Arkady Leiderman

On the product of Weak Asplund locally convex spaces.

The classical Mazur’s theorem in modern terms states that every separable Banach space is Weakly Asplund.

For locally convex spaces, we systematize several known equivalent definitions of Fréchet (Gâteaux) Differentiability Spaces and Asplund (Weak Asplund) Spaces. As an application, we extend the Mazur’s theorem as follows:

Theorem 1. 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 Fréchet spaces; then the product $E \times Y$ is Weak Asplund.

Also, we prove

Theorem 2. 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 $(E_{\alpha})$ .

This is joint work with Jerzy Ka̧kol.

Algebra, Topology, and Dynamical Systems

A conference in honor of Dikran Dikranjan on the occasion of his 75th birthday and of Ivan Prodanov on the occasion of the 90th anniversary of his birth

Arkady Leiderman