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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!

flag	asked Jan 9 at 11:51 Wolfgang Loehr 537●2●9

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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 at 12:29

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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 at 14:07

@Gerald: Thanks, I'll check Dunford & Schwartz @Theo: This proof is more elementary than mine, thank you. – Wolfgang Loehr Jan 9 at 15:11

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I think this result is also in the book A Course in Functional Analysis by John Conway. – Joseph Van Name Jan 9 at 18:03

Christian Clason · Accepted Answer · 2013-01-09 20:05:28Z UTC

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The result does appear in Dunford/Schwartz, Linear Operators Part I (page 437), but is only stated as an exercise.

Edit after @JosephVanName' comment: Conway's Functional Analysis has the result for completely regular spaces as Theorem 6.6 (page 140).

edited Jan 9 at 20:05

answered Jan 9 at 16:17

Christian Clason
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Thank you very much! I think one can cite exercises in Dunford & Schwartz ;-) – Wolfgang Loehr Jan 9 at 16:57

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I don't see why you wanted this to be a comment. It seems to be a fine answer. – S. Carnahan♦ Jan 9 at 22:35

Well, a proof should not only convince the reader that a statement is true, but also explain why it is true. For the former, an appeal to authority (like a reference to an exercise in D&S) is sufficient, but the latter requires pointing the reader to a full proof. That's why I did not think my first reply was a full answer. (Also, the references were suggested by others and I just checked them, so it seemed wrong to harvest the rep for them.) – Christian Clason Jan 10 at 8:31

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

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