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In [[mathematics]], '''Kōmura's theorem''' is a result on the [[derivative|differentiability]] of [[absolute continuity|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 : [0, ''T''] → '''R''' given by

:<math>\Phi(t) = \int_{0}^{t} \varphi(s) \, \mathrm{d} s,</math>

is differentiable at ''t'' for [[almost everywhere|almost every]] 0 < ''t'' < ''T'' when ''φ'' : [0, ''T''] → '''R''' lies in the [[Lp space|''L''''p'' space]] ''L''1([0, ''T'']; '''R''').

==Statement of the theorem==
Let (''X'', || ||) be a [[reflexive space|reflexive]] Banach space and let ''φ'' : [0, ''T''] → ''X'' be absolutely continuous. Then ''φ'' is (strongly) differentiable almost everywhere, the derivative ''φ''′ lies in the [[Bochner space]] ''L''1([0, ''T'']; ''X''), and, for all 0 ≤ ''t'' ≤ ''T'',

:<math>\varphi(t) = \varphi(0) + \int_{0}^{t} \varphi'(s) \, \mathrm{d} s.</math>

==References==
* {{cite book
| last = Showalter
| first = Ralph E.
| title = Monotone operators in Banach space and nonlinear partial differential equations
| series = Mathematical Surveys and Monographs 49
| publisher = American Mathematical Society
| location = Providence, RI
| year = 1997
| pages = 105
| isbn = 0-8218-0500-2
}} {{MathSciNet|id=1422252}} (Theorem III.1.7)

{{DEFAULTSORT:Komuras theorem}}
[[Category:Measure theory]]
[[Category:Theorems in functional analysis]]

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en>TakuyaMurata: /* Examples */ lk