I'm trying to show that a map $f$ between Banach spaces $X$ and $Y$ is Frechet differentiable at a point $u$. To do this, it is enough to calculate its Gateaux derivative at $u$ (call it $df(u)$) and show that $df(u)$ is continuous: so for every $\epsilon$, there exists a $\delta$ such that if $\lVert h_1 - h_2 \rVert_X \leq \delta,$ then $\lVert df(u)h_1 - df(u)h_2 \rVert_Y < \epsilon.$ Is that correct to show Frechet differentiability?
Secondly, my Banach spaces $X$ and $Y$ are Hölder spaces. Specifically, let $X = C^{k, \alpha}(S)$, where $S$ is the closure of a set. Let $g$ be a smooth function in its arguments (so it's continuous). Since $S$ is closed and the norm is over a space of that set, can I say that $g$ is bounded above and hence $\lVert gh_1 - gh_2 \rVert_X \leq C\rVert h_1 - h_2 \rVert_X$ for some constant?
Thanks