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Area Mathematics - Functions / General concepts

IEV ref103-01-08

en
inner product
for two complex-valued functions, f _ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaamaamaaabaqcLbwaca WGMbaaaaaa@3718@ and g _ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaamaamaaabaqcLbwaca WGNbaaaaaa@3719@ , defined on the interval [ a,b] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaamaadmaabaGaamyyai aacYcacaWGIbaacaGLBbGaayzxaaaaaa@39BD@ of ℝ, complex number f _ , g _ = a b f _ (x) g _ (x)dx MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGeaGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaqFn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaaamaabaWaaWaaaeaacaWGMbaaaiaacYcadaadaaqaaiaadEgaaaaacaGLPmIaayPkJaGaeyypa0Zaa8qmaeaadaadaaqaaiaadAgaaaGaaiikaiaadIhacaGGPaWaaWaaaeaacaWGNbaaamaaCaaaleqabaGaey4fIOcaaaqaaiaadggaaeaacaWGIbaaniabgUIiYdGccaGGOaGaamiEaiaacMcaciGGKbGaamiEaaaa@4866@ , where g _ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGeaGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaqFn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaamaaabaGaam4zaaaadaahaaWcbeqaaiabgEHiQaaaaaa@3765@ is the conjugate of g _ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaamaamaaabaGaam4zaa aaaaa@364A@

Note 1 to entry: The inner product has the following properties: f _ , g _ = g _ , f _ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGeaGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaqFn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaaamaabaWaaWaaaeaacaWGMbaaaiaacYcadaadaaqaaiaadEgaaaaacaGLPmIaayPkJaGaeyypa0ZaaaWaaeaadaadaaqaaiaadEgaaaGaaiilamaamaaabaGaamOzaaaaaiaawMYicaGLQmcadaahaaWcbeqaaiabgEHiQaaaaaa@405D@ and α _ f _ + β _ g _ , h _ = α _ f _ , h _ + β _ g _ , h _ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaamaaamaabaWaaWaaae aacqaHXoqyaaGaaGPaVpaamaaabaGaamOzaaaacqGHRaWkdaadaaqa aiabek7aIbaacaaMc8+aaWaaaeaacaWGNbaaaiaacYcadaadaaqaai aadIgaaaaacaGLPmIaayPkJaGaeyypa0ZaaWaaaeaacqaHXoqyaaWa aaWaaeaadaadaaqaaiaadAgaaaGaaiilamaamaaabaGaamiAaaaaai aawMYicaGLQmcacqGHRaWkdaadaaqaaiabek7aIbaadaaadaqaamaa maaabaGaam4zaaaacaGGSaWaaWaaaeaacaWGObaaaaGaayzkJiaawQ Yiaaaa@5053@ where α _ , β _ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaamaamaaabaGaeqySde gaaiaacYcadaadaaqaaiabek7aIbaacqGHiiIZaaa@3AE2@ ℂ.

Note 2 to entry: The inner product for complex functions is similar to the Hermitian product for vectors (see IEC 60050-102, 102-03-18). For real functions, it is similar to the scalar product (see IEC 60050-102, 102-03-17).


fr
produit interne, m
pour deux fonctions à valeurs complexes, f _ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaamaamaaabaqcLbwaca WGMbaaaaaa@3718@ et g _ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaamaamaaabaqcLbwaca WGNbaaaaaa@3719@ définies sur l'intervalle [ a,b] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaamaadmaabaGaamyyai aacYcacaWGIbaacaGLBbGaayzxaaaaaa@39BD@ de ℝ, nombre complexe f _ , g _ = a b f(x) g _ (x)dx MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGeaGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaqFn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaaamaabaWaaWaaaeaacaWGMbaaaiaacYcadaadaaqaaiaadEgaaaaacaGLPmIaayPkJaGaeyypa0Zaa8qmaeaacaWGMbGaaiikaiaadIhacaGGPaWaaWaaaeaacaWGNbaaamaaCaaaleqabaGaey4fIOcaaaqaaiaadggaaeaacaWGIbaaniabgUIiYdGccaGGOaGaamiEaiaacMcaciGGKbGaamiEaaaa@4856@ g _ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGeaGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaqFn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaamaaabaGaam4zaaaadaahaaWcbeqaaiabgEHiQaaaaaa@3765@ est le conjugué de g _ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaamaamaaabaGaam4zaa aaaaa@364A@

Note 1 à l'article: Le produit interne a les propriétés suivantes: f _ , g _ = g _ , f _ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGeaGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaqFn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaaamaabaWaaWaaaeaacaWGMbaaaiaacYcadaadaaqaaiaadEgaaaaacaGLPmIaayPkJaGaeyypa0ZaaaWaaeaadaadaaqaaiaadEgaaaGaaiilamaamaaabaGaamOzaaaaaiaawMYicaGLQmcadaahaaWcbeqaaiabgEHiQaaaaaa@405D@ et α _ f _ + β _ g _ , h _ = α _ f _ , h _ + β _ g _ , h _ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaamaaamaabaWaaWaaae aacqaHXoqyaaGaaGPaVpaamaaabaGaamOzaaaacqGHRaWkdaadaaqa aiabek7aIbaacaaMc8+aaWaaaeaacaWGNbaaaiaacYcadaadaaqaai aadIgaaaaacaGLPmIaayPkJaGaeyypa0ZaaWaaaeaacqaHXoqyaaWa aaWaaeaadaadaaqaaiaadAgaaaGaaiilamaamaaabaGaamiAaaaaai aawMYicaGLQmcacqGHRaWkdaadaaqaaiabek7aIbaadaaadaqaamaa maaabaGaam4zaaaacaGGSaWaaWaaaeaacaWGObaaaaGaayzkJiaawQ Yiaaaa@5053@ α _ , β _ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaamaamaaabaGaeqySde gaaiaacYcadaadaaqaaiabek7aIbaacqGHiiIZaaa@3AE2@ ℂ.

Note 2 à l'article: Le produit interne de deux fonctions complexes est analogue au produit hermitien de deux vecteurs (voir IEC 60050-102, 102-03-18). Pour des fonctions réelles, il est analogue au produit scalaire (voir IEC 60050-102, 102-03-17).


ar
الضرب الداخلى

de
inneres Produkt, n

es
producto interno

it
prodotto interno

ja
内積

pl
iloczyn wewnętrzny

pt
produto interno

sr
унутрашњи производ, м јд

sv
inre produkt

zh
内积

Publication date: 2009-12
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