Distinguished Property in Tensor Products and Weak* Dual Spaces

Axioms

Santiago Moll-López¹  Salvador López-Alfonso²  Manuel López-Pellicer³ 

[1] Department of Applied Mathematics, Universitat Politècnica de València, 46022 Valencia, Spain;Department of Architectural Constructions, Universitat Politècnica de València, 46022 Valencia, Spain;Emeritus and IUMPA, Universitat Politècnica de València, 46022 Valencia, Spain;

关键词: distinguished space;    injective and projective tensor product;    vector-valued continuous function;    Fréchet space;    nuclear space;   

DOI  :  10.3390/axioms10030151

来源: DOAJ

【 摘 要 】

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.

【 授权许可】

Unknown