Definition:Norm/Vector Space

Definition
Let $\struct {R, +, \circ}$ be a division ring with norm $\norm{\,\cdot\,}_R$.

Let $V$ be a vector space over $R$, with zero $0_V$.

A norm on $V$ is a map from $V$ to the nonnegative reals:
 * $\norm{\,\cdot\,}: V \to \R_{\ge 0}$

satisfying the (vector space) norm axioms:

Division Ring
When the vector space $V$ is the $R$-vector space $R$, the definition reduces to the division ring norm:

Also known as
The term length is occasionally seen as an alternative for norm.

Also see

 * Definition:Norm on Division Ring
 * Definition:Norm on Algebra
 * Definition:Norm on Bounded Linear Transformation
 * Definition:Norm on Bounded Linear Functional