On an abelian group
For an abelian group, then a norm on the group is a function
to the real numbers, such that
(linearity) for all .
Here denotes the absolute value.
A group with a norm is a normed group, see there for more.
On a vector space
For a field equipped with a valuation (most usually, a local field such as , , or a p-adic completion of a number field), a norm on a -vector space is a function
such that for all , we have
(where denotes the valuation)
if then .
If the third property is not required, one speaks of a seminorm.
If the triangle identity is strengthened to
one speaks of a non-archimedean seminorm, otherwise of an archimedean one.
A vector space equipped with a norm is a normed vector space. A vector of norm 1 is a unit vector.
Each seminorm determines a topology, which is Hausdorff precisely if it is a norm.
A topological vector space is called (semi-)normed if its topology can be induced by a (semi-)norm.
Two seminorms and are called equivalent if there are such that for all we have
Equivalent seminorms determine the same topology.
The collection of (bounded) multiplicative seminorms on a (Banach) ring is called its analytic spectrum (see there for details).
The standard absolute value is a norm on the real numbers.
More generally, on any Cartesian space the Euclidean norm is given by
Let be a vector space and an absorbing absolutely convex subset. The Minkowski functional of is the function defined by:
This is a semi-norm on .