Definition:Normed Vector Space

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

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

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

Also known as
When $\Vert{\cdot}\Vert$ is arbitrary or not directly relevant, it is customary to call $V$ a normed vector space.

Also see

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