For $(A,+)$ an abelian group, then a norm on the group is a function
to the real numbers, such that
(positivity) $(g \neq 0) \Rightarrow (\vert g\vert \gt 0)$
(triangle inequality) ${\vert g + h\vert}\leq {\vert g\vert} + {\vert h\vert}$
(linearity) ${\vert k g\vert} = {\vert k\vert} {\vert g\vert}$ for all $k \in \mathbb{Z}$.
Here ${\vert k\vert} \in \mathbb{N}$ denotes the absolute value.
A group with a norm is a normed group, see there for more.
For $k$ a field equipped with a valuation (most usually, a local field such as $\mathbb{R}$, $\mathbb{C}$, or a p-adic completion of a number field), a norm on a $k$-vector space $V$ is a function
such that for all $\lambda \in k$, $v,w \in V$ we have
${\vert \lambda v \vert} = {\vert \lambda\vert} {\vert v \vert}$ (where $\vert \lambda \vert$ denotes the valuation)
${\vert v + w\vert } \leq {\vert v \vert } + {\vert w \vert}$ (“triangle inequality”)
if ${\vert v\vert} = 0$ then $v = 0$.
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.
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 ${\vert - \vert}_1$ and ${\vert - \vert}_2$ are called equivalent if there are $0 \lt C, C' \in \mathbb{R}$ such that for all $v$ 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 $\mathbb{R}^n$ the Euclidean norm is given by
Let $V$ be a vector space and $B \subseteq V$ an absorbing absolutely convex subset. The Minkowski functional of $B$ is the function $\mu_B \colon V \to \mathbb{R}$ defined by:
This is a semi-norm on $V$.