reference for "X compact <=> C_b(X) separable" (X metric space) - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T01:54:33Z http://mathoverflow.net/feeds/question/118442 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118442/reference-for-x-compact-c-bx-separable-x-metric-space reference for "X compact <=> C_b(X) separable" (X metric space) Wolfgang Loehr 2013-01-09T11:51:38Z 2013-01-09T20:05:28Z <p>I know (and am able to prove via Stone-Čech compactification) that the following is correct:</p> <blockquote> <p><strong>Theorem:</strong> A metric space is compact if and only if its space of bounded, continuous, real-valued functions is separable in the uniform topology.</p> </blockquote> <p>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!</p> http://mathoverflow.net/questions/118442/reference-for-x-compact-c-bx-separable-x-metric-space/118461#118461 Answer by Christian Clason for reference for "X compact <=> C_b(X) separable" (X metric space) Christian Clason 2013-01-09T16:17:10Z 2013-01-09T20:05:28Z <p>The result does appear in Dunford/Schwartz, <em>Linear Operators Part I</em> (page 437), but is only stated as an exercise. </p> <p>Edit after @JosephVanName' comment: Conway's <em>Functional Analysis</em> has the result for completely regular spaces as Theorem 6.6 (page 140).</p>