# Why does this Fourier inner product equal this sum?

This is part of a derivation in a text that I am struggling to follow. It says that if we write $e_k(t) = e^{2 \pi i k t}$ then

$$\langle \sum_{n=- \infty}^{\infty} \langle f, e_n \rangle e_n,\sum_{m=- \infty}^{\infty} \langle f, e_m \rangle e_m \rangle = \sum_{n,m} \langle f, e_n\rangle \overline{\langle f, e_m\rangle} \langle e_n, e_m\rangle.$$

I don't see how this follows. Here $\langle., .\rangle$ is the L2 inner product, so $\langle f, g\rangle = \int_0^1 f(t) \overline{g(t)} dt$.

• You mean $e_n(t) = e^{2\pi i n t}$, right? – Paul K Jan 8 '16 at 17:06
• Yes, I fixed it. – Fequish Jan 8 '16 at 17:10

$$\langle \langle a,b\rangle c,\langle d,e\rangle f\rangle = \langle a,b\rangle\overline{\langle d,e\rangle} \langle c,f\rangle$$