Ref. Requst: Space of bounded Lipschitz functions is separable if the domain is separable.

I have been scouring the internet for answers for some time and would therefore appreciate a reference or a proof since i'm not able to produce one myself.

Let $(\mathcal{X},d)$ be a metric space, and let $$BL(\mathcal{X})=\{f:\mathcal{X}\to \mathbb{R}\, \, | \, \, f \text{ is Lipschitz and bounded}\}$$ denote the bounded real-valued Lipschitz countinuous functions on $\mathcal{X}$.

Does anyone know a reference for the statement:

If $(\mathcal{X},d)$ is separable then $(BL(\mathcal{X}),||\cdot||_\infty)$ is separable?

You could not find a proof because it is not true. In fact $BL(X)$ is separable in the topology of uniform convergence if and only if $X$ is totally bounded.

If $X$ is totally bounded, every Lipschitz function can be extended to its completion $\tilde{X}$, which is compact. Thus we obtain an isometric embedding $BL(X) \to C(\tilde{X})$.

If $X$ is not totally bounded, there is an $\epsilon > 0$ and an infinite $A \subset X$, such that $d(x, y) > \epsilon$ for all distinct $x, y \in A$. For every $V \subset A$ the function $f_V: x \mapsto \min \{ \epsilon, d(x, V) \}$ is a bounded Lipschitz function. Clearly $\|f_V - f_W\|_\infty = \epsilon$ for distinct $V$ and $W$, hence the set of all these functions is discrete. This shows that $w(BL(X)) \ge 2^{|A|} > \aleph_0$.

Addendum: The above also holds for spaces of uniformly continuous functions, which is the context where it is usually discussed. The fact that a uniformly continuous function on a metric space has a unique extension to the completion of the space is a special case of the result discussed in this answer. This gives us a well-defined mapping $BL(X) \to C(\tilde{X})$ which takes $f: X \to \mathbb{R}$ to $\tilde{f}: \tilde{X} \to \mathbb{R}$.

That this mapping is an isometry with respect to the sup norm follows easily from the fact that $X$ is dense in $\tilde{X}$ and therefore $f[X]$ is dense in $\tilde{f}[\tilde{X}]$ by continuity of $\tilde{f}$.

• Thanks for the answer @Niels Diepeveen. But I'm having trouble formalizing the procedure, my knowledge in topology is very sparse. The problem I have lies with constructing the isometric embedding. Can you show or refer me to a book where it is done formally? – Martin Jun 28 '16 at 12:54
• I have added some details that might help. I could not find a better reference for this specific fact than the answer I linked to. To find the relevant results in a topology textbook, you would probably have to look in the chapter on uniform spaces, which is I guess is more than you bargained for. – Niels J. Diepeveen Jun 29 '16 at 12:04
• Thanks for the help and the added information, it is very much appreciated. – Martin Jun 30 '16 at 14:32