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

Theorem 1 - Let $(X,\parallel \cdot \parallel )$ 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 ${\sum }_n{x}_n$ converges in $X$.

Theorem 2 - Let $X,Y$ be normed vector spaces, $X\ne 0$. Let $B(X,Y)$ be the space of bounded operators $X⟶Y$. 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