Let $(X,\Sigma,\mu)$ be a measure space and $B$ be a Banach space. A Bochner-measurable function $f: X \rightarrow B$ is Bochner integrable if and only if $$ \int_{X}\|f(x)\|_{B} d \mu(x)<\infty $$ In that case $$ \int_{X} f(x) d \mu(x) $$ is well defined and belongs to $B$. What if $B$ is some space of functions such as $B = \mathcal{L}(E,F)$ the space of bounded linear operators from $E$ to $F$ where $E$ and $F$ are two normed vector spaces and $F$ is complete so that $B$ is a Banach space. Is it always true that, for $e \in E$
$$ \left[\int_{X} f(x) d \mu(x)\right](e) = \int_{X} f(x)(e) d \mu(x). ? $$ I could not find a reference book explaining that property. I am particularly interested in the setting where $B$ is a Reproducing Kernel Hilbert space.