Kōmura's theorem
# 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

    $\Phi (t)=\int _{0}^{t}\varphi (s)\,\mathrm {d} s,$

is differentiable at t for almost every 0 < t < T when φ : [0, T] → R lies in the $L^p$ space $L^1([0,$ T]; R).

## Statement

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 $L^1([0,$ T]; X), and, for all 0 ≤ t ≤ T,

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