inner product space


An inner product spaceMathworldPlanetmath (or pre-Hilbert space) is a vector spaceMathworldPlanetmath (over or ) with an inner productMathworldPlanetmath ,.

For example, n with the familiar dot productMathworldPlanetmath forms an inner product space.

Every inner product space is also a normed vector spacePlanetmathPlanetmath, with the norm defined by x:=x,x. This norm satisfies the parallelogram lawMathworldPlanetmathPlanetmath.

If the metric x-y induced by the norm is complete (http://planetmath.org/Complete), then the inner product space is called a Hilbert spaceMathworldPlanetmath.

The Cauchy–Schwarz inequality

|x,y|xy (1)

holds in any inner product space.

According to (1), one can define the angle between two non-zero vectors x and y:

cos(x,y):=x,yxy. (2)

This provides that the scalars are the real numbers. In any case, the perpendiculatity of the vectors may be defined with the condition

x,y=0.
Title inner product space
Canonical name InnerProductSpace
Date of creation 2013-03-22 12:14:05
Last modified on 2013-03-22 12:14:05
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 23
Author CWoo (3771)
Entry type Definition
Classification msc 46C99
Synonym pre-Hilbert space
Related topic InnerProduct
Related topic OrthonormalBasis
Related topic HilbertSpace
Related topic EuclideanVectorSpace2
Related topic AngleBetweenTwoLines
Related topic FluxOfVectorField
Related topic CauchySchwarzInequality
Defines angle between two vectors
Defines perpendicularityPlanetmathPlanetmath