Problem: Show $||f||_2^2 = \sum\limits_{n = -\infty}^\infty |\hat{f}(n)|^2$
I do not know how this holds in general, but this identity was proposed in a section of my book called "Trigonometric Polynomials" so I assume $f$ is a trig. polynomial.
Proof: \begin{align*} ||f||_2^2 &= \int_{[0,1]} f(x)\bar{f(x)} dx\\ &= \int_{[0,1]} (\sum\limits_{n = -N}^N c_ne_n)(\sum\limits_{n = -N}^N \bar{c_ne_n}) dx \\ &= \int_{[0,1]} \sum\limits_{n = -N}^N |c_n|^2 dx\\ &= \sum\limits_{n = -N}^N |c_n|^2 dx \\ &= \sum\limits_{n = -N}^N |\hat{f}(n)|^2 dx \\ &= \sum\limits_{n = -\infty}^\infty |\hat{f}(n)|^2 dx \end{align*}
I skipped a couple of steps here for brevity, but how does this look?