If en with n E N is a set of orthonormal vectors in L[a, b] then for any function L [a, b] the following inequality holds (fn) ||f|| n=0 and equality (Parseval's formula) holds if and only if that is f = fnen, n=0 = lim ||ffnen|| = 0. N n=0 Here fn (f, en) are the Fourier coefficients of f. Hint. First, show that the following version of the Pythagoras theorem holds in infinite-dimensional Hilbert spaces: If (u, v) = 0 then ||u|2+ ||v|| = ||u+v||. ofnen is orthogonal to f - sy and apply the Pythagoras Then use that sy = theorem.