necessary and sufficient conditions for a normed vector space to be a Banach space

Theorem 1 - Let $(X,\\|\\cdot\\|)$ be a normed vector space. $X$ is a Banach space if and only if every absolutelyconvergent series in $X$ is convergent, i.e., whenever $\\sum_{n}\\|x_{n}\\|<\\infty,$ $\\sum_{n}x_{n}$ converges in $X$ .

Theorem 2 - Let $X, Y$ be normed vector spaces, $X\eq 0$ . Let $B(X,Y)$ be the space of bounded operators $X\\longrightarrow Y$ . Then $Y$ is a Banach space if and only if $B(X,Y)$ is a Banach space.

单词	NecessaryAndSufficientConditionsForANormedVectorSpaceToBeABanachSpace
释义	necessary and sufficient conditions for a normed vector space to be a Banach space Theorem 1 - Let $(X,\\\|\\cdot\\\|)$ be a normed vector space. $X$ is a Banach space if and only if every absolutelyconvergent series in $X$ is convergent, i.e., whenever $\\sum_{n}\\\|x_{n}\\\|<\\infty,$ $\\sum_{n}x_{n}$ converges in $X$ . Theorem 2 - Let $X, Y$ be normed vector spaces, $X\eq 0$ . Let $B(X,Y)$ be the space of bounded operators $X\\longrightarrow Y$ . Then $Y$ is a Banach space if and only if $B(X,Y)$ is a Banach space.
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