nLab
norm

Definition

For $k$ a field equipped with a valuation (most usually, a local field such as $ℝ$, $ℂ$, or a p-adic completion of a number field), a norm on a $k$-vector space $V$ is a function

$$\mid -\mid :V\to ℝ$${\vert-\vert} : V \to \mathbb{R}

such that for all $\lambda \in k$, $v,w\in V$ we have

1. $\mid \lambda v\mid =\mid \lambda \mid \mid v\mid$ (where $\mid \lambda \mid$ denotes the valuation)

2. $\mid v+w\mid \le \mid v\mid +\mid w\mid$ (“triangle inequality?”)

3. if $\mid v\mid =0$ then $v=0$.

If the third property is not required, one speaks of a seminorm.

If the triangle identity is strengthened to

• $\mid v+w\mid \le \max (\mid v\mid ,\mid w\mid )$

one speaks of a non-archimedean seminorm.

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 ${\mid -\mid }_1$ and ${\mid -\mid }_1$ are called equivalent if there is $0<C,C\prime \in ℝ$ such that for all $v$ we have

$$C{\mid v\mid }_1\le {\mid v\mid }_2\le C\prime {\mid v\mid }_1.$$C {\vert v \vert}_1 \leq {\vert v \vert}_2 \leq C' {\vert v \vert}_1 \,.

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).

Examples

• The standard absolute value is a norm on the real numbers.

• More generally, on any Cartesian space ${ℝ}^n$ the Euclidean norm is given by

  $$(x^1,\cdots ,x^n)\mapsto \sqrt{(}\sum _{i=1}^n(x^i{)}^2).$$(x^1, \cdots, x^n) \mapsto \sqrt(\sum_{i=1}^n (x^i)^2) \,.

Related concepts

• Banach space

• metric