--- title: Quasi-Greedy Markushevich Basis url: https://www.emergentmind.com/topics/quasi-greedy-markushevich-basis type: topic --- # Quasi-Greedy Markushevich Basis A quasi-greedy Markushevich basis is a Markushevich basis \((e_n,e_n^*)\) for which the thresholding greedy operators associated with the largest coefficients are uniformly bounded. In this setting, a Markushevich basis is a biorthogonal system that is total and point-separating, but need not be a Schauder basis; accordingly, quasi-greediness is formulated directly through finite greedy truncations rather than through norm-convergent coordinate expansions. When the Markushevich basis is in fact a semi-normalized Schauder basis, the classical Banach-space theory applies, and the extremal case \(C_{qg}=1\) coincides with suppression-unconditionality with constant \(1\) [1504.04368]. ## 1. Definition and basic framework A Markushevich basis, or M-basis, in a Banach space \(X\) consists of vectors \((e_n)_{n=1}^\infty\subset X\) and functionals \((e_n^*)_{n=1}^\infty\subset X^*\) such that \(e_n^*(e_m)=\delta_{nm}\), the linear span of \(\{e_n:n\in\mathbb N\}\) is dense in \(X\), and the functionals separate points of \(X\): if \(e_n^*(x)=0\) for all \(n\), then \(x=0\). Unlike a Schauder basis, an M-basis is not required to provide norm-convergent expansions \(x=\sum_n e_n^*(x)e_n\) for all \(x\) [1504.04368]. Given such a biorthogonal system and \(x\in X\), one selects a set \(A_N(x)\subset\mathbb N\) of \(N\) largest coefficients in modulus, meaning \[ |A_N(x)|=N,\qquad \min_{j\in A_N(x)} |e_j^*(x)| \ge \max_{j\notin A_N(x)} |e_j^*(x)|. \] The corresponding \(N\)-term greedy approximant is \[ G_N(x):=\sum_{j\in A_N(x)} e_j^*(x)e_j. \] For a general M-basis these sums are finite, so the definition makes sense without any global convergence assumption. In this general biorthogonal setting one calls the system quasi-greedy if the operators \(G_N\) are uniformly bounded; for bases, Wojtaszczyk’s characterization identifies this with convergence of the thresholding greedy algorithm [1504.04368][2009.02257]. This distinction between M-bases and Schauder bases is central. Every Schauder basis with its biorthogonal functionals is an M-basis, but not conversely. Consequently, the expression “quasi-greedy Markushevich basis” is broader than “quasi-greedy Schauder basis,” even though much of the classical greedy approximation theory was first formulated for the Schauder case. ## 2. The isometric case \(C_{qg}=1\) The sharpest structural theorem currently available in this direction is the characterization of \(1\)-quasi-greedy bases. For a semi-normalized basis \(\mathcal B=(e_n)\) in a Banach space, Albiac and Ansorena proved that \[ \mathcal B \text{ is quasi-greedy with } C_{qg}=1 \quad\Longleftrightarrow\quad \mathcal B \text{ is suppression-unconditional with } K_{su}=1. \] Equivalently, \(C_w=1\) if and only if \(K_{su}=1\), where \(C_w\) is the greedy projection constant and \(K_{su}\) is the norm of coordinate suppressions. In this extremal regime, greedy truncations are contractive and suppressing coordinates can never increase the norm [1504.04368]. The proof isolates a strong monotonicity property: if \(x\) and \(y\) are finitely supported and have disjoint supports, then \( \|x\|\le \|x+y\| \). From this one obtains contractivity of all coordinate projections. The converse direction is standard, since a suppression-unconditional basis automatically controls the greedy projections by the same constant. The same paper also shows that \(C_w=1\) forces the tail constant \(C_t\) to be \(1\), so the full quasi-greedy constant is \(1\), and that a basis admits an equivalent norm making it \(1\)-quasi-greedy if and only if it is unconditional in the original norm [1504.04368]. For general Markushevich bases, the theorem itself is not explicitly stated. The finite-support convexity argument uses only biorthogonality and greedy inequalities, so it adapts naturally on finitely supported vectors. A plausible implication is that, once an M-basis behaves like a Schauder basis on its closed span, the same argument yields suppression-unconditionality with constant \(1\) on that span. However, that extension is an extrapolation from the proof technique rather than a stated theorem of the paper [1504.04368]. ## 3. Duality and norming subspaces A major later development is the duality theory of quasi-greedy Markushevich bases. If \(\mathcal X=(x_n)\) is a \(K\)-quasi-greedy Markushevich basis of a Banach space \(\mathbb X\), then the norm-closed linear span \[ \overline{\langle x_n^*:n\in\mathbb N\rangle}\subset \mathbb X^* \] is a \(K^{-1}\)-norming subspace of \(\mathbb X^*\). Equivalently, for every \(f\in\mathbb X\) there exists \(f^*\in \langle x_n^*\rangle\) such that \[ f^*(f)=\|f\|,\qquad \|f^*\|\le K. \] This answers the problem of whether quasi-greediness alone forces the dual coordinate functionals to norm the space [2510.06398]. The same work proves substantially weaker hypotheses are sufficient. Uniformly bounded weak greedy projections along an unbounded sequence of orders already imply that the dual span is norming, and even pointwise bounded weak greedy sequences, with no uniform global constant, suffice. By contrast, semi-greediness or almost-greediness alone do not imply that the dual span is norming, and bidemocracy does not suffice either: there exist bidemocratic Markushevich bases whose dual systems do not span norming subspaces [2510.06398]. This places quasi-greedy Markushevich bases at a distinct level of rigidity. Quasi-greediness is not only a nonlinear approximation property of greedy truncations; it also enforces a linear-duality feature, namely that the coordinate functionals recover the ambient norm up to constants. ## 4. Variants with gaps, weak selection, and weighted selection One line of generalization replaces the full thresholding greedy algorithm by greedy approximants only along a subsequence \(\mathbf n=(n_k)\). For Markushevich bases, the decisive parameter is the geometry of the quotient gaps \(n_{k+1}/n_k\). If \(\mathbf n\) has bounded quotient gaps, then every \(\mathbf n\)-quasi-greedy Markushevich basis is quasi-greedy. Conversely, if \(\mathbf n\) has arbitrarily large quotient gaps, there exist Markushevich bases, indeed even Schauder bases, that are \(\mathbf n\)-quasi-greedy but not quasi-greedy. This gives a sharp characterization of when “greedy with gaps” is equivalent to ordinary quasi-greediness [2009.02257]. A second line concerns semi-greedy and weak semi-greedy algorithms. In the context of infinite-dimensional Banach spaces, semi-greedy, branch semi-greedy, weak semi-greedy, and almost greedy Markushevich bases are all equivalent. Since almost greedy is the conjunction of quasi-greediness and democracy, these equivalences show that a wide range of Chebyshevian and weak-thresholding procedures collapses to the same structural class once the Markushevich hypothesis is imposed [2004.06849]. Weighted analogues produce a similar picture. Weak weight-almost greedy bases are equivalent to weight-almost greedy bases, while weak weight-semi-greedy bases imply truncation quasi-greedy behavior and \(w\)-superdemocracy. Under additional hypotheses on the weight or on norming properties of the dual system, weak weight-semi-greedy Markushevich bases become weight-almost greedy, hence quasi-greedy in the ordinary sense [2111.00693]. ## 5. Finer structural scales around quasi-greediness Quasi-greediness admits several finer gradings. One is Schreier-indexed. For countable ordinals \(\alpha,\beta\), a basis is \((\alpha,\beta)\)-quasi-greedy when it is quasi-greedy, \(\mathcal S_\alpha\)-unconditional but not \(\mathcal S_{\alpha+1}\)-unconditional, and \(\mathcal S_\beta\)-democratic but not \(\mathcal S_{\beta+1}\)-democratic. Constructions are known for every pair with \(\beta\le \alpha+1\) except the already solved case \((0,0)\), and the region \(\beta\ge \alpha+2\) remains open. These examples are built as normalized Schauder bases, hence as quasi-greedy M-bases, and show that “unconditional depth” and “democracy depth” can be prescribed separately within the known region [2504.05533]. A second grading weakens quasi-greediness toward truncation-based and threshold-based notions. Truncation quasi-greedy bases form a strictly larger class than quasi-greedy bases, but they retain SUCC, quasi-greedy-for-large-coefficients behavior, lattice partial unconditionality, and the same Lebesgue-type optimality in many situations. Nearly truncation quasi-greedy is equivalent to Elton near unconditionality, and thus supplies a threshold-sensitive weakening of full greedy boundedness [2106.00975]. A complementary viewpoint is quasi-greediness for largest coefficients. Elton near unconditionality is equivalent to quasi-greedy for largest coefficients, and in the isometric regime one has a sharp characterization: being \(1\)-quasi-greedy for largest coefficients is equivalent to being \(1\)-truncation quasi-greedy. These results are formulated for bases with biorthogonal functionals, and they provide a threshold-free language for understanding how far a quasi-greedy system is from unconditionality [2209.03445]. ## 6. Examples, constraints, and counterexamples The canonical unit vector bases of \(\ell_p\), \(1\le p<\infty\), and orthonormal bases of Hilbert spaces are unconditional with suppression constant \(1\), hence \(1\)-quasi-greedy. At the opposite extreme, conditional quasi-greedy bases exist in separable Hilbert spaces, in \(\ell_p\) and \(L_p[0,1]\) for \(1