Function $f:(a,b)\to \mathbb{R}$ is measurable and absolutely continuous if and only if there exists the weak derivative $\frac{df}{dx}\in L^1(a,b)$. The weak derivative coincides with the classical derivative almost everywhere.
Can we prove the same theorem for $f:(a,b)\to X$, where $X$ is a Banach space?
In order to have differentiabilty almost everywhere of absolutely continuous function $f$ with the values in an abstract Banach space, we have to assume that $X$ is reflexive. I tried to copy the proof from the very first case but I stop when it comes to integrating by parts. Does the classical integrating by parts has the same form for absolutely continuous functions with values in Banach spaces?