On compositions of d.c. functions and mappings

TL;DR

This paper investigates the stability of delta-convex (d.c.) functions under composition in infinite-dimensional spaces, showing that general stability does not hold but identifying specific cases where it does.

Contribution

It extends Hartman's theorem to infinite-dimensional spaces, identifying conditions under which d.c. functions and mappings preserve their structure under composition.

Findings

General stability of d.c. functions under composition fails in infinite dimensions

Stability holds in certain particular cases

Main results are established for d.c. mappings between normed spaces

Abstract

A d.c. (delta-convex) function on a normed linear space is a function representable as a difference of two continuous convex functions. We show that an infinite dimensional analogue of Hartman's theorem on stability of d.c. functions under compositions does not hold in general. However, we prove that it holds in some interesting particular cases. Our main results about compositions are proved in the more general context of d.c. mappings between normed linear spaces.