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Let V be a vector space.

A function f:V→R (R being the set of real numbers) is called a norm when it satisfies the following axioms:

1. If α is a non-negative real number then f(αv)=αf(v) for any v in V.

1. For any vectors u,v in V, f(u+v)≤f(u)+f(v)

1. f(v)\ge;0 for any vector v in V

1. f(v)=0 if and only if v=0

The norm of a vector v is usually denoted by ||v||.

A norm is useful for measuring distance. The distance between u and v is defined to be ||u-v||

An open ball is a set of the form { u in V; ||u-v||<ε } for some fixed v in V and ε>0 in R. The set of balls defines a base for a topology. In other words, if v_k is a sequence of vectors then we say that v_k converges to v if and only if the real number sequence ||v_k-v|| converges to zero.

Categorically speaking, a homomorphism of normed vector spaces would be a linear map that preserves the norm. This isn't very useful, so a notion which may be more appropriate to topological vector spaces is often used: a homomorphism is a linear map that is continuous. When referring to a norm-preserving linear map, the term isometry is used. Note that an isometry is automatically an isomorphism (its inverse is an isometry as well.) When speaking of isomorphisms of normed spaces, one normally means an isometry, or at the very least a continuous, onto linear map with a continuous inverse.

When speaking of normed vector spaces, we augment the notion of dual (see dual space) to also include the norm. The dual V of a normed vector space V is the space of all continuous linear maps from V to the root field (the complexes or the reals) -- such linear maps are labeled "functionals." This continuity requirement destroys the self-duality property that ordinary vector spaces enjoy. Note that the norm of a functional F is defined by the sup of |F(x)| where x ranges over unit vectors in V.