# reference for “X compact <=> C_b(X) separable” (X metric space)

I know (and am able to prove via Stone-Čech compactification) that the following is correct:

Theorem: A metric space is compact if and only if its space of bounded, continuous, real-valued functions is separable in the uniform topology.

I use it in a paper for readers who are presumably not familiar with this kind of topology, so I cannot call it "obvious" or "well-known". I would be thankful for a name and/or good reference to cite this theorem!

• I don't have a reference, but I'd suggest this argument: if $X$ is not compact, there is an infinite closed discrete subset $D$ of $X$. For every $A \subset D$ choose a continuous function $f_A \colon X \to [0,1]$ such that $f_A|_A = 1$ and $f_A|_{D \setminus A} = 0$ (by Urysohn). This gives an uncountable family of continuous bounded functions such that $\|f_A - f_B\| = 1$ whenever $A \neq B$. Alternatively, embed $\ell^\infty$ using a similar trick. If $X$ is compact then $C(X)$ is separable by Stone-Weierstrass. – Theo Buehler Jan 9 '13 at 12:29
• Agreed, it is easy to prove. If you don't get a good reference, perhaps searching the index in Dunford & Schwartz will work. They generally have collected all results of this type. – Gerald Edgar Jan 9 '13 at 14:07
• @Gerald: Thanks, I'll check Dunford & Schwartz @Theo: This proof is more elementary than mine, thank you. – Wolfgang Loehr Jan 9 '13 at 15:11
• I think this result is also in the book A Course in Functional Analysis by John Conway. – Joseph Van Name Jan 9 '13 at 18:03