In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can be easily extended to any real vector space **R**^{n}. It turns out that the following properties of "vector length" are the crucial ones.

- a vector always has a strictly positive length. The only exception is the zero vector which has length zero.
- multiplying a vector by a number has the same effect on the length.
- 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.

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. ||
*a***v**|| = |*a*| ||**v**||. - 3. ||
**u**+**v**|| ≤ ||**u**|| + ||**v**||.

- 1. ||

- 1'. ||
**v**|| is non-zero if and only if**v**is non-zero.

- 1'. ||

- ||
**u**±**v**|| ≥ | ||**u**|| - ||**v**|| |

Illustrations of unit circles in different norms. |

Other norms on **R**^{n} can be constructed by combining the above; for example

All the above formulas also yield norms on **C**^{n} without modification.

Examples of infinite dimensional normed vector spaces can be found in the Banach space article. In addition, inner product space becomes a normed vector space if we define the norm as

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 the definition of 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

A normed vector space *V* is finite-dimensional if and only if the unit ball *B* = {*x* : ||*x*|| ≤ 1} is compact, which is the case if and only if *V* is locally compact.

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.