Quasi-greedy Markushevich bases, duality and norming subspaces

We prove that if $\mathcal{X}$ is a quasi-greedy Markushevich basis of a Banach space $\mathbb{X}$, its dual basis ${\mathcal{X}}^{\ast }$ spans a norming subspace of ${\mathbb{X}}^{\ast }$. We also prove this result for weaker forms of quasi-greediness, and study the cases of other greedy-like properties from the literature.