Let $E$ be the set of functions $f: \mathbb{R} \rightarrow \mathbb{R}$, that are $\mathcal{C}^1$ and such that $\forall x \in \mathbb{R}$, $f(x+1)=f(x)$. Let $g \in E$ and $F: \mathbb{R} \times E \rightarrow \mathbb{R}$ such that $$F(s,f)= \int^{1}_{0}f(x+s)g'(x)dx $$ Show that $F$ is $\mathcal{C}^1$.
Now, I found this exercise in past terms exams, so it might be possible that in order to solve it I require some notions that I haven't seen yet in my course. Anyway, I wonder, how should I approach this exercise? Is there a way to apply the Leibniz integral rule?
For Fréchet differentiability, is true that continuous partial derivatives $\implies$ differentiability even in the infinite-dimensional case. Also, fixing a norm in $E$ is required, but the reasonable norm for $E$ is $$\|f\| = \|f\|_\infty + \|f'\|_\infty.$$ ($\|\cdot\|_\infty$ = $\sup$ norm)
As for $s$ fixed $f\longmapsto F(s,f)$ is obviously linear, proving continuity is enough. Taking $h\in E$: $$ |F(s,f + h) - F(s,h)| = \left|\int^{1}_{0}(f + h)(x + s)g'(x)\,dx - \int^{1}_{0}f(x + s)g'(x)\,dx\right| = $$ $$ \left|\int^{1}_{0}h(x + s)g'(x)\,dx\right|\le\|g'\|_\infty\|h\|_\infty(1 - 0) \le\|g'\|_\infty\|h\|. $$ For the other partial derivative (the derivative of $s\longmapsto F(s,f)$), use Leibniz's rule.