Let $V$ be a vector space of all $2\pi$-periodic function from $\Bbb R$ to $\Bbb C$ with the inner product:
$$\langle f,g \rangle =\int_0^{2\pi} \overline{f(x)}g(x)dx$$
$$A:V\to V, f\to \frac{df}{dx}$$
Now, my attempt
$$\langle Af,g \rangle =\int_0^{2\pi} \frac{\overline{f(x)}}{dx}g(x)dx=\overline{f(x)}g(x) \biggr |^{2\pi}_0-\int_0^{2\pi} \overline{f(x)}\frac{g(x)}{dx}dx$$
And here i'm stuck. This is supposed to equal to $\langle f,Ag\rangle$ to complete the answer right?
and second part:
Let $$U=\text{span}\{x \rightarrow e^{inx}=\sin(nx)+i \cos(nx) \mid n\in \Bbb Z\}$$
be the induced inner product. Find a orthonormal basis of U and the eigenvalues and eigenvectors of $A_{|U}$.
I don't quite understand what the second part wants me to do, appreciate any help!