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.

Banach space

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Banach space

Set

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