) If $\tau$ is any polarly barrelled topology on $C(X, E)$ which is finer than $\tau_c$, then $\tau$ is finer than $\tau_o$. Hence $\tau_o$ is the polarly barrelled topology associated with each of the topologies $\tau_b$ and $\tau_c$.

*Proof:* (1). Since $E$ is polarly barrelled, the same is true for $\hat{E}$. The space

$F = C_c(\mu_o X, \hat{E})$ is polarly barrelled and the map

$$S : (C(X, E), \tau_o) \to F, \quad f \mapsto f^{\mu_o},$$

is a linear homeomorphism. Thus $\tau_o$ is polarly barrelled. Also, since for each bounding subset $B$ of $X$, its closure $\bar{B}^{\mu_o X}$ is compact, it follows that $\tau_o$ is finer than $\tau_b$.

(2). Let $\tau$ be a polarly barrelled topology on $C(X, E)$, which is finer than $\tau_c$, and let $G = (C(X, E), \tau)$. The identity map

$$T : G \to C_c(X, E)$$

is continuous and hence the map

$$\tilde{T} : G \to C_c(\mu_o X, \hat{E}), \quad f \mapsto f^{\mu_o},$$

is continuous. This proves that $\tau_o$ is coarser than $\tau$ and the Theorem follows.

**Theorem 4.8.** Suppose that $E$ is polar. Then $G = (C(X, E), \tau_b)$ is polarly barrelled iff $E$ is polarly barrelled and, for each compact subset $A$ of $\mu_o X$, there exists a bounding subset $B$ of $X$ such that $A \subset \bar{B}^{\mu_o X}$.
P-ADIC SPACES OF CONTINUOUS FUNCTIONS I

*Proof:* Assume that $G$ is polarly barrelled. It is easy to see that $E$ is polarly barrelled. In view of the preceding Theorem, $\tau_b = \tau_o$. Thus, for each compact subset $A$ of $\mu_o X$ and each non-zero $p \in cs(E)$, there exist a bounding subset $B$ of $X$ and $q \in cs(E)$ such that

$$ \{f \in C(X, E) : \|f\|_{B,q} \le 1\} \subset \{f : \|f^{\mu_o}\|_{A,p} \le 1\}. $$

It follows easily that $A \subset \bar{B}^{\mu_o X}$. Conversely, suppose that the condition is satisfied. The condition clearly implies that $\tau_o$ is coarser than $\tau_b$ and hence $\tau_b = \tau_o$, which implies that $G$ is polarly barrelled by the preceding Theorem.

Let us say that a family $\mathcal{F}$ of subsets of a set $Z$ is finite on a subset $F$ of $Z$ if the family of all members of $\mathcal{F}$ 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 $\mathcal{F}$ 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 $w_o$-space) if every $w$-bounded (resp. $w_o$-bounded) subset is relatively compact.

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

*Proof:* Assume that $D$ is not $w$-bounded. Then, there exists an infinite sequence $(x_n)$ of distinct elements of $D$ and a sequence $(V_n)$ of open sets such that $x_n \in V_n$ and $(V_n)$ is finite on each compact subset of $X$. By [5, Lemma 2.5], there exists a subsequence $(x_{n_k})$ and pairwise disjoint clopen sets $W_k$ with $x_{n_k} \in W_k$. We may choose $W_k \subset V_{n_k}$. 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 $E' \ne \{0\}$. Then $G = C_c(X, E)$ is polarly quasi-barrelled iff $E$ is polarly quasi-barrelled and $X$ a $w_o$-space.

*Proof:* Suppose that $E$ is polarly quasi barrelled and $X$ a $w_o$-space. Let $H$ be a strongly bounded subset of the dual space $M_c(X, E)$ of $G$. We
A. KATSARAS

show first that there exists $p \in \mathcal{c}s(E)$ such that $\sup_{m \in H} \|m\|_p < \infty$. In fact, let $B$ be a bounded subset of $E$ and consider the set

$$D = \{ms : m \in H, s \in B\}.$$

If $h \in C_{rc}(X)$, then the set $\{hs : s \in B\}$ is a bounded subset of $G$ and so

$$\sup_{m \in H} \left| \int h s \, dm \right| = \sup_{m \in H} \left| \int h \, d(ms) \right| < \infty.$$

Considering $D$ a a subset of the dual of the Banach space $F = (C_{rc}(X), \tau_u)$, we see that $D$ is a $w^*$-bounded subset of $F'$ and hence equicontinuous. Thus

$$d = \sup_{m \in H, s \in B} \|ms\| < \infty.$$

Let

$$\Phi = \bigcup_{m \in H} m(K(X)).$$

Then for $A \in K(X)$, $s \in B$, $m \in H$, we have $|m(A)s| \le \|ms\| \le d$. Hence $\Phi$ is a strongly bounded subset of $E'$. By our hypothesis, $\Phi$ is an equicontinuous subset of $E'$. Thus, there exists $p \in \mathcal{c}s(E)$ such that $|m(A)s| \le 1$ for all $m \in H$ and all $s \in E$ with $p(s) \le 1$. It follows from this that $\sup_{m \in H} \|m\|_p = r < \infty$. We may choose $p$ so that $r \le 1$. Let now

$$Y = S(H) = \overline{\bigcup_{m \in H} \text{supp}(m)}.$$

Then $Y$ is $w_o$-bounded. Assume the contrary. Then, there exists a sequence $(V_n)$ of distinct clopen subsets of $X$, such that $V_n \cap Y \ne \emptyset$ for all $n$ and $(V_n)$ is finite on each compact subset of $X$. . For each $n$ there exists $m_n \in H$ with $V_n \cap \text{supp}(m_n) \ne \emptyset$. Then $(m_n)_p(V_n) > 0$. There are a clopen subset $W_n$ of $V_n$ and $s_n \in E$, with $p(s_n) \le 1$, such that $m(W_n)s_n = \gamma_n \ne 0$. Let $|\lambda| > 1$ and take

$$M = \{\gamma_n^{-1} \lambda^n \chi_{W_n} s_n : n \in \mathbb{N}\}.$$

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 $\lambda_o \ne 0$ be such that $M \subset \lambda_o H^o$. But then

$$1 \ge |\lambda_o^{-1} \gamma_n^{-1} \lambda^n m_n(W_n)s_n| = |\lambda_o^{-1} \lambda^n|$$
P-ADIC SPACES OF CONTINUOUS FUNCTIONS I

for all $n$, which is a contradiction. So $Y$ is $w_o$-bounded and hence compact by our hypothesis. Moreover

$$ \{f \in G : \|f\|_{Y,p} \le 1\} \subset H^o. $$

Indeed, let $\|f\|_{Y,p} \le 1$. The set $V = \{x : p(f(x)) > 1\}$ is disjoint from $Y$ and hence $m_p(V) = 0$ for all $m \in H$. Thus, for $m \in H$, we have

$$ \left| \int_V f \, dm \right| \le \|f\|_p \cdot m_p(V) = 0 $$

and so

$$ \left| \int f \, dm \right| = \left| \int_{V^c} f \, dm \right| \le m_p(V^c) \le 1. $$

Conversely, suppose that $G$ is polarly quasi-barrelled. Let $\Phi$ be a strongly bounded subset of $E'$ and let $x \in X$. For $u \in E'$, define $u_x$ on $G$ by $u_x(f) = u(f(x))$. Then $u_x \in G'$. The set $H = \{u_x : u \in \Phi\}$ is a strongly bounded subset of $G'$. Indeed, let $D$ be a bounded subset of $G$. Since the set $\{f(x) : f \in D\}$ is a bounded subset of $E$, we have that

$$ \sup_{f \in D, u \in \Phi} |u_x(f)| = \sup_{f \in D, u \in \Phi} |u(f(x))| < \infty. $$

By our hypothesis, $H$ is an equicontinuous subset of $G'$. Thus, there exists a compact subset $Y$ of $X$ and $p \in cs(E)$ such that

$$ \{f \in G : \|f\|_{Y,p} \le 1\}. $$

But then $\{s \in E : p(s) \le 1\} \subset \Phi^o$ and so $\Phi$ is an equicontinuous subset of $E'$, which proves that $E$ is polarly quasi-barrelled. Finally, let $A$ be a $w_o$-bounded subset of $X$ and choose a non-zero element $x'$ of $E'$. Let $p(s) = |x'(s)|$ and consider the set

$$ Z = \{f \in G : \|f\|_{A,p} \le 1\}. $$

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 $(f_n)$ in $M$, $\|f_n\|_{A,p} > n$. Let

$$ V_n = \{x : p(f_n(x)) > n\}. $$

Then $V_n$ intersects $A$. Since $A$ is $w_o$-bounded, there exists a compact subset $Y$ of $X$ such that $(V_n)$ is not finite on $Y$, which is a contradiction since $\sup_{f \in M} \|f\|_{Y,p} < \infty$. This contradiction shows that $Z$ absorbs bounded
A. KATSARAS

subsets of $G$. In view of our hypothesis, there exist a compact subset $Y$ of $X$ and $q \in cs((E)$ such that

$$ \{f \in G : \|f\|_{Y,q} \le 1\}, $$

which implies that $A \subset Y$ and so $A$ is relatively compact. This clearly completes the proof.

**Corollary 4.13.** (1) $C_c(X)$ is polarly quasi-barrelled iff $X$ is a $w_o$-space.

(2) If $E' \ne \{0\}$, then $C_c(X, E)$ is polarly quasi-barrelled iff both $E$ and $C_c(X)$ are polarly quasi-barrelled.

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ATHANASIOS KATSARAS
Department of Mathematics
University of Ioannina
Ioannina, 45110
Greece
akatsar@cc.uoi.gr