Quasi-Greedy Markushevich Basis

• Quasi-greedy Markushevich bases are biorthogonal systems where finite greedy truncations are uniformly bounded, generalizing Schauder bases.
• They enforce a duality property by ensuring the corresponding coordinate functionals form a norming subspace of the dual space.
• Extensions to weighted, weak, and variant selection methods reveal a rich structural interplay between greedy algorithms and classical Banach space theory.

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 $1$ (Albiac et al., 2015).

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 $C_{qg}=1$0 for all $C_{qg}=1$1, then $C_{qg}=1$2. Unlike a Schauder basis, an M-basis is not required to provide norm-convergent expansions $C_{qg}=1$3 for all $C_{qg}=1$4 (Albiac et al., 2015).

Given such a biorthogonal system and $C_{qg}=1$5, one selects a set $C_{qg}=1$6 of $C_{qg}=1$7 largest coefficients in modulus, meaning

$C_{qg}=1$8

The corresponding $C_{qg}=1$9-term greedy approximant is

$1$0

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 $1$1 are uniformly bounded; for bases, Wojtaszczyk’s characterization identifies this with convergence of the thresholding greedy algorithm (Albiac et al., 2015, Berasategui et al., 2020).

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 $1$2

The sharpest structural theorem currently available in this direction is the characterization of $1$3-quasi-greedy bases. For a semi-normalized basis $1$4 in a Banach space, Albiac and Ansorena proved that

$1$5

Equivalently, $1$6 if and only if $1$7, where $1$8 is the greedy projection constant and $1$9 is the norm of coordinate suppressions. In this extremal regime, greedy truncations are contractive and suppressing coordinates can never increase the norm (Albiac et al., 2015).

The proof isolates a strong monotonicity property: if $X$0 and $X$1 are finitely supported and have disjoint supports, then $X$2. 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 $X$3 forces the tail constant $X$4 to be $X$5, so the full quasi-greedy constant is $X$6, and that a basis admits an equivalent norm making it $X$7-quasi-greedy if and only if it is unconditional in the original norm (Albiac et al., 2015).

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 $X$8 on that span. However, that extension is an extrapolation from the proof technique rather than a stated theorem of the paper (Albiac et al., 2015).

3. Duality and norming subspaces

A major later development is the duality theory of quasi-greedy Markushevich bases. If $X$9 is a $(e_n)_{n=1}^\infty\subset X$0-quasi-greedy Markushevich basis of a Banach space $(e_n)_{n=1}^\infty\subset X$1, then the norm-closed linear span

$(e_n)_{n=1}^\infty\subset X$2

is a $(e_n)_{n=1}^\infty\subset X$3-norming subspace of $(e_n)_{n=1}^\infty\subset X$4. Equivalently, for every $(e_n)_{n=1}^\infty\subset X$5 there exists $(e_n)_{n=1}^\infty\subset X$6 such that

$(e_n)_{n=1}^\infty\subset X$7

This answers the problem of whether quasi-greediness alone forces the dual coordinate functionals to norm the space (Berasategui, 7 Oct 2025).

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 (Berasategui, 7 Oct 2025).

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 $(e_n)_{n=1}^\infty\subset X$8. For Markushevich bases, the decisive parameter is the geometry of the quotient gaps $(e_n)_{n=1}^\infty\subset X$9. If $(e_n^*)_{n=1}^\infty\subset X^*$0 has bounded quotient gaps, then every $(e_n^*)_{n=1}^\infty\subset X^*$1-quasi-greedy Markushevich basis is quasi-greedy. Conversely, if $(e_n^*)_{n=1}^\infty\subset X^*$2 has arbitrarily large quotient gaps, there exist Markushevich bases, indeed even Schauder bases, that are $(e_n^*)_{n=1}^\infty\subset X^*$3-quasi-greedy but not quasi-greedy. This gives a sharp characterization of when “greedy with gaps” is equivalent to ordinary quasi-greediness (Berasategui et al., 2020).

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 (Berasategui et al., 2020).

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 $(e_n^*)_{n=1}^\infty\subset X^*$4-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 (Berasategui et al., 2021).

5. Finer structural scales around quasi-greediness

Quasi-greediness admits several finer gradings. One is Schreier-indexed. For countable ordinals $(e_n^*)_{n=1}^\infty\subset X^*$5, a basis is $(e_n^*)_{n=1}^\infty\subset X^*$6-quasi-greedy when it is quasi-greedy, $(e_n^*)_{n=1}^\infty\subset X^*$7-unconditional but not $(e_n^*)_{n=1}^\infty\subset X^*$8-unconditional, and $(e_n^*)_{n=1}^\infty\subset X^*$9-democratic but not $e_n^*(e_m)=\delta_{nm}$0-democratic. Constructions are known for every pair with $e_n^*(e_m)=\delta_{nm}$1 except the already solved case $e_n^*(e_m)=\delta_{nm}$2, and the region $e_n^*(e_m)=\delta_{nm}$3 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 (Beanland et al., 7 Apr 2025).

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 (Albiac et al., 2021).

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 $e_n^*(e_m)=\delta_{nm}$4-quasi-greedy for largest coefficients is equivalent to being $e_n^*(e_m)=\delta_{nm}$5-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 (Albiac et al., 2022).

6. Examples, constraints, and counterexamples

The canonical unit vector bases of $e_n^*(e_m)=\delta_{nm}$6, $e_n^*(e_m)=\delta_{nm}$7, and orthonormal bases of Hilbert spaces are unconditional with suppression constant $e_n^*(e_m)=\delta_{nm}$8, hence $e_n^*(e_m)=\delta_{nm}$9-quasi-greedy. At the opposite extreme, conditional quasi-greedy bases exist in separable Hilbert spaces, in $\{e_n:n\in\mathbb N\}$0 and $\{e_n:n\in\mathbb N\}$1 for $\{e_n:n\in\mathbb N\}$2, in $\{e_n:n\in\mathbb N\}$3, and in $\{e_n:n\in\mathbb N\}$4. Thus quasi-greediness is genuinely weaker than unconditionality away from the isometric constant $\{e_n:n\in\mathbb N\}$5 (Albiac et al., 2015).

In $\{e_n:n\in\mathbb N\}$6, quasi-greedy bases are strongly constrained. They satisfy a square-function equivalence

$\{e_n:n\in\mathbb N\}$7

and no normalized unconditional basis in $\{e_n:n\in\mathbb N\}$8, $\{e_n:n\in\mathbb N\}$9, can be semi-normalized in $X$0 for $X$1. For Jacobi polynomials, the $X$2-normalized system is quasi-greedy if and only if $X$3; equivalently, decreasing rearrangements of Jacobi-Fourier series fail in $X$4 unless the ambient space is Hilbertian (Albiac et al., 2015).

In the nonlocally convex range $X$5, the picture is in some respects more rigid. Every quasi-greedy basis in $X$6 is democratic with fundamental function of order $X$7, and the same holds in separable $X$8-spaces with the bounded approximation property. Consequently, quasi-greedy bases in these spaces are automatically almost greedy (Albiac et al., 2020).

Quasi-Banach counterexamples show that the category of quasi-greedy Markushevich bases is broader than the classical Schauder setting. There exists an almost greedy Markushevich basis in a nonlocally convex quasi-Banach space that fails to be a Schauder basis under any reordering, and there exists an almost greedy basis whose image in the Banach envelope ceases to be quasi-greedy. These examples demonstrate that local convexity and the Banach envelope construction can decisively alter greedy behavior (Albiac et al., 15 Oct 2025).

A quasi-greedy Markushevich basis therefore occupies an intermediate position between general biorthogonal systems and unconditional bases. It is robust enough to support a greedy approximation theory, strong enough to force norming properties of the dual coordinate functionals, and flexible enough to admit ordinal refinements, weak and weighted variants, and highly nonclassical behavior in quasi-Banach settings.