Normed and Banach spaces

A normed space $(X, \|\cdot\|)$ is a vector space with a norm; it is a metric space via $d(x, y) = \|x - y\|$. A Banach space is a complete normed space. Examples: $\mathbb{K}^n$ (any norm); $\ell^p = \{(x_n) : \sum|x_n|^p < \infty\}$; $\ell^\infty$; $C[a, b]$ with $\|\cdot\|_\infty$ (complete); $L^p$ (Riesz–Fischer). Non-example: $C[0, 1]$ with $\|\cdot\|_1$. Absolutely convergent series test: $X$ is Banach iff every series with $\sum\|x_n\| < \infty$ converges.

(i) All norms on a finite-dimensional space are equivalent, and finite-dimensional subspaces are closed and complete. (ii) Riesz's lemma: if $Y\subsetneq X$ is a closed subspace, then for every $\theta < 1$ there is a unit vector $x$ with $d(x, Y)\ge\theta$. (iii) Hence the closed unit ball of $X$ is compact iff $\dim X < \infty$.