mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Moreau's theorem is a result in

convex analysis Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex optimization, convex minimization, a subdomain of optimization (mathematics), optimization theor ...

named after French mathematician Jean-Jacques Moreau. It shows that sufficiently

well-behaved In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved or n ...

convex function In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of a function, graph of the function lies above or on the graph between the two points. Equivalently, a function is conve ...

als on

Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...

s 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 Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...

, convex and

lower semi-continuous In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semicontinuous at a point x_0 if, r ...

extended real-valued functional on ''H''. Let ''A'' stand for ∂''φ'', the

subderivative In mathematics, the subderivative (or subgradient) generalizes the derivative to convex functions which are not necessarily differentiable. The set of subderivatives at a point is called the subdifferential at that point. Subderivatives arise in c ...

of ''φ''; for ''α'' > 0 let ''J''_''α'' denote the resolvent: :

J_ = (\mathrm + \alpha A)^;

and let ''A''_''α'' denote the Yosida approximation to ''A'': :

A_ = \frac1 ( \mathrm - J_ ).

For each ''α'' > 0 and ''x'' ∈ ''H'', let :

\varphi_ (x) = \inf_ \frac1 \,  y - x \, ^ + \varphi (y).

Then :

\varphi_ (x) = \frac \,  A_ x \, ^ + \varphi (J_ (x))

and ''φ''_''α'' is convex and Fréchet differentiable with derivative d''φ''_''α'' = ''A''_''α''. Also, for each ''x'' ∈ ''H'' (pointwise), ''φ''_''α''(''x'') converges upwards to ''φ''(''x'') as ''α'' → 0.

References

* (Proposition IV.1.8) {{Functional analysis Convex analysis Theorems in functional analysis