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 banach_space [2015/02/03 10:18]nikolaj banach_space [2015/02/03 10:21] (current)nikolaj Both sides previous revision Previous revision 2015/02/03 10:21 nikolaj 2015/02/03 10:18 nikolaj 2015/02/03 10:17 nikolaj 2014/03/24 19:38 nikolaj 2014/03/24 19:37 nikolaj 2014/03/24 19:37 nikolaj 2014/03/24 19:36 nikolaj 2014/03/24 19:36 nikolaj 2014/03/24 19:35 nikolaj 2014/03/24 19:35 nikolaj 2014/03/24 19:34 nikolaj 2014/03/24 19:34 nikolaj 2014/03/24 19:34 nikolaj 2014/03/21 11:11 external edit2013/09/13 19:39 nikolaj 2013/09/13 19:36 nikolaj created 2015/02/03 10:21 nikolaj 2015/02/03 10:18 nikolaj 2015/02/03 10:17 nikolaj 2014/03/24 19:38 nikolaj 2014/03/24 19:37 nikolaj 2014/03/24 19:37 nikolaj 2014/03/24 19:36 nikolaj 2014/03/24 19:36 nikolaj 2014/03/24 19:35 nikolaj 2014/03/24 19:35 nikolaj 2014/03/24 19:34 nikolaj 2014/03/24 19:34 nikolaj 2014/03/24 19:34 nikolaj 2014/03/21 11:11 external edit2013/09/13 19:39 nikolaj 2013/09/13 19:36 nikolaj created Line 6: Line 6: | @#55EE55: postulate ​  | @#55EE55: $\exists v_\infty.\,​\mathrm{lim}_{n\to\infty}\Vert v_n-v_\infty \Vert = 0$ | | @#55EE55: postulate ​  | @#55EE55: $\exists v_\infty.\,​\mathrm{lim}_{n\to\infty}\Vert v_n-v_\infty \Vert = 0$ | + ----- === Elaboration === === 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]]