Distinguished Property in Tensor Products and Weak* Dual Spaces

Abstract

[EN] 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 $C_{p}\left( X\right) \otimes _{\varepsilon }E$ 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 $L_{p}\left(X\right) \otimes _{\varepsilon }E$ 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.