A Pointwise Lipschitz Selection Theorem

TL;DR

This paper proves a new pointwise Lipschitz selection theorem for correspondences in metric and Banach spaces, with applications to inverse operators, showing optimality of the result.

Contribution

It introduces a novel pointwise Lipschitz selection theorem for correspondences, improving classical results and demonstrating optimality through a counterexample.

Findings

Existence of continuous pointwise Lipschitz selections on dense sets

Application to positively homogeneous right inverses of surjective operators

Optimality of the theorem shown by a counterexample

Abstract

We prove that any correspondence (multi-function) mapping a metric space into a Banach space that satisfies a certain pointwise Lipschitz condition, always has a continuous selection that is pointwise Lipschitz on a dense set of its domain. We apply our selection theorem to demonstrate a slight improvement to a well-known version of the classical Bartle-Graves Theorem: Any continuous linear surjection between infinite dimensional Banach spaces has a positively homogeneous continuous right inverse that is pointwise Lipschitz on a dense meager set of its domain. An example devised by Aharoni and Lindenstrauss shows that our pointwise Lipschitz selection theorem is in some sense optimal: It is impossible to improve our pointwise Lipschitz selection theorem to one that yields a selection that is pointwise Lipschitz on the whole of its domain in general.