Separability of $B(X)$.

Question

Assume $X$ is a separable space, or even a countable one. Is then the space of bounded complex-valued functions on $X$ with the supremum norm, $\|f\|_{\infty}=\sup_{x\in X} |f(x)|$, separable as well?

Edit: Assume that $X$ is a countable set. Is the answer still no? All the counterexamples seem to consider $X$ as uncountable.

Related https://math.stackexchange.com/questions/104244/if-c-bx-separable-then-x-is-a-compact-metrizable-space and https://math.stackexchange.com/questions/954920/is-the-space-ba-b-separable. — Akira, Jul 19 '23 at 18:08

score 2 · Answer 1 · answered Jul 19 '23 at 18:11

2

The Banach space $L^{\infty }(\mathbb{R})$ is not separable, even though $\mathbb{R}$ is.

answered Jul 19 '23 at 18:11

G. Bellaard

757

Martin Argerami · Accepted Answer · 2023-07-20T08:44:22.307

2

The space $\ell^\infty(\mathbb N)$ is not separable. An easy way to see this is that it contains all sequences of zeros and ones (which can be identified with the power set of $\mathbb N$), and the distance between any two of these sequences is $1$.

edited Jul 20 '23 at 08:44

answered Jul 19 '23 at 20:01

Martin Argerami

205,756

... and these sequences have distance $1$ from each other, otherwise this would not rule out separability. – MaoWao Jul 20 '23 at 08:34
Indeed, I have edited that in. Thanks! – Martin Argerami Jul 20 '23 at 08:44

Separability of $B(X)$.

2 Answers2