Definition:Normed Vector Space

Definition
Let $\left({K, +, \circ}\right)$ be a normed division ring.

Let $V$ be a vector space over $K$. Let $\left\Vert{\,\cdot\,}\right\Vert$ be a norm on $V$.

Then $\left({V, \left\Vert{\,\cdot\,}\right\Vert}\right)$ is a normed vector space.

Also known as
When $\left\Vert{\,\cdot\,}\right\Vert$ 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.

Also see

 * Definition:Norm (Vector Space)
 * Definition:Normed Division Ring