(Most of my training is in physical sciences, so forgive my question if its trivial)
I read in my course on partial differential equations that the set of eigenfunctions of a regular Sturm-Liouville problem is "complete" and that this implies that any piece-wise smooth function can therefore be expanded in terms of linear combinations of eigenfunctions. Can someone explain in terms a physicist might understand why the completeness of a set implies that functions can be built out of linear combinations of elements of the set? (I have low proficiency in linear algebra or number theory, but evidently substantial calculus)
The basic answer is that, if $\{ e_{\alpha} \}_{\alpha\in\Lambda}$ is a complete orthonormal subset of a Hilbert space $\mathcal{H}$, then $x-\sum_{\alpha\in\Lambda}\langle x,e_{\alpha}\rangle e_{\alpha}$ is orthogonal to every $e_{\alpha}$, which would force it to be $0$, and that would give $x=\sum_{\alpha\in\Lambda}\langle x,e_{\alpha}\rangle e_{\alpha}$.
By the way, completeness in a Hilbert space is also equivalent to the Parseval identity, $\|x\|^2=\sum_{\alpha\in\Lambda}|\langle x,e_{\alpha}\rangle|^2$, holding for all $x$. I'll let you figure how this is related to the condition of the first paragraph.