Distinguished Property in Tensor Products and Weak* Dual Spaces

: A local convex space E is said to be distinguished if its strong dual E (cid:48) β has the topology β ( E (cid:48) , ( E (cid:48) β ) (cid:48) ) , i.e., if E (cid:48) β is barrelled. The distinguished property of the local convex space C p ( X ) 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 ( X ) is distinguished if and only if any function f ∈ R X belongs to the pointwise closure of a pointwise bounded set in C ( X ) . The extensively studied distinguished properties in the injective tensor products C p ( X ) ⊗ ε E and in C p ( X , E ) contrasts with the few distinguished properties of injective tensor products related to the dual space L p ( X ) of C p ( X ) 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 ( X ) ⊗ ε 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.


Introduction
In this paper, X is an infinite Tychonoff space and C(X) is the linear space of all realvalued continuous functions over X. C p (X) and C k (X) denote the space C(X) equipped with the pointwise and compact-open topology, respectively. L p (X) represents the weak* dual of C p (X), i.e., the topological dual L(X) of C p (X) endowed with the weak topology σ(L(X), C(X)) of the dual pair L(X), C(X) , i.e., L p (X) has the topology of pointwise convergence on C(X).
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 β of a local convex space E is the topological dual E of E equipped with the strong topology β(E , E), which is the topology of uniform convergence on the bounded subsets of E. E, E 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 ∈ F : | a, x | ≤ 1, ∀a ∈ A}.
A local convex space E is barrelled if for each pointwise bounded subset M of E 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 β(E, E ), i.e., [E (weak * )] β = 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 β is barrelled. In [1][2][3][4][5][6][7] the distinguished property of the space C p (X) has been extensively studied. Furthermore, [8] [Proposition 6.4] is connected with distinguished C p (X) spaces. It is observed in [3] [Theorem 10] that C p (X) is distinguished if and only if C p (X) is a large subspace of R X , i.e., if each bounded set in R X is contained in the closure in R X of a bounded set of C p (X), or, equivalently, if the strong bidual of C p (X) is R X [5]. In [7], [Theorem 2.1] it is shown that C p (X) is distinguished if and only if X is a ∆-space in the sense of Knight [9], and several applications of this fact are given. Equivalently, C p (X) is distinguished if for each countable partition {X k : k ∈ N} of X into nonempty pairwise disjoint sets, there are open sets {U k : k ∈ N} with X k ⊆ U k , for each k ∈ N, such that each point x ∈ X belongs to U n for only finitely many n ∈ N, [5] [ Theorem 5].
If E and F are local convex spaces, E ⊗ ε F and E ⊗ π F represent the injective and projective tensor product of E and F, respectively. A basis of neighborhoods of the origin in Analogously, a basis of neighborhoods of the origin in the tensor product space E β ⊗ π F β is formed by the sets π(A, B) := acx A 0 ⊗ B 0 , where A is a bounded set in E, B is a bounded set in F and acx A 0 ⊗ B 0 denotes the absolutely convex cover of the tensor The distinguished property of C p (X) 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 (X) ⊗ ε E is distinguished if both C p (X) is distinguished and R (X) ⊗ ε E β is barrelled, where R (X) the local convex direct sum of |X| real lines.
If E is a local convex space C p (X, E) 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 (X) ⊗ ε 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 (X) is distinguished by [5] [Corollary 6] and on the other hand R (X) is both barrelled and nuclear (the latter because [11] [21.2.3 Corollary]), so that R (X) ⊗ ε E β = R (X) ⊗ π E β is barrelled by [12] According to [1] [Theorem 3.9], the strong dual L β (X) of C p (X) is always distinguished. The distinguished property of the weak* dual L p (X) of C p (X) is investigated in [5], where the following theorem is proved.
Recall that a Tychonoff space X is called a µ-space if each functionally bounded set is relatively compact.
The extensively studied distinguished properties in the injective tensor products C p (X) ⊗ ε E and in C p (X, E) contrasts with the few distinguished properties related with the injective tensor products L p (X) ⊗ ε E and with the weak* dual of C p (X, E). Theorem 1 and the fact that L p (X) spaces are studied so extensively as C p (X) spaces motivated us to fill partially this gap in this paper obtaining distinguished properties of injective tensor products L p (X) ⊗ ε 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.
. Let E and F be local convex spaces, where E carries the weak topology. If τ ε and τ π denote the injective and projective topologies of E β ⊗ F β , the following properties hold

Distinguished Tensor Products of L p (X) Spaces
This section deals mainly with the injective tensor product of L p (X) 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 D(Ω), where Ω is an open set in R n , with their usual local convex inductive topologies, are examples of nuclear Fréchet spaces [11] [Section 21.6]. The strong dual of D(Ω) is the space of distributions on Ω and it is denoted by D (Ω).

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 (X) ⊗ ε E is distinguished.
Proof. The space X is a µ-space if and only if C k (X) 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 (X) is also a (DF)-space [15] [ Theorem 12]. In addition, the strong dual E β of a metrizable local convex space E it is a complete (DF)-space by [16] [see 29.3 -in "By 2(1)"-]. Moreover, nuclearity of E implies that E β is nuclear too by [11] [21.5.3 Theorem]. As E β is a nuclear (DF)-space, one has that E β is a quasi-barrelled space [11] [21.5.4 Corollary]. Finally, the completeness of the quasi-barrelled space E β implies that E β is barrelled [16] [27.1.(1)], so E is distinguished.
The projective tensor product C k (X) ⊗ π E β is barrelled by [11] [15.6.8 Proposition]. Thus, taking into consideration E β nuclearity, it can be obtained that C k (X) ⊗ ε E β is also barrelled. On the other hand, since X is a µ-space it follows from [5] [Theorem 27] that C k (X) coincides with the strong dual of L p (X), i.e., L p (X) β = C k (X), hence is barrelled. Finally, as L p (X) carries the weak topology, the first statement of Theorem 2, ensures that the space L p (X) ⊗ ε 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 (X) ⊗ ε 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 D(Ω) or s, then the injective tensor products L p (K) ⊗ ε D(Ω), L p (K) ⊗ ε s, L p ([0, 1]) ⊗ ε D(Ω) and L p ([0, 1]) ⊗ ε s are distinguished. Corollary 1. If X is a compact space and Y is a countable Tychonoff space, then the space [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 βN of the topological space N formed by the natural numbers endowed with the discrete topology and Y equal to the space Q of rational numbers endowed with the usual metrizable topology then we get that L p (βN) ⊗ ε C p (Q) is a distinguished space.
If the factor E of L p (X) ⊗ ε 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 (X) ⊗ ε E is distinguished.
Proof. If X is a µ-space with finite compact sets, the space C k (X) = C p (X) is barrelled and nuclear. As E β is a Banach space, [12] [Corollary 1.6.6] assures that C k (X) ⊗ π E β is a barrelled space, and C k (X) nuclearity yields that C k (X) ⊗ ε E β is also a barrelled space.
Bearing in mind that L p (X) β = C k (X), as a consequence of the fact that X is a µ-space (cf. [5] [Theorem 27]), Theorem 2 ensures that L p (X) ⊗ ε 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 (X) ⊗ ε E is distinguished.

Example 2.
If L(m) denotes the Lindelöfication of the discrete space of cardinal m ≥ ℵ 1 , the space L p (L(m)) ⊗ ε C k ([0, 1]) is distinguished. In this case L(m) 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 (X) ⊗ ε E (weak * ) is distinguished.
Proof. By [12] [Theorem 1.6.6] the projective tensor product C k (X) ⊗ π E is a barrelled space, hence C k (X) nuclearity yields that C k (X) ⊗ ε E is barrelled. So, the conclusion follows from the first statement of Theorem 2.
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 (X) ⊗ ε E (weak * ) is distinguished.
Proof. Clearly X is a Lindelöf space, hence it is a µ-space, and then both C k (X) and E are metrizable and barrelled spaces. Then [12] [Corollary 1.6.4] ensures that C k (X) ⊗ π E is also a (metrizable) barrelled space. This property and the E nuclearity imply that C k (X) ⊗ ε E is a barrelled space. Consequently, using that L p (X) β = C k (X) and E (weak So, Theorem 2 applies to guarantee that L p (X) ⊗ ε E (weak * ) is distinguished.
By Theorem 7 the injective tensor product L p (R) ⊗ ε D (Ω)(weak * ) is distinguished since R is hemicompact and D (Ω)(weak * ) is a nuclear Fréchet space. Theorem 7 is also applied in the next Example 4.

Example 4.
If N is equipped with the discrete topology, p ∈ βN \ N and Z = N ∪ {p} has the topology induced by βN, then L p (Z) ⊗ ε L p (Z) is distinguished.
Proof. The subspace Z = N ∪ {p} of βN is countable and has finite compact sets, so that it is hemicompact. Since Z is countable, C p (Z) is metrizable and, on the other hand, as a subspace of the nuclear space R X , the space C p (Z) is nuclear. In addition, since Z is a µ-space with finite compact sets, the space C p (Z) is barrelled [18]. So, according to the previous theorem, L p (Z) ⊗ ε L p (Z) is distinguished.

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 (X) ⊗ ε E. To get this property we need the following lemma. Lemma 1. The injective topology of the tensor product L p (X) ⊗ E (weak * ) coincides with the weak topology σ(L(X) ⊗ E , C(X) ⊗ E).
Proof. Since L p (X) carries the weak topology Hence, the injective topology τ ε of L p (X) ⊗ E (weak * ) is stronger than the weak topology σ(L(X) ⊗ E , C(X) ⊗ E). We prove that both topologies are the same. Indeed, if U is a closed absolutely convex neighborhood of the origin in L p (X) and V is a closed absolutely convex neighborhood of the origin in E (weak * ), there are finite sets Φ in C(X) and Corollary 3. If X is a hemicompact space and E is a nuclear Fréchet space, the weak* dual of C p (X) ⊗ ε E is distinguished.
Proof. According to Lemma 1 the weak* dual of C p (X) ⊗ ε E is linearly homeomorphic to L p (X) ⊗ ε E (weak * ), 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 (R) ⊗ ε C p (Z) and C p (Z) ⊗ ε D(Ω) 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 (X) ⊗ ε 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.

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 (X, E ) the weak* dual of C p (X, E). Since by Theorem 3 the dual space C p (X, E) is algebraically isomorphic to L(X) ⊗ E , one has L p X, E L(X) ⊗ E , σ L(X) ⊗ E , C(X, E) .
A completely regular topological space X is a k 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 R -space and let E be a nuclear Fréchet space. The space L p (X, E ) is distinguished if and only if the strong dual of L p (X, E ) coincides with C k (X, E).
Proof. We will denote by C β (X, E) the linear space C(X, E) equipped with the strong topology β(C(X, E), L(X) ⊗ E ), i.e., the strong dual of L p (X, E ). Since X is a k R -space and E is complete, [11] [16.6.3 Corollary] ensures that So, as both C k (X) and E are metrizable, C k (X, E) is a Fréchet space. Consequently, if C β (X, E) = C k (X, E) then C β (X, E) is barrelled and L p (X, E ) is distinguished.
Assume, conversely, that L p (X, E ) is distinguished. From C k (X, E) C k (X) ⊗ ε E it follows that C k (X, E) = (C k (X) ⊗ ε E) . Since L(X) ⊗ E is algebraically isomorphic to a subspace of (C k (X) ⊗ ε E) , it follows that the compact-open topology of C(X, E) is stronger than β(C(X, E), L(X) ⊗ E ). Hence, the identity map J : C k (X, E) → C β (X, E) is continuous.
Since X is a hemicompact, C k (X) is metrizable. As a consequence of E nuclearity, C k (X) ⊗ ε E = C k (X) ⊗ π E is a metrizable space. Hence, by (1) C k (X, E) is a Fréchet space. If L p (X, E ) is distinguished, then C β (X, E) is barrelled. So J is a linear homeomorphism by the closed graph theorem. Thus, C β (X, E) = C k (X, E).

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 (X) ⊗ ε 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 (X) of C p (X) 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 (X) ⊗ ε E are provided and in Sections 3 and 4, the distinguished property of the weak* dual of C p (X) ⊗ ε 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.