Normed Vector Space

linear-algebra

Definition

Normed Vector Space

A normed vector space $⟨V,||\cdot ||⟩$ is a vector space $V$ with a norm $||\cdot ||:V\to \mathbb{R}$.

Difference from a Vector Space

A vector space has addition and scalar multiplication, but it does not by itself define length or distance.

A normed vector space adds a norm, so one can measure the length of a vector by $\Vert x\Vert$ and the distance between two vectors by

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

Every normed vector space is a vector space, but not every vector space is normed.