Feral dual spaces and (strongly) distinguished spaces C(X)

Abstract

Following Dieudonné and Schwartz a locally convex space is distinguished if its strong dual is barrelled. The distinguished property for spaces $C_p(X)$ of continuous real-valued functions over a Tychonoff space X is a peculiar (although applicable) property. It is known that $C_p(X)$ is distinguished if and only if $C_p(X)$ is large in $\mathbb {R}^X$ if and only if X is a $\Delta$-space (in sense of Reed) if and only if the strong dual of $C_p(X)$ carries the finest locally convex topology. Our main results about spaces whose strong dual has only finite-dimensional bounded sets (see Theorems 2, 7 and Proposition 4) are used to study distinguished spaces $C_k(X)$ with the compact-open topology. We also put together several known facts (Theorem 6) about distinguished spaces $C_p(X)$ with self-contained full proofs.

1 Introduction

Recall that a locally convex space E is distinguished if its strong dual $E_{\beta }^{\prime }$ is barrelled (i.e. any absolutely convex, absorbing and closed subset of $E_{\beta }^{\prime }$ is a neighbourhood of zero). In fact (an equivalent condition [27, 23.7]), E is distinguished if and only if E is large in $\left( E^{\prime \prime },\sigma \left( E^{\prime \prime },E^{\prime }\right) \right)$. Recall also that a subspace F of a locally convex space G is large in G if every bounded subset of G is contained in the closure in G of some bounded subset of F, [34, Definition 8.3.22]. All semi-reflexive locally convex spaces are distinguished ([38, IV 5.5]).

If X is a Tychonoff space then $C_{p}(X)$ and $C_k(X)$ denote the linear space C(X) of all real-valued continuous functions defined on X equipped with the pointwise convergence and the compact-open topology, respectively.

The simplest examples of distinguished $C_{p}(X)$ spaces which are not semi-reflexive are those with X any countable nondiscrete Tychonoff space (see [12]). Several recently obtained results about distinguished spaces $C_p(X)$ have been proved in articles [12, 15, 16, 24, 25, 32]. We know that $C_p(X)$ is distinguished if and only if the strong dual $L_{\beta }(X)$ of $C_p(X)$ (i.e. the topological dual $C_p(X)'$ of $C_p(X)$ with the strong topology $\beta (C_p(X)', C_p(X))$) carries the finest locally convex topology, see [12, 15]. In [24] we proved that $C_{p}(X)$ is distinguished if and only if X is a $\Delta$-space (in the sense of Reed [35]). This theorem apparently provides a nice connection with problems from the set theory and related with $\Delta$-sets, $\lambda$-sets, and Q-sets X, and corresponding distinguished spaces $C_p (X)$.

We proved, among others, that: (i) For each metrizable scattered space X the space $C_p(X)$ is distinguished ([24, Proposition  4.1]); (ii) For each compact Eberlein scattered space X the space $C_p(X)$ is distinguished ([24, Theorem 3.7] or [15, Theorem 49]); (iii) If $X=[0,\omega _1]$ then $C_p(X)$ is not distinguished ([24, Theorem 3.12]).

A similar characterization of distinguished spaces $C_p(X)$ in term of the space X has been proved in [16].

In Sect. 2 we put together (Theorem 6) several equivalent conditions for $C_p(X)$ to be distinguished with self-contained proofs.

Being motivated by results from papers mentioned above we propose

Definition 1

A locally convex space E is strongly distinguished if its strong dual $E_{\beta }^{\prime }$ caries the finest locally convex topology (i.e. any absolutely convex and absorbing subset of $E_{\beta }^{\prime }$ is a neighbourhood of zero).

Following [22] a locally convex space is feral if every bounded set in E is finite-dimensional. Hence a locally convex space E carries the finest locally convex topology if and only if E is feral and E is bornological (i.e. every locally bounded linear map from E into any locally convex space is continuous). Recall that a linear map between locally convex spaces is locally bounded if it maps bounded sets into bounded sets; clearly, any continuous linear map is locally bounded.

In [12] we showed that $C_p(X)$ is strongly distinguished if and only if it is distinguished, see also [15]; in [13, page 392] we proved that for each Tychonoff space X the strong dual $L_{\beta }(X)$ of $C_p(X)$ is feral. Consequently, $C_p(X)$ is strongly distinguished if and only if its strong dual $L_{\beta }(X)$ is bornological.

In [26] we proved the following

Theorem 1

( [26, Theorem 1])

For a Tychonoff space X the following are equivalent:

• (1) The space $C_{k}(X)$ is strongly distinguished.

• (2) X is a $\Delta$-space and every compact subset of X is finite.

• (3) The space $C_{k}(X)$ is large in ${\mathbb {R}}^{X}$.

Since a linear space with the finest locally convex topology is feral, Theorem 1 suggests the following natural

Problem 1

Characterize locally convex spaces whose strong dual is feral.

This problem is solved in Theorem 7 providing some applications, see Corollary 7 (extending [11, Theorem 3.3]). The main application of Theorem 7 extends Theorem 1, essentially for (1) $\Rightarrow$ (2), and is formulated below:

Theorem 2

For a Tychonoff space X the following are equivalent:

• (1) The strong dual of $C_k(X)$ is feral.

• (2) Every compact subset of X is finite.

If $C_{k}(X)$ is quasi-barrelled, then the strong bidual of $C_{k}(X)$ is feral if and only if X is finite.

Recall that a locally convex space E is quasi-barrelled if every absolutely convex closed set in E absorbing bounded sets in E is a neighbourhood of zero. It is not clear if the quasi-barrelledness of $C_{k}(X)$ in Theorem 2 can be omitted. Note however that for each Tychonoff space X the strong bidual $M_{\beta }(X)$ of $C_p(X)$ is feral if and only if X is finite, see Corollary 4.

Recall that $C_k(X)$ is quasi-barrelled if and only if X is a W-space i.e. every b-bounding subset B of X is relatively compact, see [34, Theorem 10.1.21]; the b-boundedness of B means that for every bounded subset M of $C_k(X)$ we have $\sup \{|f(x)|: x\in B, f\in M\}<\infty .$

Observe that the item (2) in Theorem 2 does not guarantee that the strong dual of $C_k(X)$ is bornological, see Example 1.

Krupski and Marciszewski proved that for any infinite compact spaces X and Y the spaces $C_p(X)$ and $C_w(Y)$ are not isomorphic, where $C_w(Y)$ is the Banach space C(Y) with its weak topology i.e. $C_w(Y)= (C(Y), \sigma (C(Y), C(Y)'))$ ([29, Corollary 3.2], see also [28] for motivations and results around this line of research).

Applying Theorem 2 we extend this result by proving the following.

Corollary 1

Let X and Y be Tychonoff spaces. Let $C_w(Y)$ be the space $C_k(Y)$ with its weak topology $\sigma (C_k(Y), C_k(Y)')$. If there exists a continuous linear map from $C_p(X)$ onto $C_w(Y)$, then every compact subset of Y is finite. In particular, the spaces $C_p(X)$ and $C_w(Y)$ are not isomorphic, if Y contains an infinite compact subset.

In fact we prove a more general statement, see Corollary 5.

A collection $\{U_t: t\in T\}$ of subsets of a topological space X is called

(a) point-finite if for every $x\in X$ the set $\{t\in T: x\in U_t\}$ is finite;

(b) compact-finite if for every compact subset F of X the set $\{t\in T: U_t\cap F\ne \emptyset \}$ is finite;

(c) an open expansion of a collection $\{X_t: t\in T\}$ of subsets of X if $U_t$ is open and $U_t \supset X_t$ for every $t\in T.$

Clearly, a decreasing sequence $(X_n)$ of subsets of X is point-finite if and only if $\bigcap _{n=1}^{\infty } X_n=\emptyset$.

A topological space X is called

(a) a $\Delta$-space if every decreasing point-finite sequence $(X_n)$ of subsets of X admits a decreasing point-finite open expansion $(U_n)$, see [24];

(b) a strong $\Delta$-space if every decreasing point-finite sequence $(X_n)$ of subsets of X admits a decreasing compact-finite open expansion $(U_n)$.

Theorems 1, 2 and 6 imply the following

Corollary 2

For a Tychonoff space X the following are equivalent:

• (1) The space $C_{k}(X)$ is strongly distinguished.

• (2) The space $C_{k}(X)$ is large in ${\mathbb {R}}^{X}$.

• (3) The strong dual $C_{k}(X)'_{\beta }$ of $C_k(X)$ is feral and the strong dual $L_{\beta }(X)$ of $C_p(X)$ is bornological.

• (4) X is a $\Delta$-space and every compact subset of X is finite.

• (5) X is a strong $\Delta$-space.

• (6) Any countable disjoint collection of subsets of X admits a compact-finite open expansion in X.

• (7) Any countable partition of X admits a compact-finite open expansion in X.

Proof

(1)$\Leftrightarrow$ (2) $\Leftrightarrow$ (3) $\Leftrightarrow$ (4) follow by Theorems 1, 2 and 6.

(4) $\Rightarrow$ (5) $\Rightarrow$ (6) $\Rightarrow$ (7) $\Rightarrow$ (4) are clear (see the Proof of Theorem 6). $\square$

Note however that there exist (non-complete) Montel spaces (hence distinguished) whose strong dual is not bornological, see [34, Example 4.7.8]. Clearly, for every discrete space X and for every countable Tychonoff space X the space $C_p(X)$ is distinguished, since every discrete space and every countable Tychonoff space are $\Delta$-spaces.

On the other hand, we show that

Example 1

There exists an uncountable pseudocompact Haydon space X with all compact sets finite such that $C_k(X)$ $(=C_p(X))$ is not distinguished.

Recall that any Haydon space X is a subspace of $\beta {\mathbb {N}}$ with $X\supset {\mathbb {N}}.$

Another example is related with the paper [20].

Example 2

The space $\omega ^{*}=(\beta {\mathbb {N}}\setminus {\mathbb {N}})$ contains a dense countably compact subspace X such that every compact subset of X is finite and $C_k(X)$ $(=C_p(X))$ is not distinguished.

The subspace $(\ell _{\infty })_p=\{(x_{n})\in {\mathbb {R}}^{{\mathbb {N}}}: \sup _n |x_{n}|< \infty \}$ of ${\mathbb {R}}^{{\mathbb {N}}}$ is distinguished. For all Haydon spaces X there exists a continuous open linear map from $C_p(X)$ onto $(\ell _{\infty })_p$ (see [23, Theorem 1.3]), but for some Haydon spaces X the space $C_p(X)$ is not distinguished (by Example 1).

We note also the following more general fact involving the case $C_k(X)$.

Proposition 1

Let X be a subspace of $\beta {\mathbb {N}}$ with $X\supset {\mathbb {N}}.$ Then

• (1) $C_k(X)$ admits a continuous open linear map onto ${\mathbb {R}}^{{\mathbb {N}}}$ or $c_{0}$ or $\ell _{2}$ or $(\ell _{\infty })_{p}$.

• (2) $C_p(X)$ admits a continuous open linear map onto ${\mathbb {R}}^{{\mathbb {N}}}$ or $(\ell _{\infty })_{p}$.

In particular, the spaces $C_p(X)$ and $C_k(X)$ have infinite-dimensional quotients that are distinguished although these spaces can be not distinguished.

We assume that all locally convex spaces are Hausdorff and over the field ${\mathbb {R}}$ of real numbers.

2 A few facts about distinguished spaces

A locally convex space E is called quasi-normable [34, Definition 8.3.34] if for every neighbourhood of zero U in E there exists a neighbourhood of zero V such that for every $\lambda >0$ there exists a bounded set B in E with $V\subseteq B+\lambda U$.

The class of quasi-normable spaces contains, for example, (DF)-spaces, metrizable spaces $C_{k}(X)$, spaces $C^{n}(\Omega )$ for open subsets $\Omega$ of ${\mathbb {R}}^{{\mathbb {N}}}$, as well as all Fréchet-Montel spaces (see [19, 33]).

Grothendieck showed that a metrizable locally convex space E is distinguished if and only if its strong dual $E_{\beta }^{\prime }=(E',\beta (E',E))$ is bornological, [34, Theorem 8.3.44]. Heinrich [18] observed that each metrizable quasi-normable locally convex space satisfies the density condition what implies that every metrizable quasi-normable locally convex space is distinguished. In particular, the strong dual of a distinguished metrizable locally convex space can be described as a regular $\left( LB\right)$-space, see [34, Observation 8.5.14 (e)].

We refer to [5, 6, 15, 18, 19, 27] for several information about distinguished metrizable spaces. The most interesting example of a nondistinguished Fréchet (i.e. a metrizable and complete locally convex) space is the K öthe’s echelon space from [27, 31.7].

Since each space $C_{k}\left( X\right)$ is quasi-normable (see [19, 10.8.2 Theorem]) and any metrizable quasi-normable space is distinguished, we have

Proposition 2

Any metrizable space $C_{k}\left( X\right)$ is distinguished.

Note however that non-metrizable and distinguished spaces $C_k(X)$ which are not strongly distinguished do exist, see Example 7.

Proposition 2 implies the following

Theorem 3

$C_k(X)$ is distinguished for any locally compact paracompact space X.

Proof

It is known that X is the direct sum $\oplus _{t\in T}X_{t}$ of locally compact $\sigma$-compact spaces, see [9]. Then $C_k(X)$ is isomorphic to the product $\prod _{t\in T} C_k(X_t)$ of Fréchet spaces $C_k(X_t)$. Hence the strong dual $C_k(X)'_{\beta }$ of $C_k(X)$ is isomorphic to the direct sum $\bigoplus _{t\in T}C_k(X_{t})'_{\beta }$, see [38, Exc. 8, p.192]. By Proposition 2 each space $C_k(X_{t})'_{\beta }$ is barrelled, so $C_k(X)'_{\beta }$ is barrelled, too ([34, Corollary 4.2.7]). $\square$

Note that the strong dual $C_k(X)'_{\beta }$ of $C_k(X)$ is a complete strict (LB)-space, i.e., complete strict inductive limit of a sequence of Banach spaces what is a consequence of the fact that $C_k(X)'_{\beta }$ is bornological (by applying the Grothendieck theorem, see [27, 23.7, 29.3]).

Below we provide two concrete examples of metrizable dense subspaces of ${\mathbb {R}}^{{\mathbb {N}}}$ which are strongly distinguished.

Example 3

The spaces $(\ell _{\infty })_{p}$ and $(c_{0})_p=\{(x_{n})\in {\mathbb {R}}^{{\mathbb {N}}}: x_{n}\rightarrow 0\}$ with the topologies induced from ${\mathbb {R}}^{{\mathbb {N}}}$ are dense large subspaces of ${\mathbb {R}}^{{\mathbb {N}}}$, so they are strongly distinguished. On the other hand, does not exist a Tychonoff space X such that $(\ell _{\infty })_{p}$ is isomorphic to $C_p(X)$. In fact, as easily seen, $(\ell _{\infty })_{p}$ is $\sigma$-compact, while the space $C_p(X)$ is $\sigma$-compact if and only if X is finite, see [1, Theorem I.2.1].

Distinguished spaces $C_p(X)$ are totally different from distinguished spaces $C_k(X)$, see Theorem 6. Recall first the following general

Theorem 4

[15, 36] The following assertions are equivalent for any locally convex space E.

• (1) E has the finest locally convex topology.

• (2) Every absolutely convex absorbing subset of E is a neighbourhood of zero.

• (3) E is the strong dual of the product ${\mathbb {R}}^{\dim E}.$

• (4) E is barrelled and admits a continuous basis [36].

Clearly, any linear functional on a linear space E is continuous in the finest locally convex topology $\xi$ on E; if E is infinite-dimensional, then $\xi$ is non metrizable (since every metrizable infinite dimensional locally convex space admits a discontinuous linear functional).

On the other hand, in [15, Theorem 9] we proved

Theorem 5

The homeomorphic copy of X in the dual $C_p(X)'$ with the weak$^{*}$-topology is a continuous basis in $L_{\beta }(X)$.

Recall (see [22]) that a locally convex space E is called primitive if for any increasing sequence $(E_n)$ of linear subspaces of E with $\bigcup _{n=1}^{\infty } E_n=E$, a linear functional f on E is continuous if and only if its restrictions to $E_n, n\in {\mathbb {N}},$ are continuous.

Using [12, 15, 16, 24, 36] we put together several equivalent conditions for $C_p(X)$ to be distinguished.

Proofs of some implications are original.

Theorem 6

For a Tychonoff space X the following are equivalent:

• (1) $C_p(X)$ is distinguished.

• (2) $C_p(X)$ is strongly distinguished.

• (3) $C_p(X)$ and ${\mathbb {R}}^X$ have the same strong duals.

• (4) The strong dual $L_{\beta }(X)$ of $C_p(X)$ is the direct sum of |X|-many lines.

• (5) $L_{\beta }(X)$ is reflexive.

• (6) $L_{\beta }(X)$ is bornological.

• (7) $L_{\beta }(X)$ is quasi-barrelled.

• (8) $L_{\beta }(X)$ is primitive.

• (9) The strong bidual $M_{\beta }(X)$ of $C_p(X)$ is the product space ${\mathbb {R}}^{X}$.

• (10) $M(X)={\mathbb {R}}^X$ (as sets) i.e. $L_{\beta }(X)'=L_{\beta }(X)^*$.

• (11) $M_{\beta }(X)$ is reflexive.

• (12) $M_{\beta }(X)$ is quasi-complete.

• (13) $C_p(X)$ is large in ${\mathbb {R}}^{X}$.

• (14) For each $f\in {\mathbb {R}}^{X}$ there exists a bounded subset B of $C_p(X)$ such that $f\in \text{ cl}_{{\mathbb {R}}^X}(B)$.

• (15) X is a $\Delta$-space.

• (16) Any countable disjoint collection of subsets of X admits a point-finite open expansion in X.

• (17) X is coverable i.e. any countable partition of X admits a point-finite open expansion in X.

Proof

(1) $\Rightarrow$ (2) $\Rightarrow$ (3) follows from Theorem 4 and Theorem 5.

(3) $\Rightarrow$ (4) $\Rightarrow$ (5) $\Rightarrow$ (1) and (2) $\Rightarrow$ (6) are obvious.

(6) $\Rightarrow$ (1) since every quasi-complete bornological locally convex space is a barrelled space ([7, Theorem  3.6.19]) and the space $L_{\beta }(X)$ is quasi-complete.

(1) $\Leftrightarrow$ (7) since every barrel in $L_{\beta }(X)$ is bornivorous.

(1) $\Rightarrow$ (8) since any barrelled locally convex space is primitive ([34, Proposition 4.1.6]).

(3) $\Leftrightarrow$ (13) follows by the bipolar theorem ([38, Theorem 1.5]).

(9) $\Rightarrow$ (11) $\Rightarrow$ (12) are obvious. (12) $\Rightarrow$ (9) ${\mathbb {R}}^X$ is the quasi-completion of the subspace G consisting of all functions $f\in {\mathbb {R}}^X$ with finite support. By Theorem 5, $G\subset M_{\beta }(X) \subset {\mathbb {R}}^X$. Thus $M_{\beta }(X)={\mathbb {R}}^X.$

(3) $\Rightarrow$ (9) follows by reflexivity of ${\mathbb {R}}^X$. (9) $\Rightarrow$ (10) is obvious.

(10) $\Leftrightarrow$ (14) Put $F=C_p(X)$ and $E=M_{\beta }(X)$. Then $F'^*=L(X)^*={\mathbb {R}}^X$ and $\sigma (F'^*, F')=\sigma (L(X)^*, L(X))=\sigma ({\mathbb {R}}^X, L(X))$ is the product topology of ${\mathbb {R}}^X$. By [38, Theorem 5.4, IV], $f\in E$ if and only if there exists a bounded subset B of F such that $f\in \text{ cl}_{\sigma (F'^*, F')}(B)=\text{ cl}_{{\mathbb {R}}^X}(B).$

(8) $\Rightarrow$ (10) Let $f\in {\mathbb {R}}^X$. Put $X_n=\{x\in X: |f(x)|\le n\}$ and $L_n= \{\mu \in L(X): \,\text{ supp }(\mu ) \subset X_n\}$ for $n\in {\mathbb {N}}.$ Clearly, $(L_n)$ is an increasing sequence of linear subspaces of L(X) with $\bigcup _{n=1}^{\infty }L_n=L(X).$ Let $n\in {\mathbb {N}}$ and $f_n=f\chi _{X_n}$. The set $B_n=\{g\in C(X): |g(x)|\le n\,\text{ for } \text{ all }\, x\in X\}$ is bounded in $C_p(X)$ and $f_n\in \,\text{ cl}_{{\mathbb {R}}^X}(B_n).$ By [38, Theorem 5.4, IV], $\text{ cl}_{{\mathbb {R}}^X}(B_n) \subset M(X),$ so $f_n\in M(X).$ Clearly, $f|_{L_n}=f_n|_{L_n}$, so $f|_{L_n}$ is continuous for any $n\in {\mathbb {N}}.$ Hence $f\in M(X).$ Thus ${\mathbb {R}}^X=M(X).$

$(13) \Rightarrow (14)$ is obvious.

$(14) \Rightarrow (15)$ Let $(X_n)$ be a decreasing sequence of subsets of X with empty intersection. Put $X_0=X$. Let $f\in {\mathbb {R}}^X$ with $f(x)=n+1$ for all $x\in [X_{n-1}\setminus X_n], n\in {\mathbb {N}}.$ Let B be a bounded subset of $C_p(X)$ with $f \in \,\text{ cl}_{{\mathbb {R}}^X}(B).$ Let $x\in X$. Then $x\in [X_{n-1}{\setminus } X_n]$ for some $n\in {\mathbb {N}}.$

The set $W=\{g\in {\mathbb {R}}^X: |g(x)-f(x)|<1\}$ is a neighbourhood of f in ${\mathbb {R}}^X$, so $W\cap B\ne \emptyset .$ Let $g_x\in W\cap B.$ Then $g_x(x)>n,$ so the set $V_x=g_x^{-1}((n, \infty ))$ is an open neighbourhood of x in X. Put $U_n=\bigcup _{x\in X_n}V_x$ for $n\in {\mathbb {N}}.$ Clearly, $(U_n)$ is a decreasing sequence of open subsets of X and $X_n\subset U_n$ for $n\in {\mathbb {N}}.$ Let $n\in {\mathbb {N}}.$ For $x\in X_n$ and $z\in V_x$ we have $g_x(z)>n$, so $\sup _{g\in B}|g(z)|>n$ for $z\in U_n.$ Let $y\in X.$

For some $m\in {\mathbb {N}}$ we have $\sup _{g\in B}|g(y)|\le m,$ since B is bounded in $C_p(X).$ It follows that $y\not \in U_m$, so $y\not \in \bigcap _{n=1}^{\infty }U_n.$ Thus $\bigcap _{n=1}^{\infty }U_n = \emptyset .$

$(15) \Rightarrow (13)$ Let A be a non-empty bounded subset of ${\mathbb {R}}^X$. Let $\psi : X\rightarrow {\mathbb {R}}, \psi (x)=\sup _{f\in A}|f(x)|.$ Put $X_n=\{x\in X: |\psi (x)|\ge n-1\}$ for $n\in {\mathbb {N}}.$ Then $(X_n)$ is a decreasing sequence of subsets of X with empty intersection. Thus there exists a decreasing sequence $(U_n)$ of open subsets of X with an empty intersection such that $X_n\subset U_n$ for $n\in {\mathbb {N}}.$ For any $x\in X$ there exists $\varphi (x)\in {\mathbb {N}}$ with

$$\begin{aligned} x\in [U_{\varphi (x)}\setminus U_{\varphi (x)+1}]. \end{aligned}$$

We have $\psi (x)< \varphi (x)$ for every $x\in X.$ Indeed, let $x\in X$ and $n\in {\mathbb {N}}$ with $n-1\le |\psi (x)|<n.$ Then $x\in X_n \subset U_n$ and $x \not \in U_{\varphi (x)+1},$ so $n<\varphi (x)+1.$ Thus $\psi (x)<n\le \varphi (x).$

Clearly, the set $B=\{g\in C_p(X): |g|\le \varphi \}$ is bounded in $C_p(X).$ We shall prove that $A\subset \,\text{ cl}_{{\mathbb {R}}^X}(B).$ Let $f\in A.$ Let W be a neighbourhood of f in ${\mathbb {R}}^X.$ Then there exists a finite subset K of X such that

$$\begin{aligned} \{g\in {\mathbb {R}}^X: g|_K=f|_K\}\subset W. \end{aligned}$$

Let $\{V_x: x\in K\}$ be a family of pairwise disjoint open subsets of X with

$$\begin{aligned} x\in V_x \subset U_{\varphi (x)}, x\in K. \end{aligned}$$

For every $x\in K$ there exists a continuous function

$$\begin{aligned} h_x: X \rightarrow [-\varphi (x), \varphi (x)] \end{aligned}$$

with $h_x(x)=f(x)$ and $h_x(y)=0$ for $y\in V_x^c$ (see [9, Theorem 3.1.7]). The function $h: X \rightarrow {\mathbb {R}}, h=\sum _{x\in K} h_x$ is continuous and $h|_K=f|_K,$ so $h\in W.$

We shall prove that $h\in B.$ Clearly, $h(x)=0$ for $x\in (\bigcup _{x\in K}V_x)^c$. Let $x\in K.$ For $y\in [K{\setminus } \{x\}]$ we have $V_x\subset V_y^c,$ so $h_y|_{V_x}=0.$

Thus $h|_{V_x}=h_x|_{V_x}.$ For $t\in V_x$ we have

$$\begin{aligned} |h(t)|=|h_x(t)|\le \varphi (x) \le \varphi (t), \end{aligned}$$

since $V_x\subset U_{\varphi (x)}.$ Thus $|h|\le \varphi ,$ so $h\in B$. It follows that $W\cap B\ne \emptyset ,$ so $f \in \,\text{ cl}_{{\mathbb {R}}^X}(B).$ Hence $A\subset \,\text{ cl}_{{\mathbb {R}}^X}(B).$

$(15) \Rightarrow (16)$ Let $(S_n)$ be a disjoint collection of subsets of X. Put $X_n=\bigcup _{m=n}^{\infty } S_m$ for $n\in {\mathbb {N}}.$ Then $(X_n)$ is a decreasing point-finite sequence of subsets of X. Thus there exists a decreasing point-finite open expansion $(U_n)$ of $(X_n)$; clearly $(U_n)$ is an open expansion of $(S_n)$.

$(16) \Rightarrow (17)$ is obvious.

$(17) \Rightarrow (15)$ Let $(X_n)$ be a decreasing point-finite sequence of subsets of $X_0=X$. Put $P_n=[X_{n-1}{\setminus } X_n], n\in {\mathbb {N}}.$ Then $(P_n)$ is a partition of X. Let $(U_n)$ be a point-finite open expansion of $(P_n)$ and $V_n=\bigcup _{m=n+1}^{\infty } U_m, n\in {\mathbb {N}}.$ Then $(V_n)$ is a decreasing point-finite open expansion of $(X_n)$, since $X_n=\bigcup _{m=n+1}^{\infty } P_m \subset \bigcup _{m=n+1}^{\infty } U_m= V_n, n\in {\mathbb {N}}.$ $\square$

On the other hand, notice the following fact implying that the item (11) in Theorem 6 cannot be replaced by $M_{\beta }(X)$ is Baire.

Remark 1

Ferrando and Ka̧kol [12, Theorem 3.9] proved that the strong dual $L_{\beta }(X)$ of $C_p(X)$ is always distinguished (i.e. $M_{\beta }(X)$ is always barrelled) and then Ferrando and Saxon asked [16, Problem 11] if the Baire property of $M_{\beta }(X)$ implies that $C_p(X)$ is distinguished. The answer is negative (as noticed in [16, Addendum]) since $M(\omega _{1})$ is a Baire space [16, Corollary 21] but $C_p(\omega _1)$ is not distinguished [24].

For spaces $C_k(X)$ note the following simple

Proposition 3

Let X be a Tychonoff space. The space $C_k(X)$ is feral if and only if X is finite.

Proof

Clearly, $C_k(X)$ is feral if X is finite. Assume that X is infinite. If X is pseudocompact, then $C_k(X)$ admits a stronger normed topology. If X is not pseudocompact, then $C_k(X)$ contains an isomorphic copy of ${\mathbb {R}}^{{\mathbb {N}}}$ ([21, Theorem  2.12]). Thus $C_k(X)$ is not feral. $\square$

3 Feral dual spaces

Two locally convex spaces E and F are called bornologically isomorphic if there exists a linear bijective map $T:E\rightarrow F$ such that T and $T^{-1}$ are locally bounded; E is a free locally convex space if E carries the finest locally convex topology.

In order to prove Theorem 2 we need the following simple fact which is probably known.

Lemma 1

Let E and F be locally convex spaces. Assume that there exists a continuous linear surjection $T:E \rightarrow F$ which is bounded covering, i.e. for every bounded set B in F there exists a bounded set $A_B$ in E with $T(A_B)=B$. Then the strong dual $E'_{\beta }$ of E contains an isomorphic copy of the strong dual $F'_{\beta }$ of F.

Proof

The adjoint map $T^*: F'_{\beta }\rightarrow E'_{\beta }$ is injective. Let B be a bounded subset of F. The seminorms $p_B: F_{\beta }' \rightarrow [0, \infty ), \xi \rightarrow \sup _{f\in B}|\xi (f)|$ and $p_{A_B}: E_{\beta }' \rightarrow [0, \infty ), \eta \rightarrow \sup _{e\in A_B}|\eta (e)|$ are continuous and

$$\begin{aligned} p_{B}(\xi )=\sup _{f\in B}|\xi (f)|=\sup _{e\in A_B}|\xi (Te)|=\sup _{e\in A_B}|(T^*\xi )(e)|=p_{A_B}(T^*\xi ) \end{aligned}$$

for any $\xi \in F_{\beta }'.$ It follows that $T^*$ is an isomorphism onto its range, so $E_{\beta }'$ contains an isomorphic copy of $F_{\beta }'.$ $\square$

Dealing with locally convex spaces we have the following characterization for quasi-barrelled spaces carrying the weak topology.

Theorem 7

For a locally convex space E the following are equivalent:

• (1) $E'_{\beta }$ is feral.

• (2) $E'_{\beta }$ is bornologically isomorphic to a free locally convex space.

• (3) E is quasi-barrelled and carries the weak topology.

• (4) Every compact set in $E'_{\beta }$ has finite topological dimension.

Proof

(1) $\Leftrightarrow$ (2) is easy.

The equivalence (1) $\Leftrightarrow$ (4) has been proved in [4, Theorem 1.2].

(1) $\Rightarrow$ (3) First we show that $E_{\mu }=(E,\mu (E,E'))$ is quasi-barrelled. Let B be a bounded subset of $E_{\beta }'$. Then B is finite-dimensional, so the closure W of the absolutely convex hull of B in $(E', \sigma (E', E))$ is compact. Thus the set $^\circ W=\{x\in E: |f(x)|\le 1\, \text{ for } \text{ all }\, f\in W\}$ is a neighbourhood of zero in $E_{\mu }$ and $B\subset W\subset (^\circ W)^\circ \subset E'$; so B is $\mu (E,E')$-equicontinuous. Thus $E_{\mu }$ is quasi-barrelled.

Let $E'_{\nu }$ be the space $E'$ with the finest locally convex topology. The identity map $I: E'_{\nu } \rightarrow E'_{\beta }$ is a continuous linear surjection. Since every bouned subset of $E'_{\beta }$ is finite-dimensional, I is bounded covering. By Lemma 1 we derive that the strong bidual $(E'_{\beta })'_{\beta }$ is isomorphic to a subspace of $(E'_{\nu })'_{\beta }$. On the other hand, $E'_{\nu }$ is the direct sum $\bigoplus _{t\in X}{\mathbb {R}}$, where X is a Hamel basis of $E'$; so its strong dual $(E'_{\nu })'_{\beta }$ is isomorphic to the product ${\mathbb {R}}^X$, which clearly carries the weak topology. Hence $(E'_{\beta })'_{\beta }=(E'', \beta (E'',E')$ carries the weak topology and $\beta (E'',E')|_E=\sigma (E',E)$, since every subspace of a locally convex space with the weak topology has the weak topology. Since $E_{\mu }$ is quasi-barrelled, $\beta (E'',E')|_E=\mu (E,E')$, see [19, Proposition 11.2.2].

Thus $\sigma (E,E')=\mu (E,E')$, so E is quasi-barrelled and carries the weak topology.

(3) $\Rightarrow$ (1) Let X be a Hamel basis of $E'$ with the topology induced from $(E',\sigma (E',E))$. The linear map $\psi : E \rightarrow C_p(X), z \rightarrow \psi _z,$ where $\psi _z(x)=x(z)$ for $x\in X,$ is an isomorphism between E and $\psi (E),$ and $\psi (E)$ is dense in $C_p(X)$. Thus we can identify E with a dense subspace of the product ${\mathbb {R}}^X$ and $E'$ with $({\mathbb {R}}^X)'$.

Let B be a bounded subset of $E_{\beta }'$. Since E is quasi-barrelled, B is equicontinuous, so there exists a neighbourhood U of zero in E such that $B\subset U^\circ \subset E'=({\mathbb {R}}^X)'$. Clearly, the closure V of U in ${\mathbb {R}}^X$ is a neighbourhood of zero in ${\mathbb {R}}^X$. Let $f\in B$ and $v\in V$. Then there exists a net $(u_t)_{t\in T}$ in U convergent to v in ${\mathbb {R}}^X$. Hence the net $(f(u_t))_{t\in T}$ is convergent to f(u) in ${\mathbb {R}}$ and $|f(u_t)|\le 1, t \in T,$ so $|f(u)|\le 1.$ Thus $B\subset V^\circ \subset ({\mathbb {R}}^X)'.$ The set $V^\circ$ is bounded in $({\mathbb {R}}^X)'_{\beta }$. Indeed, let A be a bounded subset of ${\mathbb {R}}^X$. Then $sA\subset V$ for some $s>0$, so $sV^\circ \subset A^\circ .$ Hence $V^\circ$ is bounded in $({\mathbb {R}}^X)'_{\beta }=\oplus _{x\in X} {\mathbb {R}}$. Thus $V^\circ$ is finite-dimensional, since the direct sum $\oplus _{x\in X} {\mathbb {R}}$ is feral; so $B\subset V^\circ$ is finite-dimensional. Hence $E_{\beta }'$ is feral. $\square$

Since $C_p(X)$ is quasi-barrelled ([19, Corollary 11.7.3]) and carries the weak topology, Theorem 7 applies the following known result (see [13]).

Corollary 3

$L_{\beta }(X)$ is feral for any Tychonoff space X.

Corollary 4

Let X be a Tychonoff space. The strong bidual $M_{\beta }(X)$ of $C_p(X)$ is feral if and only if X is finite.

Proof

($\Rightarrow$) The strong dual $L_{\beta }(X)$ of $C_p(X)$ is quasi-barrelled (by Theorem 7) and complete (by [12, Proposition 3.10]). Consequently, $L_{\beta }(X)$ is barrelled, so $M_{\beta }(X)$ is the product ${\mathbb {R}}^{X}$ (by Theorem 6). Since $M_{\beta }(X)$ is feral, X is finite. ($\Leftarrow$) is obvious. $\square$

The following proposition provides a stronger version of Corollary 1.

Proposition 4

Let E be a locally convex space.

If there exists a Tychonoff space X and a locally bounded linear map T from $C_p(X)$ onto E, then the strong dual $E'_{\beta }$ of E is feral.

Proof

Let $\upsilon X$ be the realcompactification of the space X.

The restriction map $\pi : C_p(\upsilon X)\rightarrow C_p(X), f \rightarrow f|_X$ is a continuous linear surjection, see [21, Lemma 9.1].

The composition map ${\hat{T}}: C_{p}(\upsilon X) \rightarrow E, {\hat{T}}=T\circ \pi$ is a locally bounded linear surjection. $C_p(\upsilon X)$ is bornological by Buchwalter–Schmets theorem ([8]). Applying [34, Proposition 6.1.8] we deduce that ${\hat{T}}$ is continuous. Then the adjoint map ${\hat{T}}^{*}: E'_{\beta }\rightarrow L_{\beta }(\upsilon X)$ is injective and continuous. Hence the range ${\hat{T}}^{*}(E')$ admits a locally convex topology $\xi$, stronger than the topology restricted from $L_{\beta }(\upsilon X)$ and such that $E_{\beta }'$ is isomorphic with $({\hat{T}}^{*}(E'),\xi )$. By [14, page 392] the space $L_{\beta }(\upsilon X)$ is feral. Hence the strong dual $E'_{\beta }$ of E is feral. $\square$

Corollary 5

Let X and Y be Tychonoff spaces.

If there exists a locally bounded linear map T from $C_p(X)$ onto $C_k(Y)$, then every compact subset of Y is finite.

Proof

By Proposition 4 the strong dual of $C_{k}(Y)$ is feral. Hence, by Theorem 2, each compact subset of Y is finite. $\square$

Proposition 4 suggests the following.

Problem 2

Does there exist a locally convex space E whose strong dual is feral such that E is not continuous [not locally bounded] linear image of $C_p(X)$ for any Tychonoff space X?

Recall that the strongly distinguished space $(\ell _{\infty })_{p}$ is not isomorphic to any space $C_p(X)$ (Example 3), although if X is a Tychonoff space containing a copy of $\beta {\mathbb {N}}$, then $C_p(X)$ admits a continuous open linear map onto $(\ell _{\infty })_{p}$, see [3, Theorem 1].

Now we prove Theorem 2

Proof of Theorem 2

(I). ($\Rightarrow$) Assume that X contains an infinite compact subset K. The restriction map $R: C_k(X)\rightarrow C_k(K), R(f)=f|K$ is an open continuous surjection, see [23, Proposition 2.9]. Put $M={\textrm{ker}}R.$ Let $Q: C_k(X)\rightarrow C_k(X)/M$ be the quotient map. Then the map ${\overline{R}}: C_k(X)/M \rightarrow C_k(K), f +M \rightarrow f|_K$ is an isomorphism, since ${\overline{R}}\circ Q=R$. By Theorem 7 the space $C_{k}(X)$ carries the weak topology, so $C_k(X)/M$ carries the weak topology, too. We have a contradiction, since $C_k(X)/M$ is isomorphic to the infinite-dimensional Banach space $C_k(K)$.

($\Leftarrow$) follows by Corollary 3, since $C_k(X)=C_p(X)$.

(II). Assume that $E=C_k(X)$ is quasi-barrelled. Then E is a subspace of $(E'_{\beta })'_{\beta }$ ([19, Proposition 11.2.2]). Hence, if $(E'_{\beta })'_{\beta }$ is feral, then E is feral, too; so X is finite, by Proposition 3. The converse is obvious. $\square$

Another consequence of Theorem 7 yields the following

Corollary 6

An infinite-dimensional metrizable locally convex space E is strongly distinguished if and only if it is isomorphic to a dense subspace of ${\mathbb {R}}^{{\mathbb {N}}}$. Hence a Fréchet space F is strongly distinguished if and only if F is finite-dimensional or F is isomorphic to ${\mathbb {R}}^{{\mathbb {N}}}$.

Proof

($\Rightarrow$) By Theorem 7 and its proof, E carries the weak topology and it is isomorphic to a dense subspace of ${\mathbb {R}}^X$ for some infinite set X. E is metrizable, so X is countable.

($\Leftarrow$) By [34, Observation 8.3.23 (b)] E is large in ${\mathbb {R}}^{{\mathbb {N}}}$, so the strong duals of E and ${\mathbb {R}}^{{\mathbb {N}}}$ coincide and are isomorphic to $\varphi$, the $\aleph _{0}$-dimensional linear space with the finest locally convex topology. $\square$

Recall that a locally convex space admits a fundamental bounded resolution if there exists a family $\{B_{\alpha }:\alpha \in {\mathbb {N}}^{{\mathbb {N}}}\}$ of bounded sets such that $B_{\alpha }\subset B_{\beta }$ for all $\alpha \le \beta$ and each bounded set in E is contained in some $B_{\alpha }$. Clearly every metrizable locally convex space admits such resolution, see [21, Lemma 15.2]. A locally convex space E has a ${\mathbb {N}}^{\mathbb {N}}$-base if there exists a base $\{U_\alpha : \alpha \in {\mathbb {N}}^{\mathbb {N}}\}$ of neighborhoods of zero such that $U_\beta \subset U_\alpha$ for all elements $\alpha \le \beta$ in ${\mathbb {N}}^{\mathbb {N}}$. Clearly, every metrizable locally convex space has an ${\mathbb {N}}^{\mathbb {N}}$-base, [21].

In [11, Theorem 3.3] we proved that $C_{p}(X)$ admits a fundamental bounded resolution if and only if X is countable (if and only if $C_p(X)$ is metrizable).

Using Theorem 7 we extend the above result and supplement Corollary 6.

Corollary 7

An infinite-dimensional quasi-barrelled locally convex space E is isomorphic to a dense subspace of ${\mathbb {R}}^{{\mathbb {N}}}$ if and only if E has a fundamental bounded resolution and E caries the weak topology.

Proof

($\Rightarrow$) is clear. ($\Leftarrow$) By the proof of Theorem 7, E is isomorphic to a dense subspace of ${\mathbb {R}}^X$ for some infinite set X. Assume that X is uncountable. Then $E'$ is uncountable-dimensional, since $\dim E'=\dim ({\mathbb {R}}^X)'=\dim (\oplus _{x\in X} {\mathbb {R}})$. Let $\{B_{\alpha }:\alpha \in {\mathbb {N}}^{{\mathbb {N}}}\}$ be a fundamental bounded resolution of E. Then $\{B_{\alpha }^{\circ }: \alpha \in {\mathbb {N}}^{{\mathbb {N}}}\}$ is a ${\mathbb {N}}^{{\mathbb {N}}}$-base of neighbourhoods of zero in the strong dual $E'_{\beta }$ of E. By [4, Theorem  1.2], $E'_{\beta }$ contains an infinite-dimensional metrizable compact set; a contradiction, since $E'_{\beta }$ is feral (by Theorem 7). Thus X is countable. $\square$

4 Proof of Proposition 1 and Examples

Proof of Proposition 1

(1) Consider three cases: (1.1). X is not pseudocompact. Then the space $C_k(X)$ contains a complemented copy of the space ${\mathbb {R}}^{{\mathbb {N}}}$, see [21, Theorem 2.12] and [34, Corollary 2.6.5]. Hence there exists a continuous open linear map from $C_k(X)$ onto ${\mathbb {R}}^{{\mathbb {N}}}$.

(1.2). X is pseudocompact and every compact subset of X is finite. Since ${\mathbb {N}}\subset X\subset \beta {\mathbb {N}}$, then ${\mathbb {N}}$ is C$^{*}$-embedded into X. Applying [3, Theorem 1] we get that $C_{p}(X)$ has a quotient $C_{p}(X)/W$ isomorphic to the subspace $(\ell _\infty )_{p}$ of ${\mathbb {R}}^{\mathbb {N}}$ (endowed with the product topology), where $W=\bigcap _{n} \{f\in C_p(X): \sum _{x\in F_n} f(x)=0 \}$ and $(F_{n})$ is a sequence of non-empty, finite and pairwise disjoint subsets of ${\mathbb {N}}$ with $\lim _n |F_n|=\infty$. Hence $C_k(X)$($=C_p(X)$) admits a continuous open linear map onto $(\ell _\infty )_{p}$.

(1.3). X is pseudocompact and contains an infinite compact subset K. By [23, Proposition 2.9] the restriction map $T:C_{k}(X)\rightarrow C_k(K), f \rightarrow f|_K,$ is a continuous open linear surjection. By [37] the Banach space C(K) admits a continuous open linear map onto $\ell _{2}$ or $c_{0}$. Thus $C_k(X)$ admits a continuous open linear map onto $\ell _{2}$ or $c_{0}$.

(2) If X is not pseudocompact, then $C_p(X)$ contains a complemented copy of ${\mathbb {R}}^{{\mathbb {N}}}$ ([2, Sect. 4]), so $C_p(X)$ admits a continuous open linear map onto ${\mathbb {R}}^{{\mathbb {N}}}$.

If X is pseudocompact, then $C_p(X)$ has a quotient isomorphic to $(\ell _\infty )_{p}$ (see (1.2)), so $C_p(X)$ admits a continuous open linear map onto $(\ell _\infty )_{p}$. $\square$

Below we provide concrete situations where $C_k(X)$ is distinguished but its strong dual $C_k(X)_{\beta }'$ is not feral.

Example 4

Let X be an uncountable hemicompact space. Then the space $C_k(X)$ is distinguished but its strong dual $C_k(X)_{\beta }'$ is not feral.

Proof

X is hemicompact, i.e. X is covered by a sequence of compact sets $(K_{n})$ such that each compact set in X is contained in some $K_{n}$, so $C_k(X)$ is metrizable. Applying Proposition 2 we infer that $C_k(X)$ is distinguished. Since X is uncountable, for some $n\in {\mathbb {N}}$ the set $K_{n}$ is infinite. By Theorem 2, $C_k(X)_{\beta }'$ is not feral. $\square$

A topological space X is called a Q-space if each subset of X is $G_{\delta }$. Recall that a normal space X is a Q-space if and only if X is strongly splittable, i.e. for every $f\in {\mathbb {R}}^X$ there exists a sequence $(f_n)_{n}$ in $C_p(X)$ such that $f_n \rightarrow f$ in ${\mathbb {R}}^X$, see [39, Problems 445, 447]. Using Theorem 2 one gets the following

Proposition 5

For a normal space X the assertions are equivalent.

• (1) X is a Q-space and every compact subset of X is finite.

• (2) For each $f\in {\mathbb {R}}^X$ there exists a bounded sequence $(f_n)$ in $C_k(X)$ such that $f_n \rightarrow f$ in ${\mathbb {R}}^X$.

Proof

(1) $\Rightarrow$ (2). Then $C_k(X)=C_p(X)$. Let $f\in {\mathbb {R}}^X$. X is a Q-space, so there exists a sequence $(f_n)\subset C_p(X)$ with $f_n \rightarrow f$ in ${\mathbb {R}}^X$. Then $(f_n)$ is bounded in $C_p(X)$ ($=C_k(X)$).

(2) $\Rightarrow$ (1). Then X is a Q-space and $C_k(X)$ is dense in ${\mathbb {R}}^X$. It is known that $C_k(X)$ has a base $\{U_t: t\in T\}$ of neighbourhoods of zero such that $U_t, t\in T,$ are closed in $C_p(X)$. For any $t\in T$, the set $\text{ cl}_{{\mathbb {R}}^X}(U_t)$ is a neighbourhoods of zero in ${\mathbb {R}}^X$, so $U_t=C_p(X) \cap \text{ cl}_{{\mathbb {R}}^X}(U_t)$ is a neighbourhood of zero in $C_p(X).$ Thus the topological spaces $C_k(X)$ and $C_p(X)$ are equal, so any compact subset of X is finite. $\square$

Example 1 uses the following scheme of constructing uncountable pseudocompact spaces without infinite compact subsets due to Haydon, [17]. Let $\omega ^{*}=(\beta {\mathbb {N}}\setminus {\mathbb {N}}).$ For each infinite subset A of ${\mathbb {N}}$, choose a cluster point $u_A$ of A in $\beta {\mathbb {N}}$. Let $X = ({\mathbb {N}}\cup \{u_A: A \,\text{ is } \text{ an } \text{ infinite } \text{ subset } \text{ of }\, {\mathbb {N}}\})$ be topologized as a subspace of $\beta {\mathbb {N}}$. To simplify the notation we will call such spaces the Haydon spaces. It is known that each Haydon space X is an uncountable pseudocompact space and each compact subset of X is finite, see [17].

A point x of a topological space X is said to be a a weak P-point if for any countable subset F of $(X\setminus \{x\})$ we have $x\not \in {\overline{F}}$. Clearly, any countable set of weak P-points of X is discrete. By [30], see also [20], the set of all weak P-points of the space $\omega ^*$ is dense in $\omega ^*$.

Thus for any infinite subset A of ${\mathbb {N}}$ there exists an element $u_A \in {\overline{A}} \cap \omega ^*,$ that is a weak P-point of X. Then any countable subset of the set $Z:=\{u_A: A\; \text{ is } \text{ an } \text{ infinite } \text{ subset } \text{ of }\, {\mathbb {N}}\}$ is discrete. Thus any countable subset of the Haydon space $Y=({\mathbb {N}}\cup Z)$ is scattered.

We will use the following two results.

In [24] Ka̧kol and Leiderman proved the following

Theorem 8

[24, Theorem 4.7] If X is countably compact and $C_p(X)$ is distinguished, then X is scattered.

A related result has been proved by Leiderman and Tkachuk in [32].

Theorem 9

[32, Theorem 3.1] If X is pseudocompact and $C_p(X)$ is distinguished then every countable subset of X is scattered.

Proof of Example 1

The compact space $\omega ^{*}$ has no isolated points. It is well-known (and easy to prove), that every compact space without isolated points contains a non-empty countable subset without isolated points, too. Thus there exists a non-empty countable subset P of $\omega ^{*}$ without isolated points; clearly, P is not scattered.

Let $P=\{p_n: n\in {\mathbb {N}}\}$ and $A_n=\{k\in {\mathbb {N}}: k>n\}$ for $n\in {\mathbb {N}}$. Since $p_n \in (\overline{A_n} \setminus {\mathbb {N}}), n\in {\mathbb {N}},$ there exists a Haydon space X, that contains P. By Theorem 9, the space $C_k(X)$ ($=C_p(X)$) is not distinguished. $\square$

Proof of Example 2

In [20] Juhasz and van Mill proved that the space $\omega ^{*}$ contains a dense subspace X such that X is countably compact and non-scattered but all countable subsets of X are scattered. Observe that every compact subset of X is finite. Indeed, if a compact subset A of $X\subset \beta N$ is infinite, then A contains a copy of $\beta {\mathbb {N}}$, but $\beta N$ contains a countable subset which is not scattered; a contradiction. By Theorem 8, the space $C_k(X)$ ($=C_p(X)$) is not distinguished. $\square$

Example 5

Let ${\mathfrak {D}}'(\Omega )$ be the space of all distributions over an open set $\Omega \subset {\mathbb {R}}^{n}$. Then every uncountable-dimensional subspace of ${\mathfrak {D}}'(\Omega )$ contains an infinite-dimensional metrizable compact set. In particular, ${\mathfrak {D}}'(\Omega )$ is not feral.

Proof

The space ${\mathfrak {D}}(\Omega )$ is a strict countable inductive limit of Fréchet Montel spaces ([19, Example 4.6.3]), so ${\mathfrak {D}}(\Omega )$ admits a fundamental bounded resolution (see [21] and the proof of Proposition 16.7 (ii) there). Therefore ${\mathfrak {D}}'(\Omega )$ admits an ${\mathbb {N}}^{\mathbb {N}}$-base. Now the conclusion follows from [4, Theorem 1.2]. $\square$

Example 6

Every uncountable-dimensional subspace of $C_k({\mathbb {R}}^{{\mathbb {N}}})$ contains a metrizable compact infinite-dimensional set and the strong dual of $C_k({\mathbb {R}}^{{\mathbb {N}}})$ admits an infinite-dimensional compact set. In particular, $C_k({\mathbb {R}}^{{\mathbb {N}}})$ and its strong dual are not feral.

Proof

The first claim follows from the fact that $C_k({\mathbb {R}}^{{\mathbb {N}}})$ admits an ${\mathbb {N}}^{\mathbb {N}}$-base (by [10]) and then we apply [4, Theorem 1.2]. The other claim follows from Theorem 2. $\square$

By the theorem of Heinrich (see Sect. 2), we know that every quasi-normable metrizable locally convex space is distinguished.

Problem 3

Is every quasi-normable locally convex space with a ${\mathbb {N}}^{\mathbb {N}}$-base a distinguished space?

Note that every (LB)-space is quasi-normable ([34]) and each (LB)-space has a ${\mathbb {N}}^{\mathbb {N}}$-base ([21]). Moreover, each space $C_p(X)$ is quasi-normable ([15]) and $C_p(X)$ has an ${\mathbb {N}}^{\mathbb {N}}$-base if and only if $C_p(X)$ is metrizable ([21]). Recall also that $C_{k}({\mathbb {R}}^{{\mathbb {N}}})$ is quasi-normable and has an ${\mathbb {N}}^{\mathbb {N}}$-base by applying the main theorem of [10].

Example 7

There exists a non-metrizable distinguished space $C_k(X)$ which is not strongly distinguished.

Proof

By [21, Example 2.4], there exists a Tychonoff space X such that $C_k(X)$ is a (df)-space but not (DF)-space. Then $C_k(X)$ is not metrizable, since any metrizable (df)-space is a (DF)-space. By [21, Theorem 2.14], the strong dual of $C_k(X)$ is a Fréchet space, so $C_k(X)$ is distinguished but not strongly distinguished. $\square$

Note also that $C_k({\mathbb {R}}^{{\mathbb {N}}})$ is not a (df)-space; it is even not covered by a sequence of bounded sets. Indeed, this follows directly from [23, Lemma 2.3].

Problem 4

Is the space $C_k(X)$ distinguished when X is metrizable? In particular, are the spaces $C_k({\mathbb {R}}^{{\mathbb {N}}})$ and $C_k({\mathbb {Q}})$ distinguished?

Problem 5

Characterize distinguished spaces $C_k(X)$ in terms of X.