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Kōmura's theorem : ウィキペディア英語版
Kōmura's theorem
In mathematics, Kōmura's theorem is a result on the differentiability of absolutely continuous Banach space-valued functions, and is a substantial generalization of Lebesgue's theorem on the differentiability of the indefinite integral, which is that Φ : () → R given by
:\Phi(t) = \int_^ \varphi(s) \, \mathrm s,
is differentiable at ''t'' for almost every 0 < ''t'' < ''T'' when ''φ'' : () → R lies in the ''L''''p'' space ''L''1((); R).
==Statement of the theorem==
Let (''X'', || ||) be a reflexive Banach space and let ''φ'' : () → ''X'' be absolutely continuous. Then ''φ'' is (strongly) differentiable almost everywhere, the derivative ''φ''′ lies in the Bochner space ''L''1((); ''X''), and, for all 0 ≤ ''t'' ≤ ''T'',
:\varphi(t) = \varphi(0) + \int_^ \varphi'(s) \, \mathrm s.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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