Welcome to MathOverflow

Question

0

1

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?

flag	asked Feb 1 2011 at 21:21 Orbicular 1,794●8●18

1

Is $C^0$ the uniform norm? If so, the answer is clearly no. You can find an uncountable set such that the distance between any two is 1; for each set of integers pick a continuous function which is 1 on those integers and 0 on the rest. – Nate Eldredge Feb 1 2011 at 23:06

Er, here I am thinking of taking $O = H = \mathbb{R}$ and $E = \mathbb{R}$ as well. – Nate Eldredge Feb 1 2011 at 23:13

@Nate: I am confused. The space $C^0_b(\mathbb{R})$ is separable! The proof is simple - it follows from the separability of $C^0([0,1]).$ Fix a $\delta>0$ sufficiently small and cover $\mathbb{R}$ by intervals of the form $[n-\delta,n+1+\delta].$ On each of these there is a countable dense set. Now glue all these together on the overlapping parts to make the resulting function continuous. This is how you arrive at your countable dense set! – Orbicular Feb 2 2011 at 8:30

1

@Orbicular: With countably many choices on each of countably many intervals, there seem to be continuum many things you can get by gluing. – Andreas Blass Feb 2 2011 at 9:53

@Orbicular: But don't you end up "gluing a countable set to itself countably many times"?? That doesn't give a countable set: think about the cardinality of $\{0,1\}^{\mathbb N}$. – Matthew Daws Feb 2 2011 at 9:54

show 5 more comments

Ady · Accepted Answer · 2011-02-02 04:43:27Z UTC

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 !).

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 $$f_{A}\left(x\right):=\arctan\left(dist\left(\rho\left(x\right),A\right)\right)\cdot e$$ $\left(x\in O\right).$

separability of a certain space of continuous functions

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.