===== Banach space =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $V$ ... normed $F$-vector space |
| @#FFBB00: definiendum | @#FFBB00: $\mathcal V \in \mathrm{it}$ |
| @#FFFDDD: forall | @#FFFDDD: $v\in \mathrm{CauchySeq}(V)$ |
| @#55EE55: postulate   | @#55EE55: $\exists v_\infty.\,\mathrm{lim}_{n\to\infty}\Vert v_n-v_\infty \Vert = 0$ |

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=== 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: [[http://en.wikipedia.org/wiki/Banach_space|Banach space]]

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=== Requirements ===
[[Cauchy sequence]]
=== Subset of ===
[[Normed vector space]]