Kōmura's theorem

Template:Short description 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

$\mathrm{\Phi }(t)=\underset{0}{\overset{t}{\int }}\phi (s)\mathrm{d}s,$

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

$\phi (t)=\phi (0)+\underset{0}{\overset{t}{\int }}{\phi }^{\prime }(s)\mathrm{d}s.$

References

• Template:Cite book Template:MathSciNet (Theorem III.1.7)