On a Generalized Baillon-Haddad Theorem for Convex Functions on Hilbert Space

The Baillon-Haddad Theorem asserts that, if the gradient operator of a convex and Fréchet differentiable function on a Hilbert space is nonexpansive, then it is firmly nonexpansive. This theorem plays an important role in iterative optimization. In this note we present a short, elementary proof of a generalization of the Baillon-Haddad Theorem.