Banach space

Definition

Let $V$ be a vector space over $\mathbb{R}$ or $\mathbb{C}$ with a norm $\|\cdot\|$, and let $d(x, y) = \|x - y\|$. The space $(V, \|\cdot\|)$ is a Banach space when the metric space $(V, d)$ is complete.

Remarks

Examples are $\mathbb{R}^n$ with any norm, the spaces ell p for $1 \le p \le \infty$, and $C_b(X)$ with the sup norm. The space of polynomial functions on $[0, 1]$ with the sup norm is not a Banach space. Its completion is $C([0, 1])$. In a normed space $V$, completeness is the same as: each absolutely convergent series converges. Each finite-dimensional normed space is complete.

Explanations

L1 human

A Banach space is a vector space where you can measure lengths, and where Cauchy sequences always converge. So you can take limits in it freely, as in $\mathbb{R}^n$.