Moreau's theorem

In mathematics, Moreau's theorem is a result in convex analysis named after French mathematician Jean-Jacques Moreau. It shows that sufficiently well-behaved convex functionals on Hilbert spaces are differentiable and the derivative is well-approximated by the so-called Yosida approximation, which is defined in terms of the resolvent operator.

Statement of the theorem

Let H be a Hilbert space and let φ : H → R ∪ be a proper, convex and lower semi-continuous extended real-valued functional on H. Let A stand for ∂φ, the subderivative of φ; for α >; 0 let J_α denote the resolvent:
and let A_α denote the Yosida approximation to A:
For each α > 0 and x ∈ H, let
Then
and φ_α is convex and Fréchet differentiable with derivative dφ_α = A_α. Also, for each x ∈ H, φ_α converges upwards to φ as α → 0.