Let $X$ be a compact subset of the Euclidean space. Also let $Y$ be a separable Frechet space with the seminorms $\lVert \cdot \rVert\_n$'s.


Then let $C(X,Y)$ be the space of continuous mappings from $X$ into $Y$. It can be given the topology induced from the seminorms $\mid f \mid\_n := \sup\_{x \in X}\lVert f(x) \rVert\_n$ where $f \in C(X,Y)$.


My question is that, is $C(X,Y)$ Frechet and separable? If so, how to prove this?


Could anyone please help me?



Showing that $C(X,Y)$ is separable when $X$ is compact Hausdorff but $Y$ is just a separable Frechet space
If $d$ is a translation-invariant metric on $Y$ then
$$
D(f,g)=\sup\{d(f(x),g(x)):x\in X\}
$$
defines a translation-invariant metric on $C(X,Y)$, and $D$ is complete if $d$ is complete.
The topology induced by $D$ is the compact-open topology ([Arens, A topology for spaces of transformations, Ann. Math. 47 (1946), 480-495](https://doi.org/10.2307/1969087)).
Both $X$ and $Y$ are second-countable, hence the compact-open topology is separable ([E. A. Michael, On a theorem of Rudin and Klee, Proc. Amer. Math. Soc. 12 (1961), 921](https://doi.org/10.1090/S0002-9939-1961-0133108-4)).


