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In mathematics, Moreau's theorem is a result in convex analysis. 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 d''φ''''α'' = ''A''''α''. Also, for each ''x'' ∈ ''H'' (pointwise), ''φ''''α''(''x'') converges upwards to ''φ''(''x'') as ''α'' → 0. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Moreau's theorem」の詳細全文を読む スポンサード リンク
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