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

2022/04/01 by Daniel Wachsmuth, Gerd Wachsmuth, Wachsmuth, Daniel +1
Computer Science · Mathematics · #26B25 #47H05 #47N10 #49J50 #Advanced Banach Space Theory #FOS: Mathematics #Fixed Point Theorems Analysis #Functional Analysis (math.FA) #Optimization and Control (math.OC) #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.2204.00282

openalex publication_date 2022/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• An enhanced Baillon-Haddad theorem for convex functions on convex sets
• The Baillon-Haddad Theorem Revisited
• On the Properties of Convex Functions over Open Sets
• Characterization of Hilbertizable spaces via convex functions
• A universal bound on the variations of bounded convex functions