normed vector space

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 Rⁿ. 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 positive number has the same effect on the length.
the triangle inequality, which amounts roughly to saying that the distance from A through B to C is never shorter than going directly from A to C. I.e. the shortest distance between any two points is a straight line.

Their generalization for more abstract vector spaces, leads to the notion of norm. A vector space on which a norm is defined is then called a normed vector space.

Definition

A normed vector space is a pair (V,||·||) where V is a vector space and ||·|| a norm on V. Note: One often writes just V instead of (V,||·||) when it's known or unessential what the norm is.

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 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

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.

Finite-dimensional normed vector spaces

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. 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.

Linear maps and dual 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.