Verifying the Fourier Series is in Hilbert Space

[kostoglotov]

The text does it thusly:

imgur link: http://i.imgur.com/Xj2z1Cr.jpg

But, before I got to here, I attempted it in a different way and want to know if it is still valid.

Check that $f^{*}f$ is finite, by checking that it converges.

$$f^{*}f = a_0^2 + a_1^2 cos^2x + b_1^2sin^2x + a_2^2cos^22x + b_2^2sin^22x + ...$$

should converge provided the a's and b's are both less than 1 in absolute value.

Is this also a valid way to show that our Fourier Series is an okay vector to work with?

 

[BvU]

show that our Fourier Series is an okay vector to work with

What is it you want to show ? Orthogonality is OK, and normality is OK (provided you use the correct normalization constants). Completeness ? For all functions you can write the recipe to get the Fourier transform.

So yes, the Fourier basis is OK.

 

[Svein]

What you have here is a formulation of Parseval's theorem (https://en.wikipedia.org/wiki/Parseval's_theorem).