When Greedy Approximation Breaks: Counterexamples in Quasi-Banach Spaces

Abstract: We construct two counterexamples that resolve long-standing open problems on greedy approximation theory with respect to bases, posed in [F. Albiac et al., Dissertationes Math. 560 (2021)] and restated in [F. Albiac, J. L.\@ Ansorena, V. Temlyakov, J. Approx. Theory 307 (2025)]. Our first result exhibits a quasi-Banach space $\mathbb{X}$ with an almost greedy basis which, when transported to the Banach envelope of $\mathbb{X}$, ceases to be quasi-greedy. This shows that the passage to the Banach envelope, although it preserves linear and lattice structure, may radically disrupt the performance of the thresholding greedy algorithm, to the extent that in some respects it could perform better in a quasi-Banach space than in its Banach envelope. Our second result constructs an almost greedy Markushevich basis in a nonlocally convex quasi-Banach space $\mathbb{Y}$ which fails to be a Schauder basis under any reordering. Together, these examples highlight that local convexity and the Banach envelope construction play an unexpectedly active role in shaping greedy approximation phenomena, revealing structural differences between Banach and quasi-Banach spaces that go beyond the classical theory of bases.