Kōmura's theorem

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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 : [0, T] → R given by

\Phi(t) = \int_{0}^{t} \varphi(s) \, \mathrm{d} s,

is differentiable at t for almost every 0 < t < T when φ : [0, T] → R lies in the Lp space L1([0, T]; R).

Statement of the theorem[edit]

Let (X, || ||) be a 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 L1([0, T]; X), and, for all 0 ≤ t ≤ T,

\varphi(t) = \varphi(0) + \int_{0}^{t} \varphi'(s) \, \mathrm{d} s.

References[edit]

  • Showalter, Ralph E. (1997). Monotone operators in Banach space and nonlinear partial differential equations. Mathematical Surveys and Monographs 49. Providence, RI: American Mathematical Society. p. 105. ISBN 0-8218-0500-2.  MR 1422252 (Theorem III.1.7)