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.
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<sub>α</sub> denote the resolvent:
and let A<sub>α</sub> denote the Yosida approximation to A:
For each α > 0 and x ∈ H, let
Then
and φ<sub>α</sub> is convex and Fréchet differentiable with derivative dφ<sub>α</sub> = A<sub>α</sub>. Also, for each x ∈ H (pointwise), φ<sub>α</sub>(x) converges upwards to φ(x) as α → 0.