Quasi-greedy Markushevich bases, duality and norming subspaces

TL;DR

This work proves that every quasi-greedy Markushevich basis $\mathcal{X}$ has a dual $\mathcal{X}^*$ spanning a norming subspace of $\mathbb{X}^*$, and extends the result to weaker forms of quasi-greediness. It shows that if a sequence of greedy-like projections $P_A(f)$ remains uniformly bounded (under appropriate $t$-greedy sets and index sequences $d_j$), then $\mathcal{X}^*$ continues to be norming with constant $K^{-1}$, and in the canonical case $d_j=j$ there exists a norm-attaining dual functional $f^*$ with $\|f^*\|\le K$. The paper also clarifies limitations: norming duals need not arise from semi-greedy or bidemocratic systems in general, though truncation quasi-greedy behavior transfers to bidual systems and preserves several democracy-type properties. Collectively, these results advance duality and norming understanding in greedy approximation beyond Schauder bases, broadening the scope of norming phenomena for quasi-greedy Markushevich bases.

Abstract

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

Theorems & Definitions (24)

• Lemma 3.1
• proof
• Remark 3.2
• Corollary 3.3
• proof
• Theorem 3.4
• proof
• Remark 3.5
• Lemma 3.6
• proof