So I want to investigate the kind of convergence the basis function obtained from the solutions to the Schrödinger equation (SE) offer. But before I look at this, I'd like to learn why we say that the solutions of the SE form a basis for the set of square integrable functions $L^2$. Therefore:
Suppose $\psi_n(x)$ solve the time independent SE for all $x \in A \subset \mathbb{R}$. Why is it true that there exist a sequence $\{c_k\}_{k \in \mathbb{Z}}$ such that for any function $f \in L^2(A)$
$$ f(x) = \sum_{k \in \mathbb{Z}} c_k \psi_k(x) $$ for all $x \in A$.
