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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,

1. ||v|| ≥ 0 with equality if and only if v = 0.
2. ||av|| = |a| ||v||.
3. ||u + v|| ≤ ||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

||u ± v|| ≥ | ||u|| - ||v|| |

for all vectors u and v.

Examples of Norms

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

${\displaystyle \|x\|={\sqrt {|x_{1}|^{2}+\cdots +|x_{n}|^{2}}}.}$

This gives the ordinary distance from the origin to the point x, a consequence of the Pythagorean theorem. The Euclidean norm is by far the most commonly used norm on R, but there are other norms on this vector space as will be shown below.

Taxicab norm.

${\displaystyle \|x\|_{1}=\sum _{i=1}^{n}|x_{i}|.}$

The name comes from the fact that the norm gives the distance a taxi has to drive in a rectangular street grid to get from the origin to the point x.

Illustrations of unit circles in different norms.

p-norm. Let p≥1 be a real number.

${\displaystyle \|x\|_{p}=\left(\sum _{i=1}^{n}|x_{i}|^{p}\right)^{\frac {1}{p}}}$

Note that for p=1 we get the taxicab norm and for p=2 we get the Euclidean norm. See also L space.

Infinitiy norm or maximum norm.

${\displaystyle \|x\|_{\infty }=\max \left(|x_{1}|,\cdots ,|x_{n}|\right).}$

The concept of unit circle (the set of all vectors of norm 1) is different in different norms: for the 1-norm the unit circle in R is a romboid, for the 2-norm (Euclidian norm) it is the well-known unit circle, while for the infinity norm it is a square. See the accompanying illustration.

Other norms on R can be constructed by combining the above; for example

${\displaystyle \|x\|=2|x_{1}|+{\sqrt {3|x_{2}|^{2}+\max(|x_{3}|,2|x_{4}|)^{2}}}}$

is a norm on R.

All the above formulas also yield norms on C without modification.

Examples of infinite dimensional normed vector spaces can be found in the Banach space article.

Every inner product space becomes a normed vector space if we define the norm as

${\displaystyle \|x\|={\sqrt {<x,x>}}.}$

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 V sits as a dense subspace inside a Banach space; this Banach space is essentially uniquely defined by V and is called the completion of V.

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

${\displaystyle C\|x\|_{1}\leq \|x\|_{2}\leq D\|x\|_{1}}$

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 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. An important theorem about continuous linear functionals on normed vector spaces is the Hahn-Banach theorem.