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
Canonical name	NecessaryAndSufficientConditionsForANormedVectorSpaceToBeABanachSpace
Date of creation	2013-03-22 17:23:04
Last modified on	2013-03-22 17:23:04
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	6
Author	asteroid (17536)
Entry type	Theorem
Classification	msc 46B99