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IEVref:103-01-08ID:
Language:enStatus: Standard
Term: inner product
Synonym1:
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Definition: 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).


Publication date:2009-12
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Internal notes:2017-02-20: Editorial revisions in accordance with the information provided in C00020 (IEV 103) - evaluation. JGO
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