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:<math> \|g\| = \left(\int_a^b g^*(x)g(x) \, dx \right)^{1/2} </math>
where the ‘*’ denotes [[complex conjugate]] in the case of complex functions. The extension of Pythagoras' theorem in this manner leads to [[function space]]s and the notion of [[Lebesgue measure]], an idea of “space” more general than the original basis of Euclidean geometry. The {{nowrap|{ <math>\phi_j (x)\ </math> } }} satisfy [[Orthogonal#Orthogonal_functions|orthonormality relations]]:<ref name=Folland2>
{{cite book |title=Fourier Analysis and Its Applications|page =69 |first1=Gerald B | last1= Folland|url=http://books.google.com/books?id=ix2iCQ-o9x4C&pg=PA69 |isbn=0-8218-4790-2 |year=2009 |publisher=American Mathematical Society }}
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