Is the space of continuous functions on $\mathbb{R}$ separable/compact with respect to the topology of uniform convergence in compact sets?

Question

Consider the space of continuous functions $C(\mathbb{R}, \mathbb{R})$, equipped with the metric $$ d(f,g) := \sum_{n=1}^\infty \frac{1}{2^n} \frac{\sup_{x \in [-n,n]} |f(x)-g(x)|}{1 + \sup_{x \in [-n,n]} |f(x)-g(x)|} . $$

The topology induced by this metric should be the topology of uniform convergence in compact sets.

This space is complete.

1. Is it separable?
2. Is it compact?

EDIT: It's not a Banach space, since the metric $d$ does not induce a norm.

Answer 1

It is not a Banach space, as $d$ is no norm. It is a locally convex completely metrisable TVS, also called a Fréchet space in this context.

It is certainly not compact as the constant functions from a non-compact closed copy of $\Bbb R$.

It is separable: can use (rational) polynomials to approximate continuous functions uniformly on some $[-N,N]$ for $N$ large enough..