A simple proof of the Baillon-Haddad theorem on open subsets of Hilbert spaces

TL;DR

This paper presents a straightforward proof of the Baillon-Haddad theorem for convex functions on open subsets of Hilbert spaces, explores generalizations, and discusses Lipschitz continuity of derivatives in Banach spaces.

Contribution

It provides a simplified proof of the Baillon-Haddad theorem and extends the discussion to Banach spaces with new characterizations.

Findings

Simplified proof of Baillon-Haddad theorem for Hilbert spaces

Characterizations of Lipschitz continuity of derivatives in Banach spaces

Discussion of generalizations and limitations of the theorem

Abstract

We give a simple proof of the Baillon-Haddad theorem for convex functions defined on open and convex subsets of Hilbert spaces. We also state some generalizations and limitations. In particular, we discuss equivalent characterizations of the Lipschitz continuity of the derivative of convex functions on open and convex subsets of Banach spaces.