Let $D = d^2/dx^2$ and $V$ be the set of functions that are infinitely differentiable (and continuous), real, and $2\pi$-periodic. I've proven that $\langle Df, g \rangle = \langle f, Dg\rangle$, for any two functions $f,g \in V$.
I've also proven the result that $\sin(nx)$ and $\cos(nx)$ are eigenfunctions of $D$ with eigenvalue $\lambda_n = -n^2$.
How can we prove that $D : V \to V$ has an orthogonal eigenbasis using the pointwise convergence thoerem?
Pointwise Convergence Theorem: If $f$ and $f′$ are both piecewise continuous on the interval $[−\pi,\pi]$ then its Fourier series $F$ converges everywhere to the average of the lefthand and righthand limits of $f$.
Any guidance would be so appreciated.