Structure of sets of strong subdifferentiability in dual $L^1$-spaces

TL;DR

This paper investigates the structure of finite-dimensional subspaces of strongly subdifferentiable points in dual L1 spaces, revealing their relation to the discrete part of the Yoshida-Hewitt decomposition and implications for strong proximinality.

Contribution

It characterizes finite-dimensional subspaces of strongly subdifferentiable points in dual L1 spaces and provides new proofs for strong proximinality results.

Findings

Finite-dimensional subspaces are in the discrete part of the Yoshida-Hewitt decomposition.

Any Banach space of strongly subdifferentiable points is finite dimensional.

New streamlined proofs for strong proximinality in finite co-dimension subspaces.

Abstract

In this article, we analyse the structure of finite dimensional subspaces of the set of points of strong subdifferentiability in a dual space. In a dual $L_1(\mu)$ space, such a subspace is in the discrete part of the Yoshida-Hewitt type decomposition. In this set up, any Banach space consisting of points of strong subdifferentiability is necessarily finite dimensional. Our results also lead to streamlined and new proofs of results from the study of strong proximinality for subspaces of finite co-dimension in a Banach space.