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 : [0, T]  R given by

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

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,

References

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