Category:Absolutely Convergent Series in Normed Vector Space is Convergent iff Space is Banach

This category contains pages concerning Absolutely Convergent Series in Normed Vector Space is Convergent iff Space is Banach:

Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space.

Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be an absolutely convergent series in $X$.

Then $\ds \sum_{n \mathop = 1}^\infty a_n$ is convergent if and only if $X$ is a Banach space.