## 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. (1)

E is metrizable;

2. (2)

E is Fréchet-Urysohn;

3. (3)

E is Baire-like;

4. (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. (1)

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

2. (2)

All compact subsets in $$\varphi$$ are finite-dimensional.

3. (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. (1)

Each compact subset of E has finite topological dimension.

2. (2)

Each bounded linearly independent set in E is finite.

3. (3)

E is bornologically isomorphic to a free lcs.

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

4. (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. (1)

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

2. (2)

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

3. (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. (1)

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

2. (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. (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. (1)

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

2. (2)

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

3. (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. (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. (1)

E has an $$\omega ^\omega$$-base;

2. (2)

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

3. (3)

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

4. (4)

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

5. (5)

E has a countable radial network at zero;

6. (6)

each uncountable set in E contains an infinite bounded subset;

7. (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. (1)

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

2. (2)

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

3. (3)

X is pseudocompact.

Moreover, the following assertions are equivalent:

4. (4)

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

5. (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. (1)

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

2. (2)

X is countable.

3. (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. (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. (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:

1. (i)

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

2. (ii)

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

3. (iii)

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

4. (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.