Continuous function space separablity

Question

We know that $C[0, \infty)$ is complete metric space with sup norm. Is it also seperable? How to show it?

Thank you.

I assume $C[0,\infty)$ here means the bounded continuous functions on $[0,\infty)$; otherwise the sup norm can be infinite. — Nate Eldredge, Jul 14 '13 at 02:14

score 1 · Answer 1 · answered Jul 14 '13 at 02:13

1

Hint: If $f_n$ were countable and dense, how can you make a function that stays away from $f_n(n)$?

answered Jul 14 '13 at 02:13

Zach L.

6,823

Nate Eldredge · Answer 2 · 2013-07-14T13:14:49.807

1

A common way to show a normed space is not separable is to present an uncountable set $E$ such that any two distinct elements of $E$ are at distance 1 from each other. This means that the uncountably many open balls $\{B(x,1/2) : x \in E\}$ are pairwise disjoint. But each one must contain an element of any dense set, so any dense set must be uncountable.

For the problem at hand, hint: there are uncountably many subsets of $\mathbb{N}$.

edited Jul 14 '13 at 13:14

answered Jul 14 '13 at 02:19

Nate Eldredge

101,664

@julien: Thanks, fixed. – Nate Eldredge Jul 14 '13 at 13:14

Continuous function space separablity

2 Answers2