Let $f:[0,\infty) \rightarrow E$ be a continuous function that takes value in a Banach space $E$. I know that we can define the integral $\int^x_0 f(t)dt, \forall x \in [0,\infty)$ using Riemann procedure just like in the real case, a post that I consulted can be found here . Although I'm just new to the theory, still I would like to know that how we can prove the map
$$ [0,\infty) \ni x \mapsto \int^x_0f(t)dt \in E$$
is continuous and differentiable, with the Frechet derivative at $x_0$ equal to $f(x_0)$. This is not easy for me with the basic knowledge of Banach spaces. Any suggestions, outline of the proof or any basic reference on the properties of this integral are pretty much appreciated!
For the context, I read Brezis' book, chapter 7, where he gave a general theorem of Cauchy, Lipchitz and Picard and encountered this notion of integral in his proof.
The Frechet derivative is defined to be
$$F'(t):= \lim_{h \rightarrow 0} \frac{F(t+h)-F(t)}{h},$$
provided the limit in $E$ exists and is unique.
Thank you in advance!
It sounds like you only want a hint, so i will only give one. If you need more information i can give further hints.
Note that $F^\prime (t)$ being the derivative at $t$ is equivalent to $F(t+h) = F(t) + F^\prime(t) h + \mathcal{o}(h)$ using the little-$o$ notation.
Show that for $a,b \in [0,\infty)$ with $a \leq b$ it is true that $$\| \int_a^b f(t) dt \| \leq \max_{t \in [a,b]}\| f(t) \| (b-a) .$$
Use this and some of the intuitive rules for the integral (you may have to prove) to show that $F(t+h) - F(t) - f(t) h = o(h)$, where $F$ is defined by $F(t) = \int_0^t f(t^\prime) dt^\prime $.