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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:
These conditions essentially demand that the norm behave in the same way that we intuitively expect for it to be a notion of length:
Most of property 1 follows from the other axioms; it is enough to require that v be nonzero whenever v is nonzero.
A useful consequence of the norm axioms is the inequality

Euclidean norm. On R^{n}, the intuitive notion of length of the vector x = (x_{1}, x_{2}, ..., x_{n}) is captured by the formula
Taxicab norm.
Illustrations of unit circles in different norms. 
Infinity norm or maximum norm.
The concept of unit circle (the set of all vectors of norm 1) is different in different norms: for the 1norm the unit circle in R^{2} is a romboid[?], for the 2norm (Euclidian norm) it is the wellknown unit circle, while for the infinity norm it is a square. See the accompanying illustration.
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
Distances in Normed Vector Spaces
For any normed vector space we can define the distance between two vectors u and v as uv. (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
Finitedimensional normed vector spaces
All norms on a finitedimensional vector space V are equivalent. Since Euclidean space is complete, we can thus conclude that all finitedimensional normed vector spaces are Banach spaces.
A normed vector space V is finitedimensional 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 HahnBanach theorem.
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