0

Let $X$ be a locally compact, Hausdorff, second-countable space, and let $C(X,\mathbb{C})$ the space of continuous, complex-valued functions over X endowed with the uniform norm.

Is $C(X,\mathbb{C})$ separable?

Ángel Valencia
  • 1,546
  • Since continuous functions on $X$ can be unbounded, the uniform "norm" isn't even a norm. Perhaps you wanted the space of bounded continuous functions? – Nate Eldredge Jun 08 '18 at 01:23

1 Answers1

1

Not in general. Consider $X = \mathbb{R}$, for each subset $A$ of the integers let $f_A : \mathbb{R} \to \mathbb{C}$ be the function which has a narrow triangular spike (say of width 1/2) of height 1 centered at each integer in $A$, and is zero otherwise. The distance between any two such functions in the sup norm is one, and there are uncountably many of them, so the space cannot be separable.

Montie
  • 460
  • More generally, instead of the integers you can use any infinite set without limit points (taking a little more care in the construction). This shows that the space of bounded continuous functions is separable if and only if $X$ is compact. – Nate Eldredge Jun 08 '18 at 01:32