separability of a certain space of continuous functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T12:01:10Z http://mathoverflow.net/feeds/question/54026 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/54026/separability-of-a-certain-space-of-continuous-functions separability of a certain space of continuous functions Orbicular 2011-02-01T21:21:34Z 2011-02-02T04:43:27Z <p>Let $O$ be an open subset of the separable Hilbert space $H.$ Let $E$ be a separable Banach space. Is it true that $C^0_b(O;E),$ the space of bounded continuous maps $O\rightarrow E$, endowed with the $C^0$-norm, is separable? If YES, where can I find I proof of this fact?</p> http://mathoverflow.net/questions/54026/separability-of-a-certain-space-of-continuous-functions/54063#54063 Answer by Ady for separability of a certain space of continuous functions Ady 2011-02-02T04:43:27Z 2011-02-02T04:43:27Z <p>The answer is negative. For, pick some non-zero $e$ in $E$, and choose a surjection $\rho\in C\left(O,\mathbb{R}\right)$ (there exists !).</p> <p>Next, consider the (uncountable, uniformly discrete) family of functions { $f_{A}$; $A\subset\mathbb{Z}$ nonempty } $\subset C_{b}^{0}\left(O,E\right)$, expressed by $$f_{A}\left(x\right):=\arctan\left(dist\left(\rho\left(x\right),A\right)\right)\cdot e$$ $\left(x\in O\right).$ </p>