# Conditions for a linearly independent sequence with dense linear span to be a Schauder basis for a Banach space

Let $$(e_n)_{n \in \mathbb{N}}$$ be a linearly independent sequence in a Banach space $$X$$ such that $$X = \overline{\operatorname{span}} \{e_n : n \in \mathbb{N}\}$$ In particular, $$X$$ is separable.

However, $$(e_n)$$ may not be a Schauder basis for $$X$$, unless $$X$$ is a Hilbert space. The easiest counter-example is the sequence $$(1, x, x^2, \dots)$$ in $$C(\mathbb{R})$$. Even worse, $$X$$ may not have a Schauder basis at all.

Questions:

1. What are the conditions for $$(e_n)$$ to be a Schauder basis for $$X$$?
2. Even if $$(e_n)$$ is not a Schauder basis, what are the conditions for $$X$$ to have a Schauder basis?

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