Conditions for a linearly independent sequence with dense linear span to be a Schauder basis for a Banach space
Solution 1:
A well known necessary and sufficient condition for a countable linearly independent system of eigenfunctions, like $(e_k)$, to be a Shauder basis for its closed linear span is $$ \exists K>0\quad \forall (a_n)\subset \mathbb{C}\quad \forall n\in\mathbb{N}\quad\forall m\leq n\quad\left\Vert\sum\limits_{k=1}^m a_k e_k\right\Vert\leq K\left\Vert \sum\limits_{k=1}^n a_k e_k\right\Vert $$