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In mathematics, Mazur's lemma is a result in the theory of Banach spaces. It shows that any weakly convergent sequence in a Banach space has a sequence of convex combinations of its members that converges strongly to the same limit, and is used in the proof of Tonelli's theorem. ==Statement of the lemma== Let (''X'', || ||) be a Banach space and let (''u''''n'')''n''∈N be a sequence in ''X'' that converges weakly to some ''u''0 in ''X'': : That is, for every continuous linear functional ''f'' in ''X''∗, the continuous dual space of ''X'', : Then there exists a function ''N'' : N → N and a sequence of sets of real numbers : such that ''α''(''n'')''k'' ≥ 0 and : such that the sequence (''v''''n'')''n''∈N defined by the convex combination : converges strongly in ''X'' to ''u''0, i.e. : 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「Mazur's lemma」の詳細全文を読む スポンサード リンク
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