analysis - Is $C_0(mathbb R)$ separable?

Let $C_0$ be the Banach space of all continuous real value functions whose limits in $\pm \infty$ is zero, with the supremum norm. Is this space separable?

Answer

There are continuous bijections between $\Bbb R$ and $(-1, 1)$. One example is

$$h(x) = \frac{2}{\pi}\arctan(x)$$
This means there is an isometry between your space and the space of continuous, real valued functions $f$ on $(-1, 1)$ with $\lim_{x \to -1^+}f(x) = \lim_{x \to 1^-} f(x)= 0$ (still with sup norm). The set of polynomials with rational coefficients (and with $(x^2 - 1)$ as factor) are dense on this space.