I am reading about normal mode analysis - the idea that perturbations from a steady state can be expanded as a Fourier Series if the domain is bounded, or as a Fourier transform integral in the case the domain is unbounded.
What is the name of the theorem which states that if your domain $\mathcal{D}$ is bounded, then the superposition of normal modes spans the space of functions the perturbation lies in?
In other words, which theorem states boundedness of domain $\Rightarrow$ completeness?