Fourier series and Hilbert spaces
The idea behind Fourier series is to try and express some function on a domain \([-L,L]\) into a sum of complex exponentials of the form \(\frac{1}{\sqrt{2L}}e^{2\pi i \ nx/L}\). One of the reasons this is interesting is that the complex exponentials are orthonormal system under the dot product \(\int f(x)\overline{g(x)}\ dx\).
One can start by considering the vector space \(V\) of all square-integrable functions on \([-L,L]\) – this gives us a vector space with an inner product. Specifically, we’re interested in the subspace \(V_n\) that is the span of complex exponentials upto \(n\) and \(-n\).Then given a vector \(f\) in \(V\), we can ask for its projection \(f_n\) onto \(V_n\).
As the complex exponentials are already orthonormal, it is easy to calculate this projection in their basis:
$$\begin{gathered}
{ak} = \left⟨ {f,\frac{1}{\sqrt{2L}}{e2π i\;nx/L}} \right⟩ = ∫\limits - LL {f(x)\frac{e - 2π i\;kx/L}{\sqrt{2L}}dx} \hfill
{fn}(x) = ∑\limits|k|≤ n {{ak}\frac{e2π i\;kx/L}{\sqrt{2L}}} \hfill
\end{gathered} $$
Notably this implies by Cauchy-Schwarz that:
\[{\left| f \right|^2} \geqslant \sum\limits_{|k| \leqslant n} {{{\left| {{a_k}} \right|}^2}} \]
This really just is Cauchy-Schwarz, and is known as Bessel’s inequality. If we can show that the Fourier series approaches \(f\), i.e. that \(\left\|f-f_n\right\|\to 0\), then it would be obvious that
\[{\left| f \right|^2} = \sum\limits_{|k| \in \mathbb{Z}} {{{\left| {{a_k}} \right|}^2}} \]
Which is just the Pythagoras theorem, and is known as Parseval’s theorem. Obviously, these theorems exist in the general theory of Hilbert spaces.