Separability of function spaces

Question

I'm trying to understand separability of function spaces. Among $C^0[0,1]$, $L^\infty[0,1]$ and $C^0[\mathbb{R}]$, which spaces are separable?

Answer 1

1. $(\mathcal{C}^0([0,1]),\left\lVert \cdot\right\rVert_{\infty})$ is separable: this holds thanks to Stone-Weierstrass theorem; you can find it here https://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem
2. $(L^{\infty}([0,1]),\left\lVert \cdot\right\rVert_{L^p})$ is not separable: you can find it here (also for the case $l^{\infty }$) Why is $L^{\infty}$ not separable?
3. $(\mathcal{C}^0(\mathbb{R}),\left\lVert \cdot\right\rVert_{\infty})$ is not seperable: you can find it here The space of bounded continuous functions are not separable

Hope it helps.