Asplund spaces characterized by rich families and separable reduction of Fréchet subdifferentiability

Abstract Asplund property of a Banach space X is characterized by the existence of a rich family, in the product X × X ⁎ , consisting of some carefully chosen separable subspaces. This structural result is then used to add a lot of precision and simplicity to the known separable reductions of Frechet subdifferentials.