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

arXiv: https://arxiv.org/abs/2204.00282 · HTML: https://mlsanity.com/paper/2204.00282/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Apr 2022.

## Built on by (most cited first, 2 of 2)

- [Mirror Descent on Reproducing Kernel Banach Spaces](https://mlsanity.com/paper/2411.11242/) (2024)
- [ProofLoom: Proof-Obligation-Driven Theory Construction for Autoformalizing Research-Level Stochastic Optimization](https://mlsanity.com/paper/2609.34960/) (2026)

## Abstract (arXiv)

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.