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单词 InnerProductSpace
释义

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 productMathworldPlanetmathforms 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-yinduced 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.
Titleinner product space
Canonical nameInnerProductSpace
Date of creation2013-03-22 12:14:05
Last modified on2013-03-22 12:14:05
OwnerCWoo (3771)
Last modified byCWoo (3771)
Numerical id23
AuthorCWoo (3771)
Entry typeDefinition
Classificationmsc 46C99
Synonympre-Hilbert space
Related topicInnerProduct
Related topicOrthonormalBasis
Related topicHilbertSpace
Related topicEuclideanVectorSpace2
Related topicAngleBetweenTwoLines
Related topicFluxOfVectorField
Related topicCauchySchwarzInequality
Definesangle between two vectors
DefinesperpendicularityPlanetmathPlanetmath
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更新时间:2025/5/5 13:44:38