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Questions related to various forms of integration including the Riemann integral, Lebesgue integral, Riemann–Stieltjes integral, double integrals, line integrals, contour integrals, surface integrals, integrals of differential forms, ...

5 votes
1 answer
449 views

The Bochner integral about a semigroup of bounded linear operators on a Banach space

Let $T(t)$ be a semigroup of bounded linear operators on a Banach space $X$. When does the following hold $$ \int_0^t T(s)x ds = \Big(\int_0^t T(s) ds\Big)x, \quad x \in X \, , $$ where $ t \in (0,1)$ …
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