On the product of Weak Asplund locally convex spaces

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 classical Mazur’s theorem as follows: Let E be a separable Baire locally convex space and let Y be the product ∏α∈AEα of any family of separable Fréchet spaces; then the product E×Y is Weak Asplund. Also, we prove that the product Y of any family of Banach spaces (Eα) is an Asplund locally convex space if and only if each Eα is Asplund. Analogues of both results are valid under the same assumptions, if Y is the Σ-product of any family (Eα).