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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?

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  • $\begingroup$ Fréchet space: en.wikipedia.org/wiki/Fr%C3%A9chet_space $\endgroup$
    – Joel David Hamkins
    Commented Jul 13, 2021 at 8:30
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    $\begingroup$ The tensor product $C(X)\otimes Y$ is dense in $C(X,Y)$. $\endgroup$
    – Jochen Wengenroth
    Commented Jul 13, 2021 at 11:12
  • $\begingroup$ Never mind Frechet. When $\ X\ $ is metric and compact, and $\ Y\ $ is metric and separable then $\ C(X\ Y)\ $ is separable. #### This is textbook knowledge, no need to ask MO. $\endgroup$
    – Wlod AA
    Commented Jul 14, 2021 at 5:41
  • $\begingroup$ Oh, there is also the Frechet assertion that $\ C(X\ Y)\ $ is Frechet. It is isometrically/linearly embedded in the inverse limit of a sequence of R^n spaces (each with the max norm), with Lip_1 projections. $\endgroup$
    – Wlod AA
    Commented Jul 14, 2021 at 5:56

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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). 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).

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