# 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 absolutely convergent 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, $X\neq 0$. 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.

Title necessary and sufficient conditions for a normed vector space to be a Banach space NecessaryAndSufficientConditionsForANormedVectorSpaceToBeABanachSpace 2013-03-22 17:23:04 2013-03-22 17:23:04 asteroid (17536) asteroid (17536) 6 asteroid (17536) Theorem msc 46B99