Separable Banach space

Summary

A separable Banach space is a Banach space that has a countable dense subset. Most spaces met in analysis are of this kind, for example $\ell^p$ and $L^p[0, 1]$ for $1 \le p < \infty$, $c_0$ and $C[0, 1]$. Separability allows arguments by sequences and diagonal extraction. For instance, the closed unit ball of the dual of a separable Banach space is metrizable in the weak* topology. The Banach–Mazur theorem says that every separable Banach space is isometric to a closed subspace of $C[0, 1]$.