Kōmura's theorem - Alchetron, The Free Social Encyclopedia

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

Φ ( t ) = ∫ 0 t φ ( s ) d s ,

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

Statement of the theorem

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,

φ ( t ) = φ ( 0 ) + ∫ 0 t φ ′ ( s ) d s .

References

Kōmura's theorem Wikipedia

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