Let $K$ be a compact metric space, and $(E,d\_E)$ a complete separable metric space.
Define $C:=C(K,E)$ to be the continuous functions from $K$ to $E$ equipped with
the metric $d(f,g)=\sup\_{k\in K}\ d\_E (f(k),g(k))$. Is the space $C$ separable?


The result is true when $E$ is the real line; this is Corollary 11.2.5 in Dudley's 
book *Real Analysis and Probability*. 


The result is also true when $K=[0,1]$ (if I'm not being too careless) by 
considering $C$ as a subspace of the Skorohod space $D\_E[0,1]$, which is 
complete and separable by Theorem 5.6 in Ethier and Kurtz's book *Markov Processes:
 Characterization and Convergence*. 


For general $K$, it is not so obvious how to find an explicit countable
dense set in $C$, but I suspect one could modify Ethier and Kurtz's approach and get a proof. 


But surely this result is known, and stated in some book? I've searched through
my library without success. 




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**Update:** This result is also **Theorem 2.4.3** of S. M. Srivastava's book *A Course on Borel Sets*. His proof is the same as Kechris's. I have also found an alternative, but false, published proof using the "fact" that $C(K,E)$ is $\sigma$-compact. Beware! 



Is the space of continuous functions from a compact metric space into a Polish space Polish?
Yes, it appears e.g. as Theorem 4.19 in Chapter I of Kechris' *Classical Descriptive Set Theory*. (The relevant page is visible in Google Books if it's not in your library.)


