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With 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive. This can be extended to any Euclidean space R. For more abstract vector spaces, a norm is a generalization of this idea. A vector space on which a norm is defined is then called a normed vector space.

If V is a vector space over a field K (which must be either the real numbers or the complex numbers), a norm on V is a function from V to R, the real numbers — that is, it associates to each vector v in V a real number, which is usually denoted ||v||. The norm must satisfy the following conditions:

For all a in K and all u and v in V,

${\displaystyle \|v\|\geq 0\ {\rm {with\ equality\ if\ and\ only\ if}}\ v=0.}$

${\displaystyle \|av\|=\left|a\right|\|v\|.}$

${\displaystyle \|u+v\|\leq \|u\|+\|v\|.}$

These conditions essentially demand that the norm behave in the same way that we intuitively expect for it to be a notion of length:

1. a vector always has a strictly positive length. The only exception is the zero vector which has length zero.
2. multiplying a vector by a number has the same effect on the length
3. the triangle inequality, which amounts roughly to saying that the distance from A to B to C is never shorter than going directly from A to C.

Most of property 1 follows from the other axioms; it is enough to require that ||v|| be non-zero whenever v is non-zero.

A useful consequence of the norm axioms is the inequality

${\displaystyle \|u\pm v\|\geq \left|\,\|u\|-\|v\|\,\right|}$

for all vectors u and v.

On R, the intuitive notion of length of the vector x = (x₁, x₂, ..., x_n) is captured by the formula ||x|| = ( |x₁| + ... + |x_n| ).

Other Norms

Illustrations of unit circles in different norms.

The most common idea of a norm is in the above sense. It is by far the most commonly used norm on ${\displaystyle \mathbf {R} ^{n}}$, but there are others:

• ${\displaystyle ||x||_{1}=\sum _{i=1}^{n}|x_{i}|}$
• ${\displaystyle ||x||_{p}=\left(\sum _{i=1}^{n}|x_{i}|^{p}\right)^{\frac {1}{p}}}$ for any real number ${\displaystyle p\geq 1}$
• ${\displaystyle ||x||_{\infty }=\max \left(|x_{1}|,\cdots |x_{n}|\right)}$ called the infinity norm.

All of these are norms on ${\displaystyle \mathbf {R} ^{n}}$. Note that the Euclidean norm is ${\displaystyle ||.||_{2}}$. The norm can be said to represent the "length" of a vector, in the intuitive way that Pythagorean theorem does for any vector in the euclidan ${\displaystyle (p=2)}$ norm, i.e. ${\displaystyle l=||x||_{2}={\sqrt {x_{1}^{2}+x_{2}^{2}}}}$.

Also the concept of unit circle is different in different norms: for a 1-norm the unit circle for a vector of two elements is romboid, for a 2-norm (euclidian) it is the well-known unit circle, while for the infinity norm it resembles a square. These figures all illustrate the values in the two-dimensional plane that have ${\displaystyle ||x||_{p}=1}$ in different norms. (See illustration.)

Distances in Normed Vector Spaces

For any normed vector space we can define the distance between two vectors u and v as ||u-v||. (Note that the Euclidean norm gives rise to the Euclidean distance in this fashion.) This turns the normed space into a metric space and allows to define notions such as continuity and convergence. The norm is then a continuous map. If this metric space is complete then the normed space is called a Banach space. Every normed vector space sits as a dense subspace inside a Banach space which is called its completion.

Two norms ||.||₁ and ||.||₂ on a vector space V are called equivalent if there exist positive real numbers C and D such that

C ||x||₁ ≤ ||x||₂ ≤ D ||x||₁

for all x in V. In this case, the two norms define the same notions of continuity and convergence and do not need to be distinguished for most purposes. Importantly, all norms on a finite-dimensional vector space V are equivalent. Since Euclidean space is complete, we can thus conclude that all finite-dimensional normed vector spaces are Banach spaces.

The most important maps between two normed vector spaces are the continuous linear maps. Together with these maps, normed vector spaces form a category. An isometry between two normed vector spaces is a linear map f which preserves the norm (meaning ||f(v)|| = ||v|| for all vectors v). Isometries are always continuous and injective. A surjective isometry between the normed vector spaces V and W is called a isometric isomorphism, and V and W are called isometrically isomorphic. Isometrically isomorphic normed vector spaces are identical for all practical purposes.

When speaking of normed vector spaces, we augment the notion of dual (see dual space) to take the norm into account. The dual V ' of a normed vector space V is the space of all continuous linear maps from V to the base field (the complexes or the reals) — such linear maps are called "functionals". The norm of a functional φ is defined as the supremum of |φ(v)| where v ranges over all unit vectors (i.e. vectors of norm 1) in V. This turns V ' into a normed vector space.

See also: inner product space