I am reading Functional Analysis, Sobolev Spaces, and Partial Differential Equations by Haim Brezis, and the theorem proving the existence of spectral decompositions of compact self-adjoint operators in Hilbert spaces is prefaced with the following.
Our last statement is a fundamental result. It asserts that every compact self-adjoint operator may be diagonalized in some suitable basis.
This is the theorem in question.
Theorem 6.11. Let $H$ be a separable Hilbert space and let $T$ be a compact selfadjoint operator. Then there exists a Hilbert basis composed of eigenvectors of $T$.
My question is a naïve (and compound) one: what is the motivation for considering this, and what does this get us? I saw that it allows us to prove that all Hilbert spaces have the approximation property, but I'm wondering what other results in functional analysis/PDEs we get from it.