Banach Space

linear-algebra

Definition

Banach Space

A Banach space is a normed vector space that is complete with respect to the metric induced by its norm.

Equivalently, a Banach space is a normed vector space in which every Cauchy sequence converges in the space.

In other words, if $⟨V,\Vert \cdot \Vert ⟩$ is a normed vector space, then it is a Banach space when the metric

$$d(x,y)=\Vert x-y\Vert$$

makes $V$ into a complete metric space.