If $D$ is the operator of differentiation, prove $D^{2}$ is a self adjoint linear operator on V and find all its eigenvalues and eigenvectors

Question

Suppose $V$ is the space of infinitely differentiable complex valued functions $f$ on $[0,\pi]$ such that $D^{2k+1}f(0) = 0 = D^{2k+1}f(\pi)$ for all integers $ k \geq 0$.
Then V is a complex IPS with

$<f,g> \ = \ \int_0^\pi \! \overline{f(s)} \ g(s) \, \mathrm{d}t$

If $D$ is the operator of differentiation, show that $D^{2}$ is a self adjoint linear operator on $V$ and find all its eigenvalues and eigenvectors.

I'm not sure where to even begin with this question. The hint suggests to split the integral into three parts since apparently if the eigenvalue $\lambda \in \mathbb{R}$ then we have three cases to consider, depending on the sign on $\lambda$. I'm not sure I understand this.

Answer 1

The selfadjointness can be proven by integration by parts.

The eigenfunctions are $exp( 2 n i x )$ for each integer $n$. These span all function by Fourier series theory or more prescily the form an orthonormal basis, although your space isn't Hilbertspace.