Banach Spaces

Recall that a normed space $(X,\Vert \cdot \Vert )$ is called complete if the metric space $(X,d)$, where $d(x,y)=\Vert x-y\Vert$, is complete, i.e. if every Cauchy sequence is convergent.

Definition (Banach Space)

A Banach space is a complete normed space.