nLab norm

Definition

On an abelian group

For $(G,+)$ an abelian group, then a norm on the group is a function

$${\vert-\vert} \;\colon\; G \longrightarrow \mathbb{R}$$

to the real numbers, such that

1. (positivity) $(g \neq 0) \Rightarrow (\vert g\vert \gt 0)$

2. (triangle inequality) ${\vert g + h\vert}\leq {\vert g\vert} + {\vert h\vert}$

3. (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.

In constructive mathematics, it is common to replace the denial inequality with a tight apartness relation in the positivity condition.

On a vector space

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

$${\vert-\vert} \colon V \to \mathbb{R}$$

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

1. ${\vert \lambda v \vert} = {\vert \lambda\vert} {\vert v \vert}$ (where $\vert \lambda \vert$ denotes the valuation)

2. ${\vert v + w\vert } \leq {\vert v \vert } + {\vert w \vert}$ (“triangle inequality”)

3. 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

• ${\vert v + w\vert } \leq max ({\vert v\vert}, {\vert w\vert})$

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 ${\vert - \vert}_1$ and ${\vert - \vert}_2$ are called equivalent if there are $0 \lt C, C' \in \mathbb{R}$ such that for all $v$ we have

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

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

General

• 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

  $$\vert \vec x\vert \;\coloneqq\; \sqrt{ \underoverset{i = 1}{n}{\sum} (x_i)^2 } \,.$$

1. more generally, for $n \in \mathbb{N}$, and $p \in \mathbb{N}$, $p \geq 1$, then the Cartesian space $\mathbb{R}^n$ carries the p-norm

   $${\vert \vec x \vert}_p \coloneqq \root p {\sum_i {|x_i|^p}}$$

2. The p-norm generalizes to sequence spaces and Lebesgue spaces.

• operator norm in operator algebra/functional analysis

Minkowski Functionals

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:

$$\mu_B(v) = \inf\{t \gt 0 : v \in t B\}$$

This is a semi-norm on $V$.

Properties

• Hahn-Banach theorem

The (open or closed) unit ball of a seminormed vector space is a convex set, a balanced set and an absorbing set. The first two of these properties make the unit ball (or even any ball of positive radius) an absolutely convex set.

Uniqueness

In dream mathematics, a given real vector space (with no topological structure) can have at most one complete norm, up to topological equivalence (homeomorphism of the identity function). It can have multiple inequivalent complete seminorms and incomplete norms, but their Hausdorff quotients and completions must be different. For example, the various Lebesgue norms on a Cartesian space $\mathbb{R}^n$ for finite $n$ are complete and equivalent; on $\mathbb{R}^\infty$, they are inequivalent but incomplete.

As dream mathematics includes excluded middle and dependent choice, the existence of inequivalent complete norms on a given vector space cannot be proved without a stronger form of the axiom of choice, enough to disprove the Baire property (which is the only classically false axiom needed in the proof of uniqueness). In HAF, it is argued that this explains why, in applied mathematics, there tends to be only one norm considered on any particular vector space (after Hausdorff completion).

This theorem applies more generally to F-norms but not to G-norms (even on a real vector space).

Norms valued in another Archimedean integral domain

In constructive mathematics, the notion of “real numbers” bifurcates: the Dedekind real numbers are different from the modulated Cauchy real numbers, which are different from the HoTT book real numbers, which are different from the localic real numbers, and so on. As a result, there are multiple sets of real numbers in which a metric could be valued in.

In predicative mathematics, the issue becomes even worse: there is no longer one set of Dedekind real numbers, but a whole hierarchy of Dedekind real numbers, one set for every universe in the foundations. As a result, one cannot resort to merely using the Dedekind real numbers for defining the norm as in impredicative mathematics, one has to define norms and normed spaces more generally.

Thus, given an Archimedean integral domain $R$, for $k$ a field equipped with a valuation, an $R$-norm on a $k$-vector space $V$ is a function

$${\vert-\vert} \colon V \to R$$

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

1. ${\vert \lambda v \vert} = {\vert \lambda\vert} {\vert v \vert}$ (where $\vert \lambda \vert$ denotes the valuation)

2. ${\vert v + w\vert } \leq {\vert v \vert } + {\vert w \vert}$ (“triangle inequality”)

3. if ${\vert v\vert} = 0$ then $v = 0$.

One could define $R$-seminorms, non-archimedean $R$-norms, and $R$-normed vector spaces in the same way as above.

Related concepts

• metric

• quotient norm

• p-norm

• F-norm, G-norm

• in representation theory: norm map

algebraic structure	group	ring	field	vector space	algebra
(submultiplicative) norm	normed group	normed ring	normed field	normed vector space	normed algebra
multiplicative norm (absolute value/valuation)			valued field
completeness	complete normed group	Banach ring	complete field	Banach vector space	Banach algebra

References

• Wikipedia, Normed vector space

• Siegfried Bosch, Ulrich Güntzer, Reinhold Remmert, Non-Archimedean Analysis – A systematic approach to rigid analytic geometry, 1984 (pdf)

• HAF, §27.47.b (for uniqueness in dream mathematics)