I have a function $\phi:L^2(-\pi,\pi)\to l^2$, defined by $$(\phi(x))_k=\frac{1}{2\pi}\int_{-\pi}^{\pi}x(t)e^{-ikt}dt, k\in\mathbb{Z}$$ I have to argue that this function preserves the norm. I have to show that $$\Vert\phi(x)\Vert_{l^2}=\Vert x\Vert_{L^2}$$ My approach was this:
Let $$(\phi(x))_k=\langle x(t),e^{ikt}\rangle$$ where $\frac{1}{2\pi}e^{ikt}, k\in\mathbb{Z}$ is a orthonormal system. By the Riesz–Fischer theorem I can state that $$\sum_{k=1}^\infty\vert\langle x(t),e^{ikx}\rangle\vert^2=\Vert x(t)\Vert^2$$ so $$\Vert\phi(x)\Vert_{l^2}=\Vert x\Vert_{L^2}$$
This is my best attempt on this problem, I don't know if its correct though. I'm currently studying functional analysis and this is a problem in a book about this topic. Sorry if the way how this "proof" is done is sloppy!
Would really appreciate some comments and would be glad if someone could point out some mistakes I made. Big thanks in advance!