Bessels Inequality and Parsevals Equality

Let ( , be an inner product defined on a space X, and p an orthonormal sequence in X. Then we have the following Bessel s inequality (Yoshida 1980)  

If we recall the Fourier analysis, we understand that f = (/, are the Fourier coefficients. Note that Bessel s inequality (C.18) is reduced to an equality if we have 

It is only in this case that the Fourier series is complete i.e., fn4 n = f, 

Based on the above discussions, it is understood that we must prove the completeness of the following Fourier basis in a function space  

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