# What set $Y$ that $C_{c}(U) \subset Y$ and $Y$ separable w.r.t. $\vert \vert \cdot \vert \vert_{\infty}$

Let $$U \in \mathbb R^{d}$$ open and further $$C_{c}(U)$$ be the space of functions with compact support. Show that $$C_{c}(U)$$ is separable w.r.t. $$\vert\vert \cdot \vert \vert_{\infty}$$, and I have been given the tip that:

$$X$$ is subset of separable metric space $$(M,d)$$, then $$X$$ is also separable. This is of course logical, but which set $$M$$ can I find that is separable. I've looked at the candidates $$C(U), C_{b}(U)$$ and $$L^{\infty}(U)$$ and none of them hold for separability w.r.t. the essential supremum.

My main goal is to show that $$C_{c}(U)$$ is separable w.r.t. $$\vert \vert \cdot \vert \vert_{p}$$ for all $$p \in [1,\infty[$$ and I have been advised to go about it the way above, but I am unsure how proving separability w.r.t. $$\vert \vert \cdot \vert \vert_{\infty}$$ can help me in anyway.

Any ideas, and hints are greatly appreciated.

• Not the "space of functions". You want the "space of continuous functions". – zhw. Jun 16 at 3:48

1. First, show that $$C(X)$$ with the norm $$\| \cdot \| _\infty$$ is separable whenever $$X$$ is compact. Almost full proofs are to be found here and here.

2. Next, use (1.) and a compact exhaustion of $$U$$ to show that $$C_c (U)$$ with the norm $$\| \cdot \| _\infty$$ is separable, as shown here.

3. Next, use the well-known result that $$C_c (U)$$ is dense in $$L^p (U)$$ for $$p \in (1, \infty)$$. Since $$C_c (U)$$ is separable, and $$L^p (U)$$ is first-countable (because is is a metric space), it follows that $$L^p (U)$$ is separable (the dense countable subset being the same one that is dense in $$C_c (U)$$).

4. Finally, $$C_c (U)$$ with the norm $$\| \cdot \| _p$$ may be seen as a subspace of $$L^p (U)$$. Since the latter has been seen to be separable (and a metric space), so will be the subspace $$(C_c (U), \| \cdot \|_p)$$.

The person who gave you the hint is probably thinking of $$M$$ being the space $$C_0(U)$$ consisting of continuous functions that vanish at $$\partial U$$ and at infinity (if $$U$$ is unbounded). This is a Banach space and is well known to be separable. However, proving this isn't really any easier than proving the separability of $$C_c(U)$$ directly.

Here's a sketch as to how you might prove this directly:

• Construct an "exhaustion" sequence of open sets $$U_n \subset U$$ such that $$\overline{U_n}$$ is compact and contained in $$U_{n+1}$$, and $$\bigcup_n U_n = U$$.

• Use Urysohn's lemma to construct a sequence of functions $$\phi_n$$ such that $$\phi_n = 1$$ on $$\overline{U_n}$$ and is compactly supported in $$U_{n+1}$$.

• Consider the set of all functions of the form $$\phi_n f$$, where $$f$$ is a polynomial in $$x_1, \dots, x_d$$ with rational coefficients. Show that this set is countable.

• Given $$g \in C_c(U)$$, find a sequence of functions from the above set which converge uniformly to $$g$$. (Use the Weierstrass approximation theorem.)