Normed vector space

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A normed vector space (or simply normed space) is a vector space V over a field K (which must be either the real numbers or the complex numbers) together with a function (called a norm) that associates to each x in V a real number denoted by ||x||. The norm must have the following properties, for all a in K and all x and y in V.

  1. ||x|| >= 0, with equality if and only if x = 0.
  2. ||ax|| = |a|.||x||.
  3. ||x+y|| <= ||x|| + ||y||.   (The triangle inequality.)

A familiar example is the space Rn (where R denotes the real numbers and n is any natural number) with ||x|| being the Euclidean distance of x from the origin.

For any normed space we can define the distance between two vectors as ||x-y||. This makes the normed space into a metric space. If this metric space is complete then the normed space is called a Banach space.

Categorically speaking, a morphism 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 morphism 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, bijective 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.