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

Asplund property of a Banach space X   is characterized by the existence of a rich family, in the product X×X⁎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 Fréchet subdifferentials.