Find a basis of $(L^2((-\pi,\pi), \mathbb{R}))^2$

Question

I need to find a basis of $(L^2((-\pi,\pi), \mathbb{R}))^2$.

I believe a basis of $L^2((-\pi,\pi), \mathbb{R})$ can be produced by the eigenfunctions of $\triangle$ (see L.C. Evans: Partial Differential Equations, theorem 1 in section 6.5) and that one can use that as a starting point. But how proceed from there?

Or can one find a basis as eigenfunctions of $(\triangle, \triangle)^T$? What can we say about the corresponding eigenvalues?

Answer 1

If $\{e_n\}$ is an orthonormal basis for a Hilbert space $\mathcal{H}$ and $\{f_n\}$ is an ONB for Hilbert space $\mathcal{K}$, then the union $\{(e_n,0)\}\cup \{(0,f_n)\}$ is an ONB for the direct sum $\mathcal{H}\oplus \mathcal{K}$.