## Banach space

### Set

 context $V$ … normed $F$-vector space definiendum $\mathcal V \in \mathrm{it}$ forall $v\in \mathrm{CauchySeq}(V)$ postulate $\exists v_\infty.\,\mathrm{lim}_{n\to\infty}\Vert v_n-v_\infty \Vert = 0$

#### Elaboration

For each Cauchy sequence $(v)_{i\in\mathbb N}$, there is a limit $v_\infty\in\mathcal V$ w.r.t. the natural norm. $\Longleftrightarrow$ The space $\mathcal V$ is complete.

#### Reference

Wikipedia: Banach space