Distinguished Property in Tensor Products and Weak* Dual Spaces

Abstract

A local convex space E is said to be distinguished if its strong dual $E_{\beta }^{\prime }$ has the topology $\beta (E^{\prime },{(E_{\beta }^{\prime })}^{\prime })$, i.e., if $E_{\beta }^{\prime }$ is barrelled. The distinguished property of the local convex space $C_p\left(X\right)$ of real-valued functions on a Tychonoff space X, equipped with the pointwise topology on X, has recently aroused great interest among analysts and $C_p$-theorists, obtaining very interesting properties and nice characterizations. For instance, it has recently been obtained that a space $C_p\left(X\right)$ is distinguished if and only if any function $f\in {\mathbb{R}}^X$ belongs to the pointwise closure of a pointwise bounded set in $C\left(X\right)$. The extensively studied distinguished properties in the injective tensor products $C_p\left(X\right){\otimes }_{\epsilon }E$ and in $C_p(X,E)$ contrasts with the few distinguished properties of injective tensor products related to the dual space $L_p\left(X\right)$ of $C_p\left(X\right)$ endowed with the weak* topology, as well as to the weak* dual of $C_p(X,E)$. To partially fill this gap, some distinguished properties in the injective tensor product space $L_p\left(X\right){\otimes }_{\epsilon }E$ are presented and a characterization of the distinguished property of the weak* dual of $C_p(X,E)$ for wide classes of spaces X and E is provided.

1. Introduction

In this paper, X is an infinite Tychonoff space and $C\left(X\right)$ is the linear space of all real-valued continuous functions over X. $C_p\left(X\right)$ and $C_k\left(X\right)$ denote the space $C\left(X\right)$ equipped with the pointwise and compact-open topology, respectively. $L_p\left(X\right)$ represents the weak* dual of $C_p\left(X\right)$, i.e., the topological dual $L\left(X\right)$ of $C_p\left(X\right)$ endowed with the weak topology $\sigma \left(L\left(X\right),C\left(X\right)\right)$ of the dual pair $⟨L\left(X\right),C\left(X\right)⟩$, i.e., $L_p\left(X\right)$ has the topology of pointwise convergence on $C\left(X\right)$.

Moreover, all local convex spaces are assumed to be real and Hausdorff and the symbol ‘≃’ indicates some canonical algebraic isomorphism or linear homeomorphism. The strong dual $E_{\beta }^{\prime }$ of a local convex space $E$ is the topological dual $E^{\prime }$ of E equipped with the strong topology $\beta \left(E^{\prime },E\right)$, which is the topology of uniform convergence on the bounded subsets of E. $⟨E,E^{\prime }⟩$ is a dual pair. For a subset A of E the polar $A^0$ of A with respect to a dual pair $⟨E,F⟩$ is

$$A^0=\{x\in F:\left|⟨a,x⟩\right|\le 1,\forall a\in A\}.$$

A local convex space E is barrelled if for each pointwise bounded subset M of $E^{\prime }$ there exists a neighborhood of the origin U in E such that M is uniformly bounded on U. Hence E is barrelled if and only if its topology is the topology $\beta (E,E^{\prime })$, i.e., ${\left[E^{\prime }\left({\mathrm{weak}}^{*}\right)\right]}_{\beta }^{\prime }=E$. Roughly speaking, E is barrelled if it verifies the local convex version of the Banach–Steinhaus uniform boundedness theorem.

The local convex space E is called distinguished if $E_{\beta }^{\prime }$ is barrelled. In [1,2,3,4,5,6,7] the distinguished property of the space $C_p\left(X\right)$ has been extensively studied. Furthermore, [8] [Proposition 6.4] is connected with distinguished $C_p\left(X\right)$ spaces. It is observed in [3] [Theorem 10] that $C_p\left(X\right)$ is distinguished if and only if $C_p\left(X\right)$ is a large subspace of ${\mathbb{R}}^X$, i.e., if each bounded set in ${\mathbb{R}}^X$ is contained in the closure in ${\mathbb{R}}^X$ of a bounded set of $C_p\left(X\right)$, or, equivalently, if the strong bidual of $C_p\left(X\right)$ is ${\mathbb{R}}^X$ [5]. In [7], [Theorem 2.1] it is shown that $C_p\left(X\right)$ is distinguished if and only if X is a $\Delta$-space in the sense of Knight [9], and several applications of this fact are given. Equivalently, $C_p\left(X\right)$ is distinguished if for each countable partition $\left\{X_k:k\in \mathbb{N}\right\}$ of X into nonempty pairwise disjoint sets, there are open sets $\left\{U_k:k\in \mathbb{N}\right\}$ with $X_k\subseteq U_k$, for each $k\in \mathbb{N}$, such that each point $x\in X$ belongs to $U_n$ for only finitely many $n\in \mathbb{N}$, [5] [Theorem 5].

If E and F are local convex spaces, $E{\otimes }_{\epsilon }F$ and $E{\otimes }_{\pi }F$ represent the injective and projective tensor product of E and F, respectively. A basis of neighborhoods of the origin in $E_{\beta }^{\prime }{\otimes }_{\epsilon }{F}_{\beta }^{\prime }$ is determined by the sets $\epsilon \left(A,B\right):={\left(A^{00}\otimes B^{00}\right)}^0$, where A is a bounded set in E, B is a bounded set in F, $A^0\subseteq E^{\prime }$, $A^{00}\subseteq E^{″}$, $B^0\subseteq F^{\prime }$, $B^{00}\subseteq F^{″}$ and ${\left(A^{00}\otimes B^{00}\right)}^0\subseteq E^{\prime }\otimes F^{\prime }$. Analogously, a basis of neighborhoods of the origin in the tensor product space $E_{\beta }^{\prime }{\otimes }_{\pi }{F}_{\beta }^{\prime }$ is formed by the sets $\pi \left(A,B\right):=\mathbf{acx}\left(A^0\otimes B^0\right)$, where A is a bounded set in E, B is a bounded set in F and $\mathbf{acx}\left(A^0\otimes B^0\right)$ denotes the absolutely convex cover of the tensor product $A^0\otimes B^0$. Recall that if E carries the weak topology, then ${\left(E{\otimes }_{\epsilon }F\right)}^{\prime }\simeq {\left(E{\otimes }_{\pi }F\right)}^{\prime }\simeq E^{\prime }\otimes F^{\prime }$, [10] [41.3 (9) and 45.1 (2)]. A local convex space E is called nuclear if $E{\otimes }_{\epsilon }F=E{\otimes }_{\pi }F$ for every local convex space F, [11] [21.2].

The distinguished property of $C_p\left(X\right)$ under the formation of some tensor products is examined in [2]. Among other results it is showed in [2] [Corollary 6] that for a local convex space E the injective tensor product $C_p\left(X\right){\otimes }_{\epsilon }E$ is distinguished if both $C_p\left(X\right)$ is distinguished and ${\mathbb{R}}^{\left(X\right)}{\otimes }_{\epsilon }{E}_{\beta }^{\prime }$ is barrelled, where ${\mathbb{R}}^{\left(X\right)}$ the local convex direct sum of $\left|X\right|$ real lines.

If E is a local convex space $C_p\left(X,E\right)$ and $C_k(X,E)$ will denote the linear space of all E-valued continuous functions defined on X equipped with the pointwise topology and compact-open topology, respectively. It is also proved in [2] [Corolary 21] that, for any Tychonoff space X and any normed space E, the vector-valued function space $C_p(X,E)$ is distinguished if and only if $C_p\left(X\right){\otimes }_{\epsilon }E$ is distinguished. In particular, if X is a countable Tychonoff space and E a normed space, then $C_p(X,E)$ is distinguished. Indeed, if X is countable, on the one hand $C_p\left(X\right)$ is distinguished by [5] [Corollary 6] and on the other hand ${\mathbb{R}}^{\left(X\right)}$ is both barrelled and nuclear (the latter because [11] [21.2.3 Corollary]), so that ${\mathbb{R}}^{\left(X\right)}{\otimes }_{\epsilon }{E}_{\beta }^{\prime }={\mathbb{R}}^{\left(X\right)}{\otimes }_{\pi }{E}_{\beta }^{\prime }$ is barrelled by [12] [Theorem 1.6.6]. Thus, $C_p\left(X\right){\otimes }_{\epsilon }E$ is distinguished by the already mentioned [2] [Corolary 6] and, since E is normed, $C_p\left(X,E\right)$ is distinguished too by [2] [Corollary 21]. A corresponding result for the compact-open topology, due to Díaz and Domański [13] [Corolary 2.5], states that the space $C_k\left(K,E\right)$ of continuous functions defined on a compact Hausdorff space K and with values in a reflexive Fréchet space E is also distinguished, being its strong dual naturally isomorphic to $L_1\left(\mu \right){\widehat{\otimes }}_{\pi }{E}_{\beta }^{\prime }$.

According to [1] [Theorem 3.9], the strong dual $L_{\beta }\left(X\right)$ of $C_p\left(X\right)$ is always distinguished. The distinguished property of the weak* dual $L_p\left(X\right)$ of $C_p\left(X\right)$ is investigated in [5], where the following theorem is proved.

Theorem 1

([5] [Theorem 27]). If X is a μ-space, then the weak* dual $L_p\left(X\right)$ of $C_p\left(X\right)$ is distinguished.

Recall that a Tychonoff space X is called a $\mu$-space if each functionally bounded set is relatively compact.

The extensively studied distinguished properties in the injective tensor products $C_p\left(X\right){\otimes }_{\epsilon }E$ and in $C_p(X,E)$ contrasts with the few distinguished properties related with the injective tensor products $L_p\left(X\right){\otimes }_{\epsilon }E$ and with the weak* dual of $C_p(X,E)$. Theorem 1 and the fact that $L_p\left(X\right)$ spaces are studied so extensively as $C_p\left(X\right)$ spaces motivated us to fill partially this gap in this paper obtaining distinguished properties of injective tensor products $L_p\left(X\right){\otimes }_{\epsilon }E$ and providing a characterization of the distinguished property of the weak* dual of $C_p(X,E)$ for wide classes of spaces X and E. To reach these goals we require [2] [Theorem 5] and [2] [Proposition 19], which we include here for convenience.

Theorem 2

([2] [Theorem 5]). Let E and F be local convex spaces, where E carries the weak topology. If ${\tau }_{\epsilon }$ and ${\tau }_{\pi }$ denote the injective and projective topologies of $E_{\beta }^{\prime }\otimes F_{\beta }^{\prime }$, the following properties hold

1. 

If $E_{\beta }^{\prime }{\otimes }_{\epsilon }{F}_{\beta }^{\prime }$ is barrelled, then ${\tau }_{\epsilon }=\beta \left(E^{\prime }\otimes F^{\prime },E\otimes F\right)$ and $E{\otimes }_{\epsilon }F$ is distinguished.

2. 

If $E_{\beta }^{\prime }{\otimes }_{\pi }{F}_{\beta }^{\prime }$ is barrelled then ${\tau }_{\epsilon }\le \beta \left(E^{\prime }\otimes F^{\prime },E\otimes F\right)\le {\tau }_{\pi }$.

Theorem 3

([2] [Proposition 19]). For any local convex space E, the dual of the space $C_p\left(X,E\right)$ is algebraically isomorphic to $L\left(X\right)\otimes E^{\prime }$, i.e., $C_p{\left(X,E\right)}^{\prime }\simeq L\left(X\right)\otimes E^{\prime }$.

It should be noted that if ${\mathsf{\Sigma }}_{i=1}^n{f}_i\otimes u_i$ is a representation of $\phi \in C\left(X\right)\otimes E$ then Theorem 3 is due to the fact that the canonical map $T:C_p\left(X\right){\otimes }_{\epsilon }E\to C_p\left(X,E\right)$ given by

$$\left(T\phi \right)\left(x\right)={\mathsf{\Sigma }}_{i=1}^n{f}_i\left(x\right)u_i,$$

is a linear homeomorphism from $C_p\left(X\right){\otimes }_{\epsilon }E$ into a dense linear subspace of $C_p\left(X,E\right)$. Furthermore, ${\left(C_p\left(X\right){\otimes }_{\epsilon }E\right)}^{\prime }\simeq L\left(X\right)\otimes E^{\prime }$, because $C_p\left(X\right)$ carries the weak topology, so one has $C_p{\left(X,E\right)}^{\prime }\simeq L\left(X\right)\otimes E^{\prime }$, as stated.

2. Distinguished Tensor Products of L_p(X) Spaces

This section deals mainly with the injective tensor product of $L_p\left(X\right)$ and a nuclear metrizable space E. It should be noted that the class of nuclear metrizable spaces is large. Recall that the space s of all rapidly decreasing sequences, as well as the test space of distributions $\mathcal{D}\left(\mathsf{\Omega }\right)$, where $\mathsf{\Omega }$ is an open set in ${\mathbb{R}}^n$, with their usual local convex inductive topologies, are examples of nuclear Fréchet spaces [11] [Section 21.6]. The strong dual of $\mathcal{D}\left(\mathsf{\Omega }\right)$ is the space of distributions on $\mathsf{\Omega }$ and it is denoted by ${\mathcal{D}}^{\prime }\left(\mathsf{\Omega }\right)$.

Theorem 4.

Assume that X is a μ-space and let E be a nuclear metrizable local convex space. If every countable union of compact subsets of X is relatively compact, then $L_p\left(X\right){\otimes }_{\epsilon }E$ is distinguished.

Proof. 

The space X is a $\mu$-space if and only if $C_k\left(X\right)$ is barrelled, by the Nachbin-Shirota theorem [14] [Proposition 2.15]. On the other hand, as every countable union of compact subsets of X is assumed to be relatively compact, the space $C_k\left(X\right)$ is also a $\left(DF\right)$-space [15] [Theorem 12]. In addition, the strong dual $E_{\beta }^{\prime }$ of a metrizable local convex space E it is a complete $\left(DF\right)$-space by [16] [see 29.3 -in “By 2(1)”-]. Moreover, nuclearity of E implies that $E_{\beta }^{\prime }$ is nuclear too by [11] [21.5.3 Theorem]. As $E_{\beta }^{\prime }$ is a nuclear $\left(DF\right)$-space, one has that $E_{\beta }^{\prime }$ is a quasi-barrelled space [11] [21.5.4 Corollary]. Finally, the completeness of the quasi-barrelled space $E_{\beta }^{\prime }$ implies that $E_{\beta }^{\prime }$ is barrelled [16] [27.1.(1)], so E is distinguished.

The projective tensor product $C_k\left(X\right){\otimes }_{\pi }{E}_{\beta }^{\prime }$ is barrelled by [11] [15.6.8 Proposition]. Thus, taking into consideration $E_{\beta }^{\prime }$ nuclearity, it can be obtained that $C_k\left(X\right){\otimes }_{\epsilon }{E}_{\beta }^{\prime }$ is also barrelled. On the other hand, since X is a $\mu$-space it follows from [5] [Theorem 27] that $C_k\left(X\right)$ coincides with the strong dual of $L_p\left(X\right)$, i.e., ${\left(L_p\left(X\right)\right)}_{\beta }^{\prime }=C_k\left(X\right)$, hence

$${\left(L_p\left(X\right)\right)}_{\beta }^{\prime }{\otimes }_{\epsilon }{E}_{\beta }^{\prime }$$

is barrelled. Finally, as $L_p\left(X\right)$ carries the weak topology, the first statement of Theorem 2, ensures that the space $L_p\left(X\right){\otimes }_{\epsilon }E$ is distinguished. □

Example 1.

In particular, for each compact topological space X and for each nuclear metrizable local convex space E it follows that $L_p\left(X\right){\otimes }_{\epsilon }E$ is distinguished.

Hence, if X is the Cantor space K or the interval $[0,1]$, and if E is one of the local convex spaces $\mathcal{D}\left(\mathsf{\Omega }\right)$ or s, then the injective tensor products $L_p\left(K\right){\otimes }_{\epsilon }\mathcal{D}\left(\mathsf{\Omega }\right)$, $L_p\left(K\right){\otimes }_{\epsilon }s$, $L_p\left(\left[0,1\right]\right){\otimes }_{\epsilon }\mathcal{D}\left(\mathsf{\Omega }\right)$ and $L_p\left(\left[0,1\right]\right){\otimes }_{\epsilon }s$ are distinguished.

Corollary 1.

If X is a compact space and Y is a countable Tychonoff space, then the space $L_p\left(X\right){\otimes }_{\epsilon }{C}_p\left(Y\right)$ is distinguished.

Proof. 

Clearly, $C_p\left(Y\right)$ is metrizable (hence distinguished [1] [Theorem 3.3]) and nuclear (by [11] [21.2.3 Corollary]), so the statement follows from the previous theorem. □

If we apply this Corollary with X equal to the Stone-Čech compactification $\beta \mathbb{N}$ of the topological space $\mathbb{N}$ formed by the natural numbers endowed with the discrete topology and Y equal to the space $\mathbb{Q}$ of rational numbers endowed with the usual metrizable topology then we get that $L_p\left(\beta \mathbb{N}\right){\otimes }_{\epsilon }{C}_p\left(\mathbb{Q}\right)$ is a distinguished space.

If the factor E of $L_p\left(X\right){\otimes }_{\epsilon }E$ is a normed space, the following theorem holds true.

Theorem 5.

If X is a μ-space with finite compact sets (equivalently, if every functionally bounded subset of X is finite) and E is a normed space, then $L_p\left(X\right){\otimes }_{\epsilon }E$ is distinguished.

Proof. 

If X is a $\mu$-space with finite compact sets, the space $C_k\left(X\right)=C_p\left(X\right)$ is barrelled and nuclear. As $E_{\beta }^{\prime }$ is a Banach space, [12] [Corollary 1.6.6] assures that $C_k\left(X\right){\otimes }_{\pi }{E}_{\beta }^{\prime }$ is a barrelled space, and $C_k\left(X\right)$ nuclearity yields that $C_k\left(X\right){\otimes }_{\epsilon }{E}_{\beta }^{\prime }$ is also a barrelled space. Bearing in mind that ${\left(L_p\left(X\right)\right)}_{\beta }^{\prime }=C_k\left(X\right)$, as a consequence of the fact that X is a $\mu$-space (cf. [5] [Theorem 27]), Theorem 2 ensures that $L_p\left(X\right){\otimes }_{\epsilon }E$ is distinguished. □

A P-space in the sense of Gillman–Henriksen is a topological space in which every countable intersection of open sets is open.

Corollary 2.

If X is a P-space and E is a normed space, then $L_p\left(X\right){\otimes }_{\epsilon }E$ is distinguished.

Proof. 

Every P-space is a $\mu$-space with finite compact sets (cf. [17] [Problem 4K]). □

Example 2.

If $L\left(\mathfrak{m}\right)$ denotes the Lindelöfication of the discrete space of cardinal $\mathfrak{m}\ge {\aleph }_1$, the space $L_p\left(L\left(\mathfrak{m}\right)\right){\otimes }_{\epsilon }{C}_k\left(\left[0,1\right]\right)$ is distinguished. In this case $L\left(\mathfrak{m}\right)$ is a Lindelöf P-space.

Theorem 6.

If X is a μ-space with finite compact sets and E is normed space, then $L_p\left(X\right){\otimes }_{\epsilon }{E}^{\prime }\left({\mathrm{weak}}^{*}\right)$ is distinguished.

Proof. 

By [12] [Theorem 1.6.6] the projective tensor product $C_k\left(X\right){\otimes }_{\pi }E$ is a barrelled space, hence $C_k\left(X\right)$ nuclearity yields that $C_k\left(X\right){\otimes }_{\epsilon }E$ is barrelled. So, the conclusion follows from the first statement of Theorem 2. □

Example 3.

The space $L_p\left(L\left(\mathfrak{m}\right)\right){\otimes }_{\epsilon }{\ell }_p\left({\mathrm{weak}}^{*}\right)$ is distinguished for $1\le p<\infty$.

A topological space is said to be hemicompact if it has a sequence of compact subsets such that every compact subset of the space lies inside some compact set in the sequence.

Theorem 7.

If X is a hemicompact space and E is a nuclear metrizable barrelled space (for instance a nuclear Fréchet space), then $L_p\left(X\right){\otimes }_{\epsilon }{E}^{\prime }\left({\mathrm{weak}}^{*}\right)$ is distinguished.

Proof. 

Clearly X is a Lindelöf space, hence it is a $\mu$-space, and then both $C_k\left(X\right)$ and E are metrizable and barrelled spaces. Then [12] [Corollary 1.6.4] ensures that $C_k\left(X\right){\otimes }_{\pi }E$ is also a (metrizable) barrelled space. This property and the E nuclearity imply that $C_k\left(X\right){\otimes }_{\epsilon }E$ is a barrelled space. Consequently, using that ${\left(L_p\left(X\right)\right)}_{\beta }^{\prime }=C_k\left(X\right)$ and $E^{\prime }{\left({\mathrm{weak}}^{*}\right)}_{\beta }^{\prime }=E$, we get

$$L_p{\left(X\right)}_{\beta }^{\prime }{\otimes }_{\epsilon }{E}^{\prime }{\left({\mathrm{weak}}^{*}\right)}_{\beta }^{\prime }=C_k\left(X\right){\otimes }_{\epsilon }E.$$

So, Theorem 2 applies to guarantee that $L_p\left(X\right){\otimes }_{\epsilon }{E}^{\prime }\left({\mathrm{weak}}^{*}\right)$ is distinguished. □

By Theorem 7 the injective tensor product $L_p\left(\mathbb{R}\right){\otimes }_{\epsilon }{\mathcal{D}}^{\prime }\left(\mathsf{\Omega }\right)\left({\mathrm{weak}}^{*}\right)$ is distinguished since $\mathbb{R}$ is hemicompact and ${\mathcal{D}}^{\prime }\left(\mathsf{\Omega }\right)\left({\mathrm{weak}}^{*}\right)$ is a nuclear Fréchet space. Theorem 7 is also applied in the next Example 4.

Example 4.

If $\mathbb{N}$ is equipped with the discrete topology, $p\in \beta \mathbb{N}\backslash \mathbb{N}$ and $Z=\mathbb{N}\cup \left\{p\right\}$ has the topology induced by $\beta \mathbb{N}$, then $L_p\left(Z\right){\otimes }_{\epsilon }{L}_p\left(Z\right)$ is distinguished.

Proof. 

The subspace $Z=\mathbb{N}\cup \left\{p\right\}$ of $\beta \mathbb{N}$ is countable and has finite compact sets, so that it is hemicompact. Since Z is countable, $C_p\left(Z\right)$ is metrizable and, on the other hand, as a subspace of the nuclear space ${\mathbb{R}}^X$, the space $C_p\left(Z\right)$ is nuclear. In addition, since Z is a $\mu$-space with finite compact sets, the space $C_p\left(Z\right)$ is barrelled [18]. So, according to the previous theorem, $L_p\left(Z\right){\otimes }_{\epsilon }{L}_p\left(Z\right)$ is distinguished. □

3. Distinguished Property of the Weak* Dual of C_p(X) ⊗_ε E

The preceding theorems are going to be applied to examine the distinguished property of the weak* dual of the injective tensor product $C_p\left(X\right){\otimes }_{\epsilon }E$. To get this property we need the following lemma.

Lemma 1.

The injective topology of the tensor product $L_p\left(X\right)\otimes E^{\prime }\left({\mathrm{weak}}^{*}\right)$ coincides with the weak topology $\sigma \left(L\left(X\right)\otimes E^{\prime },C\left(X\right)\otimes E\right)$.

Proof. 

Since $L_p\left(X\right)$ carries the weak topology

$${\left(L_p\left(X\right){\otimes }_{\epsilon }{E}^{\prime }\left({\mathrm{weak}}^{*}\right)\right)}^{\prime }\simeq C\left(X\right)\otimes E.$$

Hence, the injective topology ${\tau }_{\epsilon }$ of $L_p\left(X\right)\otimes E^{\prime }\left({\mathrm{weak}}^{*}\right)$ is stronger than the weak topology $\sigma \left(L\left(X\right)\otimes E^{\prime },C\left(X\right)\otimes E\right)$. We prove that both topologies are the same. Indeed, if U is a closed absolutely convex neighborhood of the origin in $L_p\left(X\right)$ and V is a closed absolutely convex neighborhood of the origin in $E^{\prime }\left({\mathrm{weak}}^{*}\right)$, there are finite sets $\mathsf{\Phi }$ in $C\left(X\right)$ and $\Delta$ in E such that ${\mathsf{\Phi }}^0=U$ and ${\Delta }^0=V$. Setting $\mathsf{\Lambda }=\mathsf{\Phi }\otimes \Delta$, then $\mathsf{\Lambda }$ is a finite set in $C\left(X\right)\otimes E$ such that

$${\epsilon }_{U,V}\left(\psi \right)=\underset{f\in U^0,u\in V^0}{sup}\left|\sum _{i=1}^{n}f\left(x_i\right)⟨v_i,u⟩\right|=\underset{f\in U^0,u\in V^0}{sup}\left|⟨\sum _{i=1}^n{\delta }_{x_i}\otimes v_i,f\otimes u⟩\right|\le \underset{F\in \mathsf{\Lambda }}{sup}\left|⟨\psi ,F⟩\right|$$

for any $\psi ={\sum }_{i=1}^n{\delta }_{x_i}\otimes v_i\in L\left(X\right){\otimes }_{\epsilon }{E}^{\prime }$, since

$$U^0\otimes V^0={\mathsf{\Phi }}^{00}\otimes {\Delta }^{00}=\mathbf{acx}\left(\mathsf{\Phi }\right)\otimes \mathbf{acx}\left(\Delta \right)\subseteq \mathbf{acx}\left(\mathsf{\Lambda }\right).$$

Therefore ${\tau }_{\epsilon }=\sigma \left(L\left(X\right)\otimes E^{\prime },C\left(X\right)\otimes E\right)$. □

Corollary 3.

If X is a hemicompact space and E is a nuclear Fréchet space, the weak* dual of $C_p\left(X\right){\otimes }_{\epsilon }E$ is distinguished.

Proof. 

According to Lemma 1 the weak* dual of $C_p\left(X\right){\otimes }_{\epsilon }E$ is linearly homeomorphic to $L_p\left(X\right){\otimes }_{\epsilon }{E}^{\prime }\left({\mathrm{weak}}^{*}\right)$, so Theorem 7 applies. □

The space Z considered in Example 4 is hemicompact, hence from Corollary 3 we have that the weak* duals of $C_p\left(\mathbb{R}\right){\otimes }_{\epsilon }{C}_p\left(Z\right)$ and $C_p\left(Z\right){\otimes }_{\epsilon }\mathcal{D}\left(\mathsf{\Omega }\right)$ are distinguished.

Corollary 4.

If X is a μ-space with finite compact sets and E is a normed space, the weak* dual of $C_p\left(X\right){\otimes }_{\epsilon }E$ is distinguished.

Proof. 

The proof is analogous to the proof of Corollary 3, with the difference of using Theorem 6 instead of Theorem 7. □

Example 5.

The weak* dual of $C_p\left(L\left(\mathfrak{m}\right)\right){\otimes }_{\epsilon }{C}_k\left(\left[0,1\right]\right)$ is distinguished.

4. A Characterization of the Distinguished Weak* Dual of C_p(X, E)

Let E be a local convex space. We will denote by $L_p\left(X,E^{\prime }\right)$ the weak* dual of $C_p\left(X,E\right)$. Since by Theorem 3 the dual space $C_p{\left(X,E\right)}^{\prime }$ is algebraically isomorphic to $L\left(X\right)\otimes E^{\prime }$, one has

$$L_p\left(X,E^{\prime }\right)\simeq \left(L\left(X\right)\otimes E^{\prime },\sigma \left(L\left(X\right)\otimes E^{\prime },C\left(X,E\right)\right)\right).$$

A completely regular topological space X is a $k_{\mathbb{R}}$-space if every real function f defined on X whose restriction to every compact subset K of X is continuous, is continuous on X.

Theorem 8.

Let X be a hemicompact $k_{\mathbb{R}}$-space and let E be a nuclear Fréchet space. The space $L_p\left(X,E^{\prime }\right)$ is distinguished if and only if the strong dual of $L_p\left(X,E^{\prime }\right)$ coincides with $C_k\left(X,E\right)$.

Proof. 

We will denote by $C_{\beta }\left(X,E\right)$ the linear space $C\left(X,E\right)$ equipped with the strong topology $\beta \left(C\left(X,E\right),L\left(X\right)\otimes E^{\prime }\right)$, i.e., the strong dual of $L_p\left(X,E^{\prime }\right)$. Since X is a $k_{\mathbb{R}}$-space and E is complete, [11] [16.6.3 Corollary] ensures that

$$C_k\left(X,E\right)\simeq C_k\left(X\right){\widehat{\otimes }}_{\epsilon }E.$$

(1)

So, as both $C_k\left(X\right)$ and E are metrizable, $C_k\left(X,E\right)$ is a Fréchet space. Consequently, if $C_{\beta }\left(X,E\right)=C_k\left(X,E\right)$ then $C_{\beta }\left(X,E\right)$ is barrelled and $L_p\left(X,E^{\prime }\right)$ is distinguished.

Assume, conversely, that $L_p\left(X,E^{\prime }\right)$ is distinguished. From $C_k\left(X,E\right)\simeq C_k\left(X\right){\widehat{\otimes }}_{\epsilon }E$ it follows that $C_k{\left(X,E\right)}^{\prime }={\left(C_k\left(X\right){\otimes }_{\epsilon }E\right)}^{\prime }$. Since $L\left(X\right)\otimes E^{\prime }$ is algebraically isomorphic to a subspace of ${\left(C_k\left(X\right){\otimes }_{\epsilon }E\right)}^{\prime }$, it follows that the compact-open topology of $C\left(X,E\right)$ is stronger than $\beta \left(C\left(X,E\right),L\left(X\right)\otimes E^{\prime }\right)$. Hence, the identity map $J:C_k\left(X,E\right)\to C_{\beta }\left(X,E\right)$ is continuous.

Since X is a hemicompact, $C_k\left(X\right)$ is metrizable. As a consequence of E nuclearity, $C_k\left(X\right){\otimes }_{\epsilon }E=C_k\left(X\right){\otimes }_{\pi }E$ is a metrizable space. Hence, by (1) $C_k\left(X,E\right)$ is a Fréchet space. If $L_p\left(X,E^{\prime }\right)$ is distinguished, then $C_{\beta }\left(X,E\right)$ is barrelled. So J is a linear homeomorphism by the closed graph theorem. Thus, $C_{\beta }\left(X,E\right)=C_k\left(X,E\right)$. □

5. Conclusions and Two Open Problems

This paper has been motivated by the contrast between the extensively distinguished properties obtained recently in the injective tensor products $C_p\left(X\right){\otimes }_{\epsilon }E$ and in the spaces $C_p(X,E)$ with the few distinguished properties of injective tensor products related to the dual space $L_p\left(X\right)$ of $C_p\left(X\right)$ endowed with the weak* topology, as well as to the weak* dual of $C_p(X,E)$. In Section 2, distinguished properties in the injective tensor product space $L_p\left(X\right){\otimes }_{\epsilon }E$ are provided and in Section 3 and Section 4, the distinguished property of the weak* dual of $C_p\left(X\right){\otimes }_{\epsilon }E$ and a characterization of the distinguished property of the weak* dual of $C_p(X,E)$ for wide classes of spaces X and E are provided.

We do not know the answer for the following two problems when the Tychonoff space X is uncountable. It is easy to prove that the answer of these two problems is positive if X is countable.

Problem 1.

Is it true that if X is an uncountable P-space and E is a Fréchet space, then $L_p\left(X\right){\otimes }_{\epsilon }{E}^{\prime }\left({\mathrm{weak}}^{*}\right)$ is distinguished?

Problem 2.

Is it true that if X is an uncountable P-space and E is a Fréchet space, then the weak* dual of $C_p\left(X\right){\otimes }_{\epsilon }E$ is distinguished?