On the product of Weak Asplund locally convex spaces

TL;DR

This paper explores properties of Weak Asplund locally convex spaces, extending classical theorems like Mazur's and characterizing when products of Banach spaces are Asplund, with implications for differentiability and space structure.

Contribution

It extends Mazur's theorem to certain locally convex spaces and characterizes when products of Banach spaces are Asplund, providing new insights into their structure.

Findings

Product of a separable Baire locally convex space and a product of separable Fréchet spaces is Weak Asplund.

Product of any family of Banach spaces is Asplund iff each factor is Asplund.

Analogous results hold for the $ ext{Sigma}$-product of Banach spaces.

Abstract

For locally convex spaces, we systematize several known equivalent definitions of Fr\'echet (G\^ ateaux) Differentiability Spaces and Asplund (Weak Asplund) Spaces. As an application, we extend the classical Mazur's theorem as follows: Let $E$ be a separable Baire locally convex space and let $Y$ be the product $\prod_{\alpha\in A} E_{\alpha}$ of any family of separable Fr\'echet spaces; then the product $E \times Y$ is Weak Asplund. Also, we prove that the product $Y$ of any family of Banach spaces $(E_{\alpha})$ is an Asplund locally convex space if and only if each $E_{\alpha}$ is Asplund. Analogues of both results are valid under the same assumptions, if $Y$ is the $\Sigma$-product of any family $(E_{\alpha})$.