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banach_space [2014/03/24 19:38] nikolaj |
banach_space [2015/02/03 10:21] nikolaj |
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| | @#55EE55: postulate | @#55EE55: $\exists v_\infty.\,\mathrm{lim}_{n\to\infty}\Vert v_n-v_\infty \Vert = 0$ | | | @#55EE55: postulate | @#55EE55: $\exists v_\infty.\,\mathrm{lim}_{n\to\infty}\Vert v_n-v_\infty \Vert = 0$ | | ||
| - | ==== Discussion ==== | + | ----- |
| + | === 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. | 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 === | === Reference === | ||
| Wikipedia: [[http://en.wikipedia.org/wiki/Banach_space|Banach space]] | Wikipedia: [[http://en.wikipedia.org/wiki/Banach_space|Banach space]] | ||
| - | ==== Parents ==== | + | |
| - | === Subset of === | + | ----- |
| - | [[Normed vector space]] | + | |
| === Requirements === | === Requirements === | ||
| [[Cauchy sequence]] | [[Cauchy sequence]] | ||
| + | === Subset of === | ||
| + | [[Normed vector space]] | ||
