I have to prove a larger fact about Fredholm Operators. One of the facts along the way which I don't need to prove is the question posed in the title. But I would like to know none-the-less
I want to show that the set $\{ e_n(x) \}_{n \in \mathbb{Z}}$, where $e_n (x) = e^{inx}$, forms an orthonormal basis for $L_2 (S_1)$. Here $S_1$ is the unit circle, and $L_2 (S_1)$ is the Hilbert Space of complex-valued square-absolutely-integrable functions on $S_1$ with inner product $$ (f,g) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) \overline{g(x)} dx $$ It is useful to note that the fourier series $$\sum_{n=-\infty}^{\infty} \hat{f}(n) e^{inx} \mbox{ converges to }f \quad (**)$$ for $f \in L_2 (S_1)$, where the fourier coefficient at frequency $n$ is given by $\hat{f}=(f,e_n) $
To show orthonormality, I think we need to prove that:
- The $e_n$ have norm $1$
- The $e_n$ are orthogonal
- The $e_n$ span $L_2 (S_1)$
The first 2 conditions I think will be easy once I try it out. But the 3rd condition bothers me. I think it follows directly from $(**)$, but my question is why is $(**)$ true? I thought only periodic functions have fourier series? Please help with an answer or even jus a reference