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 (, 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

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