If $X$ is a real-valued random variable with characteristic function $\varphi_X$, then we know that $\operatorname E\left[|X|^n\right]<\infty$ implies that $\varphi_X$ is $n$-times continuously differentiable with derivative $$\varphi_X^{(k)}=\operatorname E\left[({\rm i}X)^ke^{{\rm i}tX}\right]\tag1$$ for all $k=0,\ldots,n$.
Are we able to generalize this result to random variables $Y$ with values in a Banach (or Hilbert) space $E$?
In this case, the characteristic function is given by $$\varphi_Y(x')=\operatorname E\left[e^{{\rm i}\langle Y,\:x'\rangle}\right]\;\;\;\text{for all }x'\in E'.$$