$\omega ^\omega$-Base and infinite-dimensional compact sets in locally convex spaces

Abstract

A locally convex space (lcs) E is said to have an $\omega ^{\omega }$-base if E has a neighborhood base $\{U_{\alpha }:\alpha \in \omega ^\omega \}$ at zero such that $U_{\beta }\subseteq U_{\alpha }$ for all $\alpha \le \beta$. The class of lcs with an $\omega ^{\omega }$-base is large, among others contains all (LM)-spaces (hence (LF)-spaces), strong duals of distinguished Fréchet lcs (hence spaces of distributions $D^{\prime }(\Omega )$). A remarkable result of Cascales-Orihuela states that every compact set in an lcs with an $\omega ^{\omega }$-base is metrizable. Our main result shows that every uncountable-dimensional lcs with an $\omega ^{\omega }$-base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional vector space $\varphi$ endowed with the finest locally convex topology has an $\omega ^\omega$-base but contains no infinite-dimensional compact subsets. It turns out that $\varphi$ is a unique infinite-dimensional locally convex space which is a $k_{\mathbb {R}}$-space containing no infinite-dimensional compact subsets. Applications to spaces $C_{p}(X)$ are provided.

1 Introduction

A topological space X is said to have a neighborhood $\omega ^{\omega }$-base at a point $x\in X$ if there exists a neighborhood base $(U_{\alpha }(x))_{\alpha \in \omega ^{\omega }}$ at x such that $U_{\beta }(x)\subseteq U_{\alpha }(x)$ for all $\alpha \le \beta$ in $\omega ^{\omega }$. We say that X has an $\omega ^{\omega }$-base if it has a neighborhood $\omega ^{\omega }$-base at each point of X. Evidently, a topological group (particularly topological vector space (tvs)) has an $\omega ^{\omega }$-base if it has a neighborhood $\omega ^{\omega }$-base at the identity. The classical metrization theorem of Birkhoff and Kakutani states that a topological group G is metrizable if and only if G is first-countable. Then, as easily seen, if $(U_{n})_{n\in \omega }$ is a neighborgood base at the identity of G, then the family $\{U_{\alpha }:\alpha \in \omega ^{\omega }\}$ formed by sets $U_{\alpha }= U_{\alpha (0)}$ forms an $\omega ^{\omega }$-base (at the identity) for G. Locally convex spaces (lcs) with an $\omega ^{\omega }$-base are known in Functional Analysis since 2003 when Cascales, Ka̧kol, and Saxon [7] characterized quasi-barreled lcs with an $\omega ^{\omega }$-base. In several papers (see [16] and the references therein) spaces with an $\omega ^{\omega }$-base were studied under the name lcs with a ${\mathfrak {G}}$-base, but here we prefer (as in [4]) to use the more self-suggesting terminology of $\omega ^{\omega }$-bases.

In [8] Cascales and Orihuela proved that compact subsets of any lcs with an $\omega ^{\omega }$-base are metrizable. This refers, among others, to each (LM)-space, i.e. a countable inductive limit of metrizble lcs, since (LM)-spaces have an $\omega ^{\omega }$-base. Also the following metrization theorem holds together a number of topological conditions.

Theorem 1.1

[16, Corollary 15.5] For a barrelled lcs E with an $\omega ^{\omega }$-base, the following conditions are equivalent.

• (1) E is metrizable;

• (2) E is Fréchet-Urysohn;

• (3) E is Baire-like;

• (4) E does not contain a copy of $\varphi$, i.e. an $\aleph _{0}$-dimensional vector space endowed with the finest locally convex topology.

Hence every Baire lcs with an $\omega ^{\omega }$-base is metrizable. The space $\varphi$ appearing in Theorem 1.1 has the following properties:

• (1) $\varphi$ is the strong dual of the Fréchet-Schwartz space ${\mathbb {R}}^{\omega }$.

• (2) All compact subsets in $\varphi$ are finite-dimensional.

• (3) $\varphi$ is a complete bornological space,

see [16, 21, 23].

Being motivated by above’s results, especially by a remarkable theorem of Cascales-Oruhuela mentioned above, one can ask for a possible large class of lcs E for which every infinite-dimensional subspace of E contains an infinite-dimensional compact (metrizable) subset. Surely, each metrizable lcs trivially fulfills this request. We prove however the following general

Theorem 1.2

Every uncountably-dimensional lcs E with $\omega ^{\omega }$-base contains an infinite-dimensional metrizable compact subset.

Theorem 1.2 will be proved in Sect. 4. An alternative proof will be presented in Sect. 5 as a consequence of Theorem 5.2.

The uncountable dimensionality of the space E in Theorem 1.2 cannot be replaced by the infinite-dimensionality of E: the space $\varphi$ is infinite-dimensional, has an $\omega ^\omega$-base and contains no infinite-dimensional compact subsets. However, $\varphi$ is a unique locally convex $k_{\mathbb {R}}$-space with this property. Recall [20] that a topological space X is a $k_{\mathbb {R}}$-space if a function $f:X\rightarrow {\mathbb {R}}$ is continuous whenever for every compact subset $K\subseteq X$ the restriction $f{\restriction }K$ is continuous. We prove the following

Theorem 1.3

An lcs E is topologically isomorphic to the space $\varphi$ if and only if E is a $k_{\mathbb {R}}$-space containing no infinite-dimensional compact subsets.

Theorem 1.3 implies that an lcs is topologically isomorphic to $\varphi$ if and only if it is homeomorphic to $\varphi$. This topological uniqueness property of the space $\varphi$ was first proved by the first author in [2].

The following characterization of the space $\varphi$ can be derived from Theorems 1.2 and 2.1. It shows that $\varphi$ is a unique bornological space for which the uncountable dimensionality in Theorem 1.2 cannot be weakened to infinite dimensionality.

Theorem 1.4

An lcs E is topologically isomorphic to the space $\varphi$ if and only if E is bornological, has an $\omega ^\omega$-base and contains no infinite-dimensional compact subset.

Theorem 1.2 provides a large class of concrete (non-metrizable) lcs containing infinite-dimensional compact sets.

Corollary 1.5

Every uncountable-dimensional subspace of an (LM)-space contains an infinite-dimensional compact set.

Let X be a Tychonoff space. By $C_{p}(X)$ and $C_{k}(X)$ we denote the space of continuous real-valued functions on X endowed with the pointwise and the compact-open topology, respectively. The problem of characterization of Tychonoff spaces X whose function spaces $C_{p}(X)$ and $C_{k}(X)$ admit an $\omega ^{\omega }$-base is already solved. Indeed, by [16, Corollary 15.2] $C_{p}(X)$ has an $\omega ^{\omega }$-base if and only if X is countable. The space $C_{k}(X)$ has an $\omega ^{\omega }$-base if and only if X admits a fundamental compact resolution [11], for necessary definitions see below. Since every Čech-complete Lindelöf space X is a continuous image of a Polish space under a perfect map (and the latter space admits a fundamental compact resolution), the space $C_{p}(X)$ has an $\omega ^{\omega }$-base. So, we have another concrete application of Theorem 1.2.

Example 1.6

Let X be an infinite Čech-complete Lindelöf space. Then every uncountable-dimensional subspace of $C_{k}(X)$ contains an infinite-dimensional metrizable compact set.

In Sect. 2 we show that all (bornological) lcs containing no infinite-dimensional compact subsets are bornologically (and topologically) isomorphic to a free lcs over discrete topological spaces. Consequently, in Sects. 3 and 4 we study the free lcs $L(\kappa )$ over infinite cardinals $\kappa$, including $L(\omega )=\varphi$. We introduce two concepts: the $(\kappa ,\lambda )$-tall bornology and the $(\kappa ,\lambda )_p$-equiconvergence, which will be used to obtain bornological and topological characterizations of $L(\kappa )$. Both concepts apply to prove Theorem 1.2. To this end, we shall prove that each topological (vector) space with an $\omega ^\omega$-base is $(\omega _1,\omega )_p$-equiconvergent (and has $(\omega _1,\omega )$-tall bornology). Another property implying the $(\omega _1,\omega )_{p}$-equiconvergence is the existence of a countable ${\mathsf {cs}}^\cdot$-network (see Theorem 4.2), which follows from the existence of an $\omega ^\omega$-base according to Proposition 3.3. Linear counterparts of ${\mathsf {cs}}^\cdot$-networks are radial networks introduced in Sect. 5, whose main result is Theorem 5.2 implying Theorem 1.2. Some applications of Theorem 1.2 to function spaces $C_{p}(X)$ are provided in Sect. 6.

2 Locally convex spaces containing no infinite-dimensional compact subsets

In this section we study lcs containing no infinite-dimensional compact subsets. We shall show that all such (bornological) spaces are bornologically (and topologically) isomorphic to free lcs over discrete topological spaces.

Recall that for a topological space X its free locally convex space is an lcs L(X) endowed with a continuous function $\delta :X\rightarrow L(X)$ such that for any continuous function $f:X\rightarrow E$ to an lcs E there exists a unique linear continuous map $T:L(X)\rightarrow E$ such that $T\circ \delta =f$. The set X forms a Hamel basis for L(X) and $\delta$ is a topological embedding, see [22]; we also refer to [5] and [4] for several results and references concerning this concept; [5, Theorem 5.4] characterizes those X for which L(X) has an $\omega ^{\omega }$-base.

Let E be a tvs. A subset $B\subseteq E$ is called bounded if for every neighborhood $U\subseteq E$ of zero there exists $n\in {\mathbb {N}}$ such that $B\subseteq nU$. The family of all bounded sets of E is called the bornology of E. A linear operator $f:E\rightarrow F$ between two tvs is called bounded if for any bounded set $B\subseteq E$ its image f(B) is bounded in F.

Two tvs E and F are

• topologically isomorphic if there exists a linear bijective function $f:E\rightarrow F$ such that f and $f^{-1}$ are continuous;

• bornologically isomorphic if there exists a linear bijective function $f:E\rightarrow F$ such that f and $f^{-1}$ are bounded.

An lcs E is called bornological if each bounded linear operator from E to an lcs F is continuous. A linear space E is called $\kappa$-dimensional if E has a Hamel basis of cardinality $\kappa$. In this case we write $\kappa =\dim (E)$.

An lcs E is free if it carries the finest locally convex topology. In this case E is topologically isomorphic to the free lcs $L(\kappa )$ over the cardinal $\kappa =\dim (E)$ endowed with the discrete topology.

The study around the free lcs $L(\omega )=\varphi$ has attracted specialists for a long time. For example, Nyikos observed [21] that each sequentially closed subset of $L(\omega )$ is closed although the sequential closure of a subset of $\varphi$ need not be closed. Consequently, $L(\omega )$ is a concrete “small” space without the Fréchet-Urysohn property. Applying the Baire category theorem one shows that $L(\omega )$ is not a Baire-like space (in sense of Saxon [23]) and a barrelled lcs E is Baire-like if E does not contain a copy of $L(\omega )$, see [23]. Although $L(\omega )$ is not Fréchet-Urysohn, it provides some extra properties since all vector subspaces in $L(\omega )$ are closed. In [17] we introduced the property for an lcs E (under the name $C_{3}^{-}$) stating that the sequential closure of every linear subspace of E is sequentially closed, and we proved [17, Corollary 6.4] that the only infinite-dimensional Montel (DF)-space with property $C_{3}^{-}$ is $L(\omega )$ (yielding a remarkable result of Bonet and Defant that the only infinite-dimensional Silva space with property $C_{3}^{-}$ is $L(\omega )$). This implies that barrelled (DF)-spaces and (LF)-spaces satisfying property $C_{3}^{-}$ are exactly of the form M, $L(\omega )$, or $M\times L(\omega )$ where M is metrizable, [17, Theorems 6.11, 6.13].

The following simple theorem characterizes lcs containing no infinite-dimensional compact subsets.

Theorem 2.1

For an lcs E the following conditions are equivalent:

• (1) Each compact subset of E has finite topological dimension.

• (2) Each bounded linearly independent set in E is finite.

• (3) E is bornologically isomorphic to a free lcs.

  If E is bornological, then the conditions (1)–(3) are equivalent to

• (4) E is free.

Proof

$(1)\Rightarrow (2)$ Suppose that each compact subset of E has finite topological dimension. Assuming that E contains an infinite bounded linearly indendent set, we can find a bounded linearly independent set $\{x_n\}_{n\in \omega }$ consisting of pairwise distinct points $x_n$. Then the sequence $(2^{-n}x_n)_{n\in \omega }$ converges to zero and

$$\begin{aligned} K=\textstyle \bigcup _{n\in \omega }\big \{\sum \limits _{k=n}^{2n}t_kx_k:(t_k)_{k=n}^{2n}\in \prod \limits _{k=n}^{2n}[0,2^{-k}]\big \} \end{aligned}$$

is an infinite-dimensional compact set in E, which contradicts our assumption.

$(2)\Rightarrow (3)$ Let $\tau$ be the finest locally convex topology on E. Then the identity map $(E,\tau )\rightarrow E$ is continuous and hence bounded. If each bounded linearly independent set in E is finite, then each bounded set $B\subseteq E$ is contained in a finite-dimensional subspace of E and hence is bounded in the topology $\tau$. This means that the identity map $E\rightarrow (E,\tau )$ is bounded and hence E is bornologically isomorphic to the free lcs $(E,\tau )$.

$(3)\Rightarrow (1)$ If E is bornologically isomorphic to a free lcs F then each bounded linearly independent set in E is finite, since the free lcs F has this property.

The implication $(4)\Rightarrow (3)$ is trivial. If E is bornological then the implication $(3)\Rightarrow (4)$ follows from the continuity of bounded linear operators on bornological spaces. $\square$

The free lcs over discrete topological spaces are not unique lcs possessing no infinite-dimensional compact sets. A subset B of a topological space X is called functionally bounded if for any continuous real-valued function $f:X\rightarrow {\mathbb {R}}$ the set f(B) is bounded.

Proposition 2.2

For a Tychonoff space X the following conditions are equivalent:

• (1) each compact subset of the free lcs L(X) has finite topological dimension;

• (2) each bounded linearly independent set in L(X) is finite;

• (3) each functionally bounded subset of X is finite.

Proof

The equivalence $(1)\Leftrightarrow (2)$ follows from the corresponding equivalence in Theorem 2.1. The implication $(3) \Rightarrow (1)$ follows from [6, Lemma 10.11.3], and $(2) \Rightarrow (3)$ follows from the observation that each functionally bounded set in an lcs is bounded. $\square$

3 Bornological and topological characterizations of the spaces $L(\kappa )$

In this section, given an infinite cardinal $\kappa$ we characterize the free lcs $L(\kappa )$ using some specific properties of the bornology and the topology of the space $L(\kappa )$.

Let $\kappa ,\lambda$ be two cardinals. An lcs E is defined to have $(\kappa ,\lambda )$-tall bornology if every subset $A\subseteq E$ of cardinality $|A|=\kappa$ contains a bounded subset $B\subseteq A$ of cardinality $|A|=\lambda$.

Theorem 3.1

Let $\kappa$ be an infinite cardinal. For an lcs E the following conditions are equivalent:

• (1) E is bornologically isomorphic to the free lcs $L(\kappa )$;

• (2) each bounded linearly independent set in E is finite and the bornology of E is $(\kappa ^+,\omega )$-tall but not $(\kappa ,\omega )$-tall.

  If E is bornological, then the conditions (1)–(2) are equivalent to

• (3) E is topologically isomorphic to $L(\kappa )$.

Proof

$(1) \Rightarrow (2)$: Assume that E is bornologically isomorphic to $L(\kappa )$. Then E has algebraic dimension $\kappa$ and each bounded linearly independent set in E is finite (since this is true in $L(\kappa )$).

To see that the bornology of E is $(\kappa ^+,\omega )$-tall, take any set $K\subseteq E$ of cardinality $|K|=\kappa ^+$. Since E has algebraic dimension $\kappa$, there exists a cover $(B_\alpha )_{\alpha \in \kappa }$ of E by $\kappa$ many compact sets. By the Pigeonhole Principle, there exists $\alpha \in \kappa$ such that $|K\cap B_\alpha |=\kappa ^+$. This means that the bornology of E is $(\kappa ^+,\kappa ^+)$-tall and hence $(\kappa ^+,\omega )$-tall.

To see that the bornology of the space E is not $(\kappa ,\omega )$-tall, observe that the Hamel basis $\kappa$ of $L(\kappa )$ has the property that no infinite subset of $\kappa$ is bounded in $L(\kappa )$. Since E is bornologically isomorphic to $L(\kappa )$, the image of $\kappa$ in E is a subset of cardinality $\kappa$ containing no bounded infinite subsets and witnessing that E is not $(\kappa ,\omega )$-tall.

$(2) \Rightarrow (1)$: Assume that each bounded linearly independent set in E is finite and the bornology of E is $(\kappa ^+,\omega )$-tall but not $(\kappa ,\omega )$-tall. Let B be a Hamel basis of E. We claim that $|B|=\kappa$. Assuming that $|B|>\kappa$, we conclude that E is not $(\kappa ^+,\omega )$-tall, which is a contradiction. Assuming that $|B|<\kappa$, we conclude that E is the union of $<\kappa$ many bounded sets and hence is $(\kappa ,\kappa )$-tall by the Pigeonhole Principle. But this contradicts our assumption. Therefore $|B|=\kappa$. Let $h:\kappa \rightarrow B$ be any bijection and ${{\bar{h}}}:L(\kappa )\rightarrow E$ be the unique extension of h to a linear continuous operator. Since B is a Hamel basis for E, the operator ${{\bar{h}}}$ is bijective. Since each bounded set in E is contained in a finite-dimensional linear subspace, the operator ${{\bar{h}}}^{-1}:E\rightarrow L(\kappa )$ is bounded and hence ${{\bar{h}}}:L(\kappa )\rightarrow E$ is a bornological isomorphism.

If the space E is bornological, then the equivalence $(1)\Leftrightarrow (3)$ follows from the bornological property of E and $L(\kappa )$. $\square$

The $(\kappa ,\omega )$-tallness of the bornology of an lcs E has topological counterparts introduced in the following definition.

Definition 3.2

Let $\kappa ,\lambda$ be cardinals. We say that a topological space X is

• $(\kappa ,\lambda )_p$-equiconvergent at a point $x\in X$ if for any indexed family $\{x_\alpha \}_{\alpha \in \kappa }\subseteq \{s\in X^\omega :\lim _{n\rightarrow \infty }s(n)=x\}$, there exists a subset $\Lambda \subseteq \kappa$ of cardinality $|\Lambda |=\lambda$ such that for every neighborhood $O_x\subseteq X$ of x there exists $n\in \omega$ such that the set $\{\alpha \in \Lambda :x_\alpha (n)\notin O_x\}$ is finite;

• $(\kappa ,\lambda )_k$-equiconvergent at a point $x\in X$ if for any indexed family $\{x_\alpha \}_{\alpha \in \kappa }\subseteq \{s\in X^\omega :\lim _{n\rightarrow \infty }s(n)=x\}$, there exists a subset $\Lambda \subseteq \kappa$ of cardinality $|\Lambda |=\lambda$ such that for every neighborhood $O_x\subseteq X$ of x there exists $n\in \omega$ such that for every $m\ge n$ and $\alpha \in \Lambda$ we have $x_\alpha (m)\in O_x$;

• $(\kappa ,\lambda )_p$-equiconvergent if X is $(\kappa ,\lambda )_p$-equiconvergent at every point $x\in X$;

• $(\kappa ,\lambda )_k$-equiconvergent if X is $(\kappa ,\lambda )_k$-equiconvergent at every point $x\in X$.

It is easy to see that every $(\kappa ,\lambda )_k$-equiconvergent space is $(\kappa ,\lambda )_p$-equiconvergent. The following observation will be used below.

Proposition 3.3

If an lcs E is $(\kappa ,\lambda )_p$-equiconvergent, then its bornology is $(\kappa ,\lambda )$-tall.

Proof

Given a subset $K\subseteq E$ of cardinality $|K|=\kappa$, for every $\alpha \in K$ consider the convergent sequence $x_\alpha \in X^\omega$ defined by $x_\alpha (n)=2^{-n}\alpha$. Assuming that the lcs E is $(\kappa ,\lambda )_p$-equiconvergent, we can find a subset $L\subseteq K$ of cardinality $|L|=\lambda$ such that for every neighborhood of zero $U\subseteq E$ there exists $n\in \omega$ such that the set $\{\alpha \in L:2^{-n}\alpha \notin U\}$ is finite. We claim that the set L is bounded. Indeed, for every neighborhood $U\subseteq E$ of zero, we find a neighborhood $V\subseteq E$ of zero such that $[0,1]\cdot V\subseteq U$. By our assumption, there exists $n\in \omega$ such that the set $F=\{\alpha \in K:2^{-n}\alpha \notin V\}$ is finite. Find $m\ge n$ such that $2^{-m}\alpha \in U$ for every $\alpha \in F$. Then $2^{-m}L\subseteq 2^{-m}(L\setminus F)\cup 2^{-m}F\subseteq ([0,1]\cdot V)\cup U=U$, and hence the set L is bounded. $\square$

Nevertheless, it seems that the following question remains open.

Problem 3.4

Assume that the bornology of an lcs E is $(\omega _1,\omega )$-tall. Is it true that E is $(\omega _1,\omega )_p$-equiconvergent?

Below we prove the following topological counterpart to Theorem 3.1.

Theorem 3.5

Let $\kappa$ be an infinite cardinal. For an lcs E the following conditions are equivalent:

• (1) E is bornologically isomorphic to $L(\kappa )$;

• (2) each compact subset of E has finite topological dimension, E is $(\kappa ^+,\omega )_k$-equiconvergent but not $(\kappa ,\omega )_p$-equiconvergent.

• (3) each compact subset of E has finite topological dimension, E is $(\kappa ^+,\omega )_p$-equiconvergent but not $(\kappa ,\omega )_k$-equiconvergent.

  If E is bornological, then the conditions (1)–(3) are equivalent to

• (4) E is topologically isomorphic to $L(\kappa )$.

Proof

$(1) \Rightarrow (2)$: Assume that E is bornologically isomorphic to $L(\kappa )$. By Theorems 3.1 each bounded linearly independent set in E is finite, and by Theorem 2.1, each compact subset of E is finite-dimensional. The linear space E has algebraic dimension $\kappa$, being isomorphic to the linear space $L(\kappa )$. Let B be a Hamel basis for the space E.

To show that E is $(\kappa ^+,\omega )_k$-equiconvergent, fix an indexed family $\{x_\alpha \}_{\alpha \in \kappa ^+}\subseteq \{s\in E^\omega :\lim _{n\rightarrow \infty }s(n)=0\}.$ Since bounded linearly independent sets in E are finite, for every $\alpha \in \kappa ^+$ there exists a finite set $F_\alpha \subseteq B$ such that the bounded set $x_\alpha [\omega ]$ is contained in the linear hull of $F_\alpha$. Since $|B|=\kappa <\kappa ^+$, by the Pigeonhole Principle, for some finite set $F\subseteq B$ the set $A=\{\alpha \in \kappa ^+:F_\alpha =F\}$ is uncountable. Let [F] be the linear hull of the finite set F in the linear space E.

Consider the ordinal $\omega +1=\omega \cup \{\omega \}$ endowed with the compact metrizable topology generated by the linear order. For every $\alpha \in A$ let ${{\bar{x}}}_\alpha :\omega +1\rightarrow [F]$ be the continuous function such that ${{\bar{x}}}_\alpha {\restriction }\omega =x_\alpha$ and ${{\bar{x}}}_\alpha (\omega )=0$. Let $C_k(\omega +1,[F])$ be the space of continuous functions from $\omega +1$ to [F], endowed with the compact-open topology. Since A is uncountable and the space $C_k(\omega +1,[F])\supseteq \{{{\bar{x}}}_\alpha \}_{\alpha \in A}$ is Polish, there exists a sequence $\{\alpha _n\}_{n\in \omega }\subseteq A$ of pairwise distinct ordinals such that the sequence $(\bar{x}_{\alpha _n})_{n\in \omega }$ converges to ${{\bar{x}}}_{\alpha _0}$ in the function space $C_k(\omega +1,[F])$. Then the set $\Lambda =\{\alpha _n\}_{n\in \omega }\subseteq \kappa ^+$ witnesses that E is $(\kappa ^+,\omega )_k$-equiconvergent to zero and by the topological homogeneity, E is $(\kappa ^+,\omega )$-equiconvergent. By Theorem 3.1, the bornology of the space E is not $(\kappa ,\omega )$-tall. By Proposition 3.3, the space E is not $(\kappa ,\omega )_p$-equiconvergent.

The implication $(2) \Rightarrow (3)$ is trivial. To prove that $(3) \Rightarrow (1)$, assume that each compact subset of E has finite topological dimension and E is $(\kappa ^+,\omega )_p$-equiconvergent but not $(\kappa ,\omega )_k$-equiconvergent. Let B be a Hamel basis in E. By Theorem 2.1, the space E is bornologically isomorphic to L(|B|). Applying the (already proved) implication $(1) \Rightarrow (2)$, we conclude that E is $(|B|^+,\omega )_k$-equiconvergent, which implies that $|B|\ge \kappa$ (as E is not $(\kappa ,\omega )_k$-equiconvergent). Assuming that $|B|>\kappa$, we can see that the family $\{x_b\}_{b\in B}\subseteq E^\omega$ of the sequences $x_b(n)=2^{-n}b$ witnesses that E is not $(|B|,\omega )_p$-equiconvergent and hence not $(\kappa ^+,\omega )_p$-equiconvergent, which contradicts our assumption. So, $|B|=\kappa$ and E is bornologically isomorphic to $L(\kappa )$. If the space E is bornological, then the equivalence $(1)\Leftrightarrow (4)$ follows from the bornological property of E and $L(\kappa )$. $\square$

Observe that the purely topological properties (2), (3) in Theorem 3.5 characterize the free lcs $L(\kappa )$ up to bornological equivalence. We do not know whether the topological structure of the space $L(\kappa )$ determines this lcs uniquely up to a topological isomorphism.

Problem 3.6

Assume that an lcs E is homeomorphic to the free lcs $L(\kappa )$ for some cardinal $\kappa$. Is E topologically isomorphic to $L(\kappa )$?

By [2] the answer to this problem is affirmative for $\kappa =\omega$. This affirmative answer can also be derived from the following topological characterizations of the space $L(\omega )=\varphi$. This characterization has been announced in the introduction as Theorem 1.3.

Theorem 3.7

An lcs E is topologically isomorphic to the free lcs $L(\omega )$ if and only if E is an infinite-dimensional $k_{\mathbb {R}}$-space containing no infinite-dimensional compact subset.

Proof

The “only if” part follows from known topological properties of the space $L(\omega )=\varphi$ mentioned in the introduction. To prove the “if” part, assume that an lcs E is a $k_{\mathbb {R}}$-space and each compact subset of E is finite-dimensional. Choose a Hamel basis B in E and consider the linear continuous operator $T:L(B)\rightarrow E$ such that $T(b)=b$ for each $b\in B$. Since B is a Hamel basis, the operator T is injective. We claim that the operator $T^{-1}:E\rightarrow L(B)$ is bounded. By Theorem 2.1 the linear hull of each compact subset $K\subseteq E$ is finite-dimensional, which implies that the restriction $T^{-1}{\restriction }K$ is continuous. Since E is a $k_{\mathbb {R}}$-space, $T^{-1}$ is continuous and hence T is a topological isomorphism. Then the free lcs L(B) is a $k_{\mathbb {R}}$-space. Applying [15], we conclude that B is countable and hence E is topologically isomorphic to $L(\omega )$. $\square$

A Tychonoff space X is called Ascoli if the canonical map $\delta :X\rightarrow C_k(C_k(X))$ assigning to each point $x\in X$ the Dirac functional $\delta _x:C_k(X)\rightarrow {\mathbb {R}}$, $\delta _x:f\mapsto f(x)$, is continuous. By [3], the class of Ascoli spaces includes all Tychonoff $k_{\mathbb {R}}$-spaces. By [15] a Tychonoff space X is countable and discrete if and only if its free lcs L(X) is Ascoli.

Problem 3.8

Assume that an infinite-dimensional lcs E is Ascoli and contains no infinite-dimensional compact subsets. Is E topologically isomorphic to the space $L(\omega )$?

4 Equiconvergence of topological spaces and proof of Theorem 1.2

In this section we establish two results related to equiconvergence in topological spaces.

Theorem 4.1

If a topological space X admits an $\omega ^\omega$-base at a point $x\in X$, then X is $(\omega _1,\omega )_k$-equiconvergent at the point x.

Proof

Let $(U_f)_{f\in \omega ^\omega }$ be an $\omega ^\omega$-base at x. To show that X is $(\omega _1,\omega )_k$-equiconvergent at x, fix an indexed family

$$\begin{aligned} \{x_\alpha \}_{\alpha \in \omega _1}\subseteq \{s\in X^\omega :\lim _{n\rightarrow \infty }s(n)=x\} \end{aligned}$$

of sequences that converge to x. For every $\alpha \in \omega _1$ consider the function $\mu _\alpha :\omega ^\omega \rightarrow \omega$ assigning to each $f\in \omega ^\omega$ the smallest number $n\in \omega$ such that $\{x_\alpha (m)\}_{m\ge n}\subseteq U_f.$ It is easy to see that the function $\mu _\alpha :\omega ^\omega \rightarrow \omega$ is monotone.

For every $n\in \omega$ and finite function $t\in \omega ^n$, let $\omega ^\omega _t=\{f\in \omega ^\omega :f{\restriction }n=t\}.$ By [4, Lemma 2.3.5], for every $f\in \omega ^\omega$ there exists $n\in \omega$ such that $\mu _\alpha [\omega ^\omega _{f{\restriction }n}]$ is finite. Let $T_\alpha$ be the set of all finite functions $t\in \omega ^{<\omega }=\bigcup _{n\in \omega }\omega ^n$ such that $\mu _\alpha [\omega ^\omega _t]$ is finite but for any $\tau \in \omega ^{<\omega }$ with $\tau \subset t$ the set $\mu _\alpha [\omega ^\omega _\tau ]$ is infinite. It follows from [4, Lemma 2.3.5] that for every $f\in \omega ^\omega$ there exists a unique $t_f\in T_{\alpha }$ such that $t_f\subset f$.

Let $\delta _\alpha (f)=\max \mu _\alpha [\omega ^\omega _{t_f}]\ge \mu _\alpha (f)$. It is clear that the function $\delta _\alpha :\omega ^\omega \rightarrow \omega$ is continuous and hence $\delta _\alpha$ is an element of the space $C_p(\omega ^\omega ,\omega )$ of continuous functions from $\omega ^\omega$ to $\omega$. Here we endow $\omega ^\omega$ with the product topology. The function space $C_p(\omega ^\omega ,\omega )$ is endowed with the topology of poitwise convergence. By Michael’s Proposition 10.4 in [19], the space $C_p(\omega ^\omega ,\omega )$ has a countable network.

Consider the function $\delta :\omega _1\rightarrow C_p(\omega ^\omega ,\omega ), \,\,\,\delta :\alpha \mapsto \delta _\alpha ,$ and observe that $\delta _\alpha (f)\ge \mu _\alpha (f)$ for any $\alpha \in \omega _1$ and $f\in \omega ^\omega$.

Since the space $C_p(\omega ^\omega ,\omega )$ has countable network, there exists a sequence $\{\alpha _n\}_{n\in \omega }\subseteq \omega _1$ of pairwise distinct ordinals such that the sequence $(\delta _{\alpha _n})_{n\in \omega }$ converges to $\delta _{\alpha _0}$ in the function space $C_p(\omega ^\omega ,\omega )$. We claim that the sequence $(x_{\alpha _n})_{n\in \omega }$ witnesses that X is $(\omega _1,\omega )_k$-equiconvergent at x. Given any open neighborhood $O_x\subseteq X$ of x, find $f\in \omega ^\omega$ such that $U_f\subseteq O_x$. Since the sequence $(x_{\alpha _0}(n))_{n\in \omega }$ converges to x, there exists $m\in \omega$ such that $\{x_{\alpha _0}(n)\}_{n\ge m}\subseteq U_f$. Since the sequence $(\delta _{\alpha _n})_{n\in \omega }$ converges to $\delta _{\alpha _0}$ in $C_p(\omega ^\omega ,\omega )$ we can replace m by a larger number and additionally assume that $\delta _{\alpha _n}(f)=\delta _{\alpha _0}(f)$ for all $n\ge m$. Choose a number $l\ge \delta _{\alpha _0}(f)$ such that for every $n<m$ and $k\ge l$ we have $x_{\alpha _n}(k)\in O_x$. On the other hand, for every $n\ge m$ and $k\ge l$ we have $k\ge l\ge \delta _{\alpha _0}(f)=\delta _{\alpha _n}(f)\ge \mu _{\alpha _n}(f)$ and hence $x_{\alpha _n}(k)\in U_f\subseteq O_x$. $\square$

Another property implying the $(\omega _1,\omega )_{p}$-equiconvergence is the existence of a countable ${\mathsf {cs}}^\bullet$-network. First we introduce the necessary definitions.

Let x be a point of a topological space X. We say that a sequence $\{x_n\}_{n\in \omega }\subseteq X$ accumulates at x if for each neighborhood $U\subseteq X$ of x the set $\{n\in \omega :x_n\in U\}$ is infinite.

A family ${\mathcal {N}}$ of subsets of X is defined to be

• an ${\mathsf {s}}^*$-network at x if for any neighborhood $O_x\subseteq X$ of x and any sequence $\{x_n\}_{n\in \omega }\subseteq X$ that accumulates at x there exists $N\in {\mathcal {N}}$ such that $N\subseteq O_x$ and the set $\{n\in \omega :x_n\in N\}$ is infinite;

• a $\mathsf {cs}^*$-network at $x\in X$ if for any neighborhood $O_x\subseteq X$ of x and any sequence $\{x_n\}_{n\in \omega }\subseteq X$ that converges to x there exists $N\in {\mathcal {N}}$ such that $N\subseteq O_x$ and the set $\{n\in \omega :x_n\in N\}$ is infinite;

• a $\mathsf {cs}^\bullet$-network at x if for any neighborhood $O_x\subseteq X$ of x and any sequence $\{x_n\}_{n\in \omega }\subseteq X$ that converges to x there exists $N\in {\mathcal {N}}$ such that $N\subseteq O_x$ and N contains some point $x_n$.

• a network at x if for any neighborhood $O_x\subseteq X$ the union $\bigcup \{N\in {\mathcal {N}}:N\subseteq O_x\}$ is a neighborhood of x;

It is clear that for any family ${\mathcal {N}}$ of subsets of a topological space X and any $x\in X$ we have the following implications.

Theorem 4.2

If a topological space X has a countable ${\textsf {cs} }^\cdot$-network at a point $x\in X$, then X is $(\omega _1,\omega )_p$-equiconvergent at x.

Proof

Let ${\mathcal {N}}$ be a countable ${\mathsf {cs}}^\bullet$-network at x and

$$\begin{aligned} \{x_\alpha \}_{\alpha \in \omega _1}\subseteq \{s\in X^\omega :\lim _{n\rightarrow \infty }s(n)=x\}. \end{aligned}$$

Endow the ordinal $\omega +1=\omega \cup \{\omega \}$ with the discrete topology. For every $\alpha \in \omega _1$ consider the function $\delta _\alpha :{\mathcal {N}}\rightarrow \omega +1$ assigning to each $N\in \mathcal N$ the smallest number $n\in \omega$ such that $x_\alpha (n)\in N$ if such number n exists, and $\omega$ if $x_n\notin N$ for all $n\in \omega$. Since $(\omega +1)^{{\mathcal {N}}}$ is a metrizable separable space, the uncountable set

$$\begin{aligned} \{\delta _\alpha \}_{\alpha \in \omega _1}\subseteq (\omega +1)^{{\mathcal {N}}} \end{aligned}$$

contains a non-trivial convergent sequence. Consequently, we can find a sequence $(\alpha _n)_{n\in \omega }$ of pairwise distinct countable ordinals such that the sequence $(\delta _{\alpha _n})_{n\in \omega }$ converges to $\delta _{\alpha _0}$ in the Polish space $(\omega +1)^{{\mathcal {N}}}$. We claim that the sequence $(x_{\alpha _n})_{n\in \omega }$ witnesses that the space X is $(\omega _1,\omega )_p$-equiconvergent. Fix any neighborhood $U\subseteq X$ of zero.

Since ${\mathcal {N}}$ is an ${\mathsf {cs}}^\bullet$-network, there exists $N\in {\mathcal {N}}$ and $n\in \omega$ such that $x_n\in N\subseteq U$. Hence

$$\begin{aligned} d:=\delta _{\alpha _0}(N)\le n. \end{aligned}$$

Since the sequence $(\delta _{\alpha _n})_{n\in \omega }$ converges to $\delta _{\alpha _0}$, there exists $l\in \omega$ such that

$$\begin{aligned} \delta _{\alpha _k}(N)=\delta _{\alpha _0}(N)=d \end{aligned}$$

for all $k\ge l$. Then for every $k\ge l$ we have $x_{\alpha _k}(d)\in N\subseteq U$. $\square$

The following proposition (connecting $\omega ^\omega$-bases with networks) is a corollary of Theorem 6.4.1 in [4].

Proposition 4.3

If $(U_\alpha )_{\alpha \in \omega ^\omega }$ is an $\omega ^\omega$-base at a point x of a topological space X, then $(\bigcap _{\beta \in {\uparrow }\alpha }U_\beta )_{\alpha \in \omega ^{<\omega }}$ is a countable ${\textsf {s} }^*$-network at x. Here ${\uparrow }\alpha =\{\beta \in \omega ^\omega :\alpha \subset \beta \}$ for any $\alpha \in \omega ^{<\omega }=\bigcup _{n\in \omega }\omega ^n$.

As a consequence of the results presented above about the $(\kappa ,\lambda )_p$-equiconvergence and the $(\kappa ,\lambda )$-tall bornology for an lcs E, we propose the following proof of Theorem 1.2.

Proof of Theorem 1.2

If an lcs E has an $\omega ^\omega$-base, then by Theorem 4.1, the space E is $(\omega _1,\omega )_k$-equiconvergent and hence $(\omega _1,\omega )_p$-equiconvergent. The $(\omega _1,\omega )_p$-equiconvergence of E also follows from Proposition 4.3 and Theorem 4.2. Next, by Proposition 3.3, the space E has $(\omega _1,\omega )$-tall bornology, which means that each uncountable set in E contains an infinite bounded set. If E has an uncountable Hamel basis H, then H contains an infinite bounded linearly independent set, and by Theorem 2.1 the space E contains an infinite-dimensional compact set. $\square$

5 Radial networks and another proof of Theorem 1.2

A family ${\mathcal {N}}$ of subsets of a linear topological space E is called a radial network if for every neighborhood of zero $U\subseteq E$ and every every $x\in E$ there exist a set $N\in {\mathcal {N}}$ and a nonzero real number $\varepsilon$ such that $\varepsilon \cdot x\in N\subseteq U$.

The following theorem is a “linear” modification of Theorem 4.2.

Theorem 5.1

If an lcs E has a countable radial network, then each uncountable subset in E contains an infinite bounded subset.

Proof

Let ${\mathcal {N}}$ be a countable radial network in E, and let A be an uncountable set in E. Endow the ordinal $\omega +1=\omega \cup \{\omega \}$ with the discrete topology.

For every $\alpha \in A$ consider the function $\delta _\alpha :{\mathcal {N}}\rightarrow \omega +1$ assigning to each $N\in \mathcal N$ the ordinal

$$\begin{aligned} \delta _\alpha (N)=\min \{n\in \omega +1:2^{-n}\cdot \alpha \in [-1,1]\cdot N\}. \end{aligned}$$

Here we assume that $2^{-\omega }=0$.

Since $(\omega +1)^{{\mathcal {N}}}$ is a metrizable separable space, the uncountable set $\{\delta _\alpha \}_{\alpha \in A}\subseteq (\omega +1)^{{\mathcal {N}}}$ contains a non-trivial convergent sequence. Consequently, we can find a sequence $\{\alpha _n\}_{n\in \omega }\subseteq A$ of pairwise distinct points of A such that the sequence $(\delta _{\alpha _n})_{n\in \omega }$ converges to $\delta _{\alpha _0}$ in the Polish space $(\omega +1)^{{\mathcal {N}}}$.

We claim that the set $\{\alpha _n\}_{n\in \omega }$ is bounded in X. Fix any neighborhood $U\subseteq X$ of zero.

Since ${\mathcal {N}}$ is a radial network, there exist a set $N\in {\mathcal {N}}$ and a nonzero real number $\varepsilon$ such that $\varepsilon \cdot \alpha _0\in N\subseteq U$. Then $d:=\delta _{\alpha _0}(N)\in \omega$. Since the sequence $(\delta _{\alpha _n})_{n\in \omega }$ converges to $\delta _{\alpha _0}$, there exists $l\in \omega$ such that $\delta _{\alpha _k}(N)=\delta _{\alpha _0}(N)$ for all $k\ge l$. Then for every $k\ge l$ we have

$$\begin{aligned} 2^{-d}\cdot \alpha _k\in [-1,1]\cdot N\subseteq [-1,1]\cdot U \end{aligned}$$

and hence $\{\alpha _k\}_{k\ge l}\subseteq [-2^d,2^d]\cdot U$, which implies that the family $(\alpha _n)_{n\in \omega }$ is bounded in X. $\square$

The implication $(1)\Rightarrow (7)$ in the following theorem provides an alternative proof of Theorem 1.2, announced in the introduction.

Theorem 5.2

For an lcs E consider the following properties:

• (1) E has an $\omega ^\omega$-base;

• (2) E has a countable ${\textsf {s} }^*$-network at zero;

• (3) E has a countable ${\textsf {cs} }^*$-network at zero;

• (4) E has a countable ${\textsf {cs} }^\bullet$-network at zero;

• (5) E has a countable radial network at zero;

• (6) each uncountable set in E contains an infinite bounded subset;

• (7) E contains an infinite-dimensional compact set.

Then $(1)\Rightarrow (2)\Rightarrow (3)\Rightarrow (4)\Rightarrow (5)\Rightarrow (6)$. If E has uncountable Hamel basis, then $(6)\Rightarrow (7)$.

Proof

The implication $(1)\Rightarrow (2)$ follows from Proposition 4.3. The implications $(2)\Rightarrow (3)\Rightarrow (4)$ are trivial and $(4)\Rightarrow (5)$ follows from the observation that every ${\mathsf {cs}}^\bullet$-network at zero in the space E is a radial network for E. The implication $(5)\Rightarrow (6)$ is proved by Theorem 5.1.

If E has an uncountable Hamel basis H, then by (6), there exists an infinite bounded set $B\subseteq H$. By Theorem 2.1, the space E contains an infinite-dimensional compact set. $\square$

Problem 5.3

Is there an lcs E that has a countable radial network but does not have a countable ${\textsf {cs} }^\bullet$-network at zero?

6 Applications to spaces $C_{p}(X)$

A family $\{B_{\alpha }:\alpha \in \omega ^\omega \}$ of bounded (compact) sets covering an lcs E is called a bounded (compact) resolution if $B_{\alpha }\subseteq B_{\beta }$ for each $\alpha \le \beta$. If additionally every bounded (compact) subset of E is contained in some $B_{\alpha }$, we call the family $\{B_{\alpha }:\alpha \in \omega ^\omega \}$ a fundamental bounded (compact) resolution of E.

Example 6.1

Let E be a metrizable lcs with a decreasing countable base $(U_{n})_{n\in \omega }$ of absolutely convex neighbourhoods of zero. For $\alpha =(n_{k})_{k\in \omega }\in \omega ^{\omega }$ put $B_{\alpha }=\bigcap _{k\in \omega }n_{k}U_{k}$ and observe that $\{B_{\alpha }: \alpha \in \omega ^{\omega }\}$ is a fundamental bounded resolution in E.

A Tychonoff space X is called pseudocompact if each continuous real-valued function on X is bounded.

The first part of the following (motivating) result has been proved in [18]; since this is not published yet, we add a short proof.

Proposition 6.2

For a Tychonoff space X the following assertions are equivalent:

• (1) The space $C_{k}(X)$ is covered by a sequence of bounded sets.

• (2) The space $C_{p}(X)$ is covered by a sequence of bounded sets.

• (3) X is pseudocompact.

  Moreover, the following assertions are equivalent:

• (4) $C_{p}(X)$ is covered by a sequence of bounded sets but is not covered by a sequence of functionally bounded sets.

• (5) X is pseudocompact and contains a countable subset which is not closed in X or not $C^{*}$-embedded in X.

Proof

(1) $\Rightarrow$ (2) is clear. (2) $\Rightarrow$ (3): Assume $C_{p}(X)$ is covered by a sequence of bounded sets but X is not psudocompact. Then $C_{p}(X)$ contains a complemented copy of ${\mathbb {R}}^{\omega }$, see [1]. But ${\mathbb {R}}^{\omega }$ cannot be covered by a sequence of bounded sets, otherwise would be $\sigma$-compact. (3) $\Rightarrow$ (1): If X is pseudocompact, then for every $n\in {\mathbb {N}}$ the set $B_n=\{f\in C(X):\sup _{x\in X} |f(x)|\le n\}$ is bounded in $C_k(X)$ and $\bigcup _{n\in {\mathbb {N}}}B_n=C_k(X)$.

The equivalence $(4)\Leftrightarrow (5)$ follows from [24, Problem 399]: $C_{p}(X)$ is covered by a sequence of functionally bounded subsets o $C_{p}(X)$ if and only if X is pseudocompact and every countable subset of X is closed and $C^{*}$-embedded in X. $\square$

Example 6.3

$C_{p}([0,\omega _{1}))$ is covered by a sequence of bounded sets but is not covered by a sequence of functionally bounded sets.

By [10], $C_{p}(X)$ has a bounded resolution if and only if there exists a K-analytic space L such that $C_{p}(X)\subseteq L\subseteq {\mathbb {R}}^{X}$. The problem when $C_{p}(X)$ has a fundamental bounded resolution is easier. As a simple application of Theorem 1.2 we prove the following

Proposition 6.4

For a Tychonoff space X consider the following assertions:

• (1) $C_{p}(X)$ admits a fundamental bounded resolution $\{B_{\alpha }: \alpha \in \omega ^{\omega }\}$.

• (2) X is countable.

• (3) ${\mathbb {R}}^{X}=\bigcup _{\alpha \in \omega ^{\omega }}\overline{B_{\alpha }}^{{\mathbb {R}}^{X}}$ for a fundamental bounded resolution $\{B_{\alpha }: \alpha \in \omega ^{\omega }\}$ in $C_{p}(X)$.

• (4) The strong (topological) dual $L_{\beta }(X)$ of $C_{p}(X)$ is a cosmic space, i.e. a continuous image of a metrizable separable space.

• (5) $C_{p}\left( X\right)$ is a large subspace of ${\mathbb {R}}^{X}$, i.e. for every mapping $f \in {{\mathbb {R}}}^X$ there is a bounded set $B \subseteq C_p(X)$ such that $f \in {\overline{B}}^{{\mathbb {R}}^X}$.

Then $(1)\Leftrightarrow (2)\Leftrightarrow (3)\Leftrightarrow (4)\Rightarrow (5)$ but $(5)\Rightarrow (2)$ fails even for compact spaces X.

The implication $(1)\Rightarrow (2)$ was recently proved by Ferrando, Gabriyelyan and Ka̧kol [9] (with the help of ${\mathsf {cs}}0^{*}$-networks). We will derive this implication from Theorem 1.2.

Proof

(1) $\Rightarrow$ (2): If $C_{p}(X)$ has a fundamental bounded resolution $\{B_{\alpha }:\alpha \in \omega ^{\omega }\}$, then the sets $U_{\alpha }=\{\xi \in L_{\beta }(X): \sup _{f\in B_\alpha }|\xi (f)|\le 1\}$ form an $\omega ^{\omega }$-base in $L_{\beta }(X)$. By [14], every bounded set in $L_\beta (X)$ is finite-dimensional. Applying Theorem 1.2, we conclude that the Hamel basis X of the lcs $L_{\beta }(X)$ is countable. (2) $\Rightarrow$ (1) is clear. (2) $\Rightarrow$ (3)$\wedge$(5): Since $C_{p}(X)$ is dense in the metrizable space ${\mathbb {R}}^X$, the claims hold. (2) $\Rightarrow$ (4): If X is countable, then $L_{\beta }(X)$ has a fundamental sequence of compact sets covering $L_{\beta }(X)$ and [19, Proposition 7.7] implies that $L_{\beta }(X)$ is an $\aleph _{0}$-space, hence cosmic. (4) $\Rightarrow$ (2): If $L_{\beta }(X)$ is cosmic, then it is separable, and [12, Corollary 2.5] shows that X is countable. (5) $\nRightarrow$ (2): $C_{p}(X)$ over every Eberlein scattered, compact X satisfies (5), see [13]. $\square$

Item (5) in Proposition 6.4 is strictly connected with the following result.

Theorem 6.5

[12, 13] For a Tychonoff space X, the following conditions are equivalent:

• (i) $C_p(X)$ is distinguished, i.e. the strong dual $L_{\beta }(X)$ of the space $C_{p}(X)$ is bornological.

• (ii) The strong dual $L_{\beta }(X)$ of the space $C_{p}(X)$ is a Montel space.

• (iii) $C_{p}\left( X\right)$ is a large subspace of ${\mathbb {R}}^{X}$.

• (iv) The strong dual $L_{\beta }(X)$ of the space $C_{p}(X)$ carries the finest locally convex topology.

The following is a linear counterpart to item (4) in Proposition 6.4.

Remark 6.6

A Tychonoff space X is finite if and only if $L_{\beta }(X)$ is a continuous linear image of a metrizable lcs.

Indeed, if X is finite, nothing is left to prove. Conversely, assume that $L_{\beta }(X)$ is a continuous linear image of a metrizable lcs E (by a one-to-one map). But $L_{\beta }(X)$ has only finite-dimensional bounded sets and E fails this property. Hence X is finite.