Two Velichko-like Theorems for C(X) ^†

Abstract

This paper provides two new Velichko-like theorems for the weak counterpart of the locally convex space $C_k\left(X\right)$ of all real-valued functions defined on a Tychonoff space X equipped with the compact-open topology ${\tau }_k$.

1. Preliminaries

Henceforth, unless otherwise stated, X will be a nonempty completely regular Hausdorff space. We represent by $C_p\left(X\right)$ the linear space $C\left(X\right)$ of real-valued continuous functions defined on X equipped with the pointwise topology ${\tau }_p$. The topological dual of $C_p\left(X\right)$ is denoted by $L\left(X\right)$, or by $L_p\left(X\right)$ when provided with the weak* topology $\sigma \left(L\left(X\right),C\left(X\right)\right)$, so that ${\tau }_p=\sigma \left(C\left(X\right),L\left(X\right)\right)$. The linear space $C\left(X\right)$ equipped with the compact-open topology ${\tau }_k$ is represented by $C_k\left(X\right)$. In what follows, we shall denote by $C_w\left(X\right)$ the weak counterpart of the locally convex space $C_k\left(X\right)$, i.e., the space $C\left(X\right)$ equipped with the weak topology $\sigma \left(C\left(X\right),E\right)$, where E stands for the dual of $C_k\left(X\right)$.

The current work supplements the research carried out in [1,2] and is related to [3], Chapter 9, and [4,5,6]. It must be regarded as a part of the important growth of $C_p$-theory and $C_k$-theory that is taking place nowadays (see, for example, [7,8,9,10]).

2. Introduction

A set M of $C_p\left(X\right)$ is called (relatively) sequentially complete if each Cauchy sequence ${\left\{f_n\right\}}_{n=1}^{\infty }$ of $C_p\left(X\right)$ contained in M converges in $C_p\left(X\right)$ to a function $f\in M$ (respectively, to some $f\in C\left(X\right)$). The classical theorem of N. V. Velichko ([11], 1.2.1 Theorem) together with two different generalizations reads as follows:

Theorem 1.

Each of the following statements implies that X is finite:

1. 

The space $C_p\left(X\right)$ is σ-compact (Velichko).

2. 

The space $C_p\left(X\right)$ is σ-countably compact (Tkachuk and Shakhmatov, [12]).

3. 

The space $C_p\left(X\right)$ is σ-bounded relatively sequentially complete (Ferrando, Ka̧kol and Saxon, [2], Corollary 3.2).

Here, $C_p\left(X\right)$ is said to have a $\sigma$-topological property if there is a sequence $\left\{A_n:n\in \mathbb{N}\right\}$ of subsets of $C\left(X\right)$, such that each set $A_n$ enjoys this ${\tau }_p$-topological property. Also, the term bounded is meant in the locally convex sense ([13], 1.4.5 Definition). The first and second statements of the previous theorem are equivalent by virtue of Theorem 2 below, which is a part of [14], Proposition 1 (see also [3], Proposition 9.6).

In the next example, $C_p^b\left(X\right)$ denotes the linear subspace of $C\left(X\right)$ consisting of bounded functions equipped with the relative pointwise topology. This example shows that the third statement of Theorem 1 does not work for $C_p^b\left(X\right)$ instead of $C_p\left(X\right)$.

Example 1.

If $C_p^b\left(X\right)$ is covered by a sequence of pointwise-bounded sequentially complete bounded sets, X need not be finite.

Proof. 

If B stands for the closed unit ball of the Banach space $\left(C^b\left(\mathbb{N}\right),{∥·∥}_{\infty }\right)$, then $C^b\left(\mathbb{N}\right)={\bigcup }_{n=1}^{\infty }nB$. So, $\{nB:n\in \mathbb{N}\}$ is a sequence of pointwise-bounded sets. Since $C_p\left(\mathbb{N}\right)={\mathbb{R}}^{\mathbb{N}}$ is sequentially complete, if ${\left\{f_n\right\}}_{n=1}^{\infty }$ is a Cauchy sequence in $C_p^b\left(\mathbb{N}\right)$ contained in B, there exists $f\in {\mathbb{R}}^{\mathbb{N}}$, such that $f_n\to f$ in ${\mathbb{R}}^{\mathbb{N}}$. But, as $\left|f_n\left(y\right)\right|\le 1$ for all $\left(n,y\right)\in \mathbb{N}\times Y$, we obtain $\left|f\left(y\right)\right|\le 1$ for all $y\in Y$. Thus, $f\in B$, which shows that B is sequentially complete in $C_p^b\left(\mathbb{N}\right)$. Consequently, each set $nB$ is sequentially complete. But $X=\mathbb{N}$ is infinite. □

Let us recall that a resolution for a topological space X is a covering $\{A_{\alpha }:\alpha \in {\mathbb{N}}^{\mathbb{N}}\}$ of X, such that $A_{\alpha }\subseteq A_{\beta }$ whenever $\alpha \le \beta$ coordinate-wise, i.e., if $\alpha \left(i\right)\le \beta \left(i\right)$ for every $i\in \mathbb{N}$. If E is a locally convex space, a resolution $\{A_{\alpha }:\alpha \in {\mathbb{N}}^{\mathbb{N}}\}$ for E is called bounded if each $A_{\alpha }$ is a bounded set in E. An increasing covering $\left\{Q_n:n\in \mathbb{N}\right\}$ of a locally convex space E consisting of absolutely convex bounded sets is a trivial example of a bounded resolution $\{A_{\alpha }:\alpha \in {\mathbb{N}}^{\mathbb{N}}\}$ for E, by setting $A_{\alpha }=Q_{\alpha \left(1\right)}$ for each $\alpha \in {\mathbb{N}}^{\mathbb{N}}$. The next result was obtained in [14], Proposition 1.

Theorem 2

(Ferrando–Ka̧kol). The space $C_p\left(X\right)$ has a bounded resolution if and only if $C_p\left(X\right)$ is both K-analytic-framed in ${\mathbb{R}}^X$ and angelic.

The last statement of Theorem 1 is actually a consequence of [3], Lemma 9.5 together with the following characterization for X to be a P-space, in which the requirement of pointwise boundedness for the covering sequence, which is required in the third statement of Theorem 1, has been dropped ([2], Theorem 3.1).

Theorem 3

(Ferrando–Ka̧kol–Saxon). The space $C_p\left(X\right)$ is σ-relatively sequentially complete if and only if X is a P-space.

In this paper, we are going to prove the following two Velichko-type theorems for the weak counterpart $C_w\left(X\right)$ of $C_k\left(X\right)$.

Theorem 4.

The space $C_w\left(X\right)$ is covered by a sequence of relatively sequentially complete sets if and only if X is a P-space.

Theorem 5.

The space $C_w\left(X\right)$ is covered by a sequence of pointwise-bounded relatively sequentially complete sets if and only if X is finite.

Theorems 4 and 5 are, respectively, the $C_w\left(X\right)$-version of Theorem 3 and of the third statement of Theorem 1.

3. An Auxiliary Result

Recall that a sequence ${\left\{f_n\right\}}_{n=1}^{\infty }$ in $C\left(X\right)$ is called pointwise eventually constant (cf. [2]) if for each $x\in X$ there exists $f\left(x\right)\in \mathbb{R}$, such that $f_n\left(x\right)=f\left(x\right)$ for all but finitely many $n\in \mathbb{N}$. So, if ${\left\{f_n\right\}}_{n=1}^{\infty }$ is a pointwise eventually constant sequence in $C\left(X\right)$, there always exists some $f\in {\mathbb{R}}^X$, such that $f_n\to f$ pointwise on X. We shall require the following theorem, which is contained (but not explicitly stated) in [2], it being a consequence of [2], Theorem 1.1, and of the equivalence of statements (7) and (7’) of paper [2] (see [2], p. 910).

Theorem 6.

The following statements are equivalent.

1. 

Every uniformly bounded pointwise eventually constant sequence converges in $C_p\left(X\right)$.

2. 

X is a P-space.

If X is a compact space and $\mu$ is a regular countably additive real-valued measure defined on the Borel algebra $\mathfrak{B}\left(X\right)$ of X, we shall denote by $L_0\left(\mu \right)$ the linear space of all (classes of) real-valued $\mu$-measurable functions defined on X, and we shall denote by $rca\left(\mathfrak{B}\left(X\right)\right)$ the linear space of regular countably additive Borel real measures on $\mathfrak{B}\left(X\right)$. As for the pointwise topology, a subset M of $C_w\left(X\right)$ is called relatively sequentially complete if each Cauchy sequence ${\left\{f_n\right\}}_{n=1}^{\infty }$ of $C_w\left(X\right)$ contained in M converges in $C_w\left(X\right)$ to some $f\in C\left(X\right)$.

Theorem 7.

Let X be a compact set. The space $C_w\left(X\right)$ is σ-relatively sequentially complete if and only if X is finite.

Proof. 

First, let us show that every uniformly bounded sequence in $C\left(X\right)$ that is pointwise convergent in ${\mathbb{R}}^X$ is a Cauchy sequence in $C_w\left(X\right)$. So, let ${\left\{g_n\right\}}_{n=1}^{\infty }$ be a uniformly bounded sequence in $C\left(X\right)$ pointwise convergent in ${\mathbb{R}}^X$. If B denotes the closed unit ball of the Banach space $C_k\left(X\right)$, there is $\delta >0$, such that $g_n\in \delta B$ for every $n\in \mathbb{N}$, so that ${sup}_{x\in X}\left|g_n\left(x\right)\right|\le \delta$ for all $n\in \mathbb{N}$. As $g_n\to g$ pointwise on X, then $g\in L_0\left(\mu \right)$ and

$$⟨g_n-g,\mu ⟩={\int }_X\left(g_n-g\right)d\mu \to 0$$

for every $\mu \in rca\left(\mathfrak{B}\left(X\right)\right)$. Thus, using the fact that $rca\left(\mathfrak{B}\left(X\right)\right)=C_k{\left(X\right)}^{*}$, we establish that ${\left\{g_n\right\}}_{n=1}^{\infty }$ is a Cauchy sequence in $C_w\left(X\right)$, as stated.

The following argument is based on (but not identical to) the proof of [2], Theorem 3.1. Let $\left\{A_n:n\in \mathbb{N}\right\}$ be a sequence of weakly relatively sequentially complete subsets of $C_k\left(X\right)$. As $C_k\left(X\right)=\left(C\left(X\right),{∥·∥}_{\infty }\right)$ is a Banach space, the Baire category theorem provides $m\in \mathbb{N}$, such that $\overline{A_m}$ has an interior point $h\in C\left(X\right)$ where the closure is in the norm-topology of $C_k\left(X\right)$. So, if B stands again for the closed unit ball of $C_k\left(X\right)$, there is $ϵ>0$, such that

$$h+ϵB\subseteq \overline{A_m}.$$

If ${\left\{f_n\right\}}_{n=1}^{\infty }$ is a uniformly bounded pointwise eventually constant sequence in $C\left(X\right)$, there is $f\in {\mathbb{R}}^X$, such that $f_n\to f$ in ${\mathbb{R}}^X$ and there exists $\delta >0$, such that $\delta f_n\in ϵB$ for every $n\in \mathbb{N}$. So, ${\left\{h+\delta f_n\right\}}_{n=1}^{\infty }$ is a uniformly bounded pointwise eventually constant sequence in $\overline{A_m}$ that converges to $h+\delta f$ in ${\mathbb{R}}^X$. Clearly, for each $n\in \mathbb{N}$ there is $g_n\in A_m$, such that

$${∥h+\delta f_n-g_n∥}_{\infty }<n^{-1},$$

where ${∥·∥}_{\infty }$ denotes the norm of $C_k\left(X\right)$.

All this implies that the sequence ${\left\{g_n\right\}}_{n=1}^{\infty }$ is uniformly bounded and converges pointwise to $h+\delta f$ in ${\mathbb{R}}^X$, so that, according to the first part of the proof, ${\left\{g_n\right\}}_{n=1}^{\infty }$ is a Cauchy sequence in $C_w\left(X\right)$. But, as ${\left\{g_n\right\}}_{n=1}^{\infty }\subseteq A_m$ and $A_m$ is weakly relatively sequentially complete, it follows that $g_n\to h+\delta f$ in $C_w\left(X\right)$. Particularly, one has $h+\delta f\in C\left(X\right)$. As $h\in C\left(X\right)$, this means that $f\in C\left(X\right)$. Therefore, ${\left\{f_n\right\}}_{n=1}^{\infty }$ converges in $C_p\left(X\right)$.

Hence, according to Theorem 6, X must be a P-space. But every compact P-space is finite (see [15], Problem 4K). □

Remark 1.

Theorem 7 clearly implies the well-known fact that if X is a compact space the Banach space $C_k\left(X\right)=\left(C\left(X\right),{∥·∥}_{\infty }\right)$ is weakly sequentially complete if and only if X is finite (note that if the compact set X is infinite, then $\left(C\left(X\right),{∥·∥}_{\infty }\right)$ contains an isomorphic copy of $c_0$, which is not weakly sequentially complete).

4. Proofs of Theorems 4 and 5

In the present section, we prove Theorems 4 and 5, which are concerned with the weak topology ${\tau }_w$ of $C_k\left(X\right)$ rather than the pointwise topology ${\tau }_p$. We shall need the following result, which is an extension of a classic result of $C_p$-theory (see [16], Proposition 4.1), which we have borrowed from [17].

Lemma 1

([17], Lemma 1). Let X be completely regular. If Q is a metrizable and compact subspace of X there exists a continuous-linear-extender map $\phi :C_k\left(Q\right)\to C_k\left(X\right)$, i.e., such that ${\left.\phi \left(f\right)\right|}_Q=f$ for every $f\in C\left(Q\right)$.

4.1. Proof of Theorem 4

Proof. 

Suppose that $C\left(X\right)$ is covered by a sequence $\left\{A_n:n\in \mathbb{N}\right\}$ consisting of weakly relatively sequentially complete sets. We claim that all compact sets of X are finite.

Assume for the sake of contradiction that there is an infinite compact set Q in X that is, hence, metrizable by the Urysohn metrizability theorem. By Lemma 1, there is a linear-continuous map $\phi :C_k\left(Q\right)\to C_k\left(X\right)$, such that ${\left.\phi \left(f\right)\right|}_Q=f$, i.e., a continuous linear extender when $C\left(Q\right)$ is regarded as a Banach space. Consequently, the linear map $\phi$ is weak-to-weak continuous and is, hence, uniformly continuous for the weak topologies. Let us observe that $\left\{{\phi }^{-1}\left(A_n\right):n\in \mathbb{N}\right\}$ is a countable covering of $C_k\left(Q\right)$ consisting of weakly relatively sequentially complete sets. In fact, if ${\left\{f_n\right\}}_{n=1}^{\infty }$ is a weakly Cauchy sequence in ${\phi }^{-1}\left(A_m\right)$ then ${\left\{\phi \left(f_n\right)\right\}}_{n=1}^{\infty }$ is a weakly Cauchy sequence in $A_m$, due to $\phi$ being uniformly continuous, so that there is $h\in C\left(X\right)$, such that $\phi \left(f_n\right)\to h$ in $C_w\left(X\right)$. Now the restriction map $T:C_k\left(X\right)\to C_k\left(Q\right)$ defined by $Tf={\left.f\right|}_Q$ is continuous, which implies that T is also weak-to-weak continuous. Therefore, $T\phi \left(f_n\right)\to Th$ in $C_w\left(Q\right)$. Hence, setting $f:=Th={\left.h\right|}_Q\in C\left(Q\right)$ and employing $T\phi \left(f_n\right)={\left.\phi \left(f_n\right)\right|}_Q=f_n$, it follows that $f_n\to f$ in $C_w\left(Q\right)$. Thus, the set ${\phi }^{-1}\left(A_m\right)$ is relatively sequentially complete in $C_w\left(Q\right)$, which shows, as stated, that the sequence $\left\{{\phi }^{-1}\left(A_n\right):n\in \mathbb{N}\right\}$ is a covering of $C_k\left(Q\right)$ consisting of weakly relatively sequentially complete sets. Now the application of Theorem 7 guarantees that Q is finite, as desired.

As every compact set of X is finite, we obtain $C_p\left(X\right)=C_w\left(X\right)=C_k\left(X\right)$, which means that the space $C_p\left(X\right)$ is $\sigma$-relatively sequentially complete. Hence, Theorem 3 yields that X is a P-space.

Conversely, if X is a P-space, the compact sets of X are finite and, consequently, we obtain $C_p\left(X\right)=C_w\left(X\right)$. So, it follows from Theorem 3 that $C_w\left(X\right)$ is $\sigma$-relatively sequentially complete. □

4.2. Proof of Theorem 5

Proof. 

Finite X ensures a suitable sequence, with each $A_n=nB$, where B is the closed unit ball in the finite-dimensional Banach space $C_p\left(X\right)=C_w\left(X\right)=C_k\left(X\right)$. To prove the converse, recall the well-known property that a Tychonoff space X is pseudocompact if and only if $C_p\left(X\right)$ is $\sigma$-bounded (in the locally convex sense). Hence, if we suppose that space $C\left(X\right)$ is covered by a sequence $\left\{A_n:n\in \mathbb{N}\right\}$ of pointwise-bounded weakly relatively sequentially complete sets, then X is pseudocompact, and by Theorem 4 we obtain that X is a P-space. Therefore, X must be finite. □

We thank our anonymous reviewer for the simplification of this proof by employing the mentioned characterization of pseudocompactness of a Tychonoff space X.

Remark 2.

It is shown in [3], Proposition 9.18, that if $C_p\left(X\right)$ has a fundamental sequence consisting of bounded sets (i.e., a sequence of bounded sets that swallows the bounded sets) then X is finite. Of course, this property does not hold for $C_w\left(X\right)$, because if X is any compact set and B stands for the closed unit ball of $C_k\left(X\right)$ then $\left\{nB:n\in \mathbb{N}\right\}$ is a fundamental sequence of bounded sets for $C_w\left(X\right)$.

5. Conclusions

In this paper, we have provided a complement to the research of [1,2] with Theorems 4 and 5, stating that if X is a Tychonoff space, then $C_k\left(X\right)$ is covered by a sequence of

• Weakly relatively sequentially complete sets if and only if X is a P-space.
• Pointwise-bounded weakly relatively sequentially complete sets if and only if X is finite.