# Non-separable Banach space

The vector space $C_b(\mathbb R)$ of bounded continuous functions on $\mathbb R$ is non-separable: it is possible to produce a direct proof of this fact, mimicking the standard proof for the non-separability of $L^\infty(\mathbb R)$ (reductio ad absurdum: given a sequence $(\phi_n)_{n\in \mathbb N}$ in $L^\infty(\mathbb R)$, construct a function in $L^\infty(\mathbb R)$ at distance 1 of that sequence).

Is it possible to avoid that type of proof and get the non-separability of $C_b(\mathbb R)$ by a more abstract (structural) fact?

• For instance the discrete uncountable metric space $P(\mathbb{N})$ endowed with the Hausdorff distance, embeds isometrically into $C_b(\mathbb{R})$ via $S\mapsto v_S$ s.t. $v_S(x):=\mathrm{dist}_S(x)$. – Pietro Majer Feb 10 '15 at 12:33
• What would be an example of a non-direct proof? – Ian Morris Feb 10 '15 at 15:06
• This reminds me a little of this question. One sophisticated way to prove this might be that $C_b(\mathbb{R})$ is isometrically isomorphic to $C(\beta \mathbb{R})$ where $\beta \mathbb{R}$ is the Stone-Cech compactification. If $C(\beta \mathbb{R})$ were separable then $\beta \mathbb{R}$ would be second countable (and in fact metrizable), but that isn't true. – Nate Eldredge Feb 10 '15 at 19:25
• For what it's worth (if anything), I wouldn't prove $L^\infty(\mathbb{R})$ to be non-separable in the way you suggest: I'd instead do it by observing that the subset $\{\chi_{(-\infty,t)}\colon t \in \mathbb{R}\}$ is isometric to $\mathbb{R}$ with the discrete metric. – Ian Morris Feb 11 '15 at 9:40

Let $\varphi$ be a continuous function supported on $[0,1]$. Then continuum many combinations $\sum c_k \varphi(x+k)$, $c_k\in \{0,1\}$ are separated in our space.
Consider the functions $f_c(x):=sin(c \pi x)$ for $c\in\mathbb{R}$. If $\frac{c_1}{c_2}\notin\mathbb{Q}$, then $f_{c_1}$ and $f_{c_2}$ have distance 2. This is basically just a restatement of the fact $\{nc + \mathbb{Z} | n\in\mathbb{Z}\}$ is dense in $\mathbb{R}/\mathbb{Z}$ if $c$ is irrational. Since we can find a continuum of real numbers that are pairwise $\mathbb{Q}$-linear independent (even without the full force of the AC), this proves the existence of a continuum-sized discrete subspace of $C_b(\mathbb{R})$.
There is a natural continuous linear surjection $C_b(\mathbb{R}) \to \ell_\infty(\mathbb{Z})$ defined by restricting the function $f \in C_b(\mathbb{R})$ to the subdomain $\mathbb{Z}\subseteq \mathbb{R}$. The space $\ell_\infty(\mathbb{Z})$ is well-known to be non-separable, and a non-separable topological space cannot be the continuous image of a separable one.
To look at the same argument in a different way, define $X$ to be the closed linear subspace of $C_b(\mathbb{R})$ consisting of functions which are affine on each of the intervals $[n,n+1]$. Then $X$ is isometrically isomorphic to $\ell_\infty(\mathbb{Z})$ via the restriction $f \mapsto f|_{\mathbb{Z}}$.