Definition:Normed Vector Space

Definition
Let $\struct {K, +, \circ}$ be a normed division ring.

Let $V$ be a vector space over $K$.

Let $\norm {\,\cdot\,}$ be a norm on $V$.

Then $\struct {V, \norm {\,\cdot\,} }$ is a normed vector space.

Also known as
When $\norm {\,\cdot\,}$ is arbitrary or not directly relevant, it is usual to denote a normed vector space merely by the symbol $V$.

A normed vector space is also known as a normed linear space.

Some sources refer just to a normed space.

Also see

 * Definition:Banach Space
 * Definition:Norm on Vector Space
 * Definition:Normed Division Ring