Banach space

A normed space is a Banach space if it is complete with respect to the metric induced by the norm.

ie, Cauchy sequences in the space converge in the space.

$\mathbb{Q}$ is not complete, but $\mathbb{R}$ is. ${\mathbb{R}}^n,{\mathbb{C}}^n$ are complete wrt any of the $||\cdot |{|}_p$ norms.