Separable reduction revisited

Abstract

We prove a new version of the separable reduction theorem for Fréchet subdifferentials. Namely, we show that, given Banach spaces X and Y, a lower semicontinuous function f on Y and a linear bounded operator A : X → Y, the relation holds if and only if a similar relation holds for restrictions of f and A on sufficiently big separable subspaces of the corresponding spaces. The result is further applied to the subdifferential calculus in Asplund spaces.