normed vector space

Let 𝔽 be a field which is either or . A over 𝔽 is a pair (V,) where V is a vector space over 𝔽 and :V is a function such that

  1. 1.

    v0 for all vV and v=0 if and only if v=0 in V (positive definiteness)

  2. 2.

    λv=|λ|v for all vV and all λ𝔽

  3. 3.

    v+wv+w for all v,wV (the triangle inequalityMathworldMathworldPlanetmath)

The function is called a norm on V.

Some properties of norms:

  1. 1.

    If W is a subspaceMathworldPlanetmath of V then W can be made into a normed spaceMathworldPlanetmath by simply restricting the norm on V to W. This is called the induced norm on W.

  2. 2.

    Any normed vector space (V,) is a metric space under the metric d:V×V given by d(u,v)=u-v. This is called the metric induced by the norm .

  3. 3.

    It follows that any normed space is a locally convex topological vector space, in the topologyMathworldPlanetmath induced by the metric defined above.

  4. 4.

    In this metric, the norm defines a continuous mapMathworldPlanetmath from V to - this is an easy consequence of the triangle inequality.

  5. 5.

    If (V,,) is an inner product spaceMathworldPlanetmath, then there is a natural induced norm given by v=v,v for all vV.

  6. 6.

    The norm is a convex function of its argument.

Title normed vector space
Canonical name NormedVectorSpace
Date of creation 2013-03-22 12:13:45
Last modified on 2013-03-22 12:13:45
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 14
Author rspuzio (6075)
Entry type Definition
Classification msc 46B99
Synonym normed space
Synonym normed linear space
Related topic CauchySchwarzInequality
Related topic VectorNorm
Related topic PseudometricSpace
Related topic MetricSpace
Related topic UnitVector
Related topic ProofOfGramSchmidtOrthogonalizationProcedure
Related topic EveryNormedSpaceWithSchauderBasisIsSeparable
Related topic EveryNormedSpaceWithSchauderBasisIsSeparable2
Related topic FrobeniusProduct
Defines norm
Defines metric induced by a norm
Defines metric induced by the norm
Defines induced norm