Let $\mathcal {F}$ be a hereditary collection of finite subsets of $\mathbb {N}$. In this paper, we introduce and characterize $\mathcal {F}$-(almost) greedy bases. Given such a family $\mathcal {F}$, a basis $(e_n)_n$ for a Banach space X is called $\mathcal {F}$-greedy if there is a constant $C\geqslant 1$ such that for each $x\in X$, $m \in \mathbb {N}$, and $G_m(x)$, we have

$$ \begin{align*} \|x - G_m(x)\|\ \leqslant\ C \inf\left\{\left\|x-\sum_{n\in A}a_ne_n\right\|\,:\, |A|\leqslant m, A\in \mathcal{F}, (a_n)\subset \mathbb{K}\right\}. \end{align*} $$

$G_m(x)$

$\mathbb {K}$

$\mathcal {F}$

$\mathcal {F}$

$\mathcal {F}$

$\mathcal {F}$

$\mathcal {F}$

$\mathcal {F}$

Furthermore, we provide several examples of bases that are nontrivially $\mathcal {F}$-greedy. For a countable ordinal $\alpha $, we consider the case $\mathcal {F}=\mathcal {S}_{\alpha }$, where $\mathcal {S}_{\alpha }$ is the Schreier family of order $\alpha $. We show that for each $\alpha $, there is a basis that is $\mathcal {S}_{\alpha }$-greedy but is not $\mathcal {S}_{\alpha +1}$-greedy. In other words, we prove that none of the following implications can be reversed: for two countable ordinals $\alpha < \beta $,

$$ \begin{align*} \mbox{quasi-greedy}\ \Longleftarrow\ \mathcal{S}_{\alpha}\mbox{-greedy}\ \Longleftarrow\ \mathcal{S}_{\beta}\mbox{-greedy}\ \Longleftarrow\ \mbox{greedy}. \end{align*} $$

We use new methods, specific for non-locally convex quasi-Banach spaces, to investigate when the quasi-greedy bases of a $p$-Banach space for $0< p<1$ are democratic. The novel techniques we obtain permit to show in particular that all quasi-greedy bases of the Hardy space $H_p({\mathbb {D}})$ for $0< p<1$ are democratic while, in contrast, no quasi-greedy basis of $H_p({\mathbb {D}}^d)$ for $d\ge 2$ is, solving thus a problem that was raised in [7]. Applications of our results to other spaces of interest both in functional analysis and approximation theory are also provided.

Suppose that X and Y are two real normed spaces. A map $f:X\rightarrow Y$ is called a min-phase-isometry if it satisfies

$$ \begin{align*} \min\{\|f(x)+f(y)\|,\|f(x)-f(y)\|\}=\min\{\|x+y\|,\|x-y\|\} \quad (x,y\in X). \end{align*} $$

We present properties of min-phase-isometries in the case that Y is strictly convex and show that if a min-phase-isometry f (not necessarily surjective) fixes the origin, then it is phase-equivalent to a linear isometry, that is, $f(x)=\varepsilon (x)g(x)$ for $x\in X$, where $g:X\rightarrow Y$ is a linear isometry and $\varepsilon $ is a map from X to $\{-1,1\}$.

We extend Katok’s result on ‘the approximation of hyperbolic measures by horseshoes’ to Banach cocycles. More precisely, let f be a $C^r(r>1)$ diffeomorphism of a compact Riemannian manifold M, preserving an ergodic hyperbolic measure $\mu $ with positive entropy, and let $\mathcal {A}$ be a Hölder continuous cocycle of bounded linear operators acting on a Banach space $\mathfrak {X}$. We prove that there is a sequence of horseshoes for f and dominated splittings for $\mathcal {A}$ on the horseshoes, such that not only the measure theoretic entropy of f but also the Lyapunov exponents of $\mathcal {A}$ with respect to $\mu $ can be approximated by the topological entropy of f and the Lyapunov exponents of $\mathcal {A}$ on the horseshoes, respectively. As an application, we show the continuity of sub-additive topological pressure for Banach cocycles.

Motivated by Altshuler’s famous characterization of the unit vector basis of $c_0$ or $\ell _p$ among symmetric bases (Altshuler [1976, Israel Journal of Mathematics, 24, 39–44]), we obtain similar characterizations among democratic bases and among bidemocratic bases. We also prove a separate characterization of the unit vector basis of $\ell _1$.

Celebrating 100 years of the Banach contraction principle, we prove some fixed point theorems having all ingredients of the principle, but dealing with common fixed points of a contractive semigroup of nonlinear mappings acting in a modulated topological vector space. This research follows the ideas of the author’s recent papers [‘On modulated topological vector spaces and applications’, Bull. Aust. Math. Soc. 101 (2020), 325–332, and ‘Normal structure in modulated topological vector spaces’, Comment. Math. 60 (2020), 1–11]. Modulated topological vector spaces generalise, among others, Banach spaces and modular function spaces. The interest in modulars reflects the fact that the notions of ‘norm like’ but ‘noneuclidean’ (and not even necessarily convex) constructs to measure a level of proximity between complex objects are frequently used in science and technology. To prove our fixed point results in this setting, we introduce a new concept of Opial sets using analogies with the norm-weak and modular versions of the Opial property. As an example, the results of this work can be applied to spaces like $L^p$ for $p> 0 $, variable Lebesgue spaces $L^{p(\cdot )}$ where $1 \leq p(t) < + \infty $, Orlicz and Musielak–Orlicz spaces.

We construct a nonseparable Banach space $\mathcal {X}$ (actually, of density continuum) such that any uncountable subset $\mathcal {Y}$ of the unit sphere of $\mathcal {X}$ contains uncountably many points distant by less than $1$ (in fact, by less then $1-\varepsilon $ for some $\varepsilon>0$). This solves in the negative the central problem of the search for a nonseparable version of Kottman’s theorem which so far has produced many deep positive results for special classes of Banach spaces and has related the global properties of the spaces to the distances between points of uncountable subsets of the unit sphere. The property of our space is strong enough to imply that it contains neither an uncountable Auerbach system nor an uncountable equilateral set. The space is a strictly convex renorming of the Johnson–Lindenstrauss space induced by an $\mathbb {R}$-embeddable almost disjoint family of subsets of $\mathbb {N}$. We also show that this special feature of the almost disjoint family is essential to obtain the above properties.

For $1\le p <\infty $, we present a reflexive Banach space $\mathfrak {X}^{(p)}_{\text {awi}}$, with an unconditional basis, that admits $\ell _p$ as a unique asymptotic model and does not contain any Asymptotic $\ell _p$ subspaces. Freeman et al., Trans. AMS. 370 (2018), 6933–6953 have shown that whenever a Banach space not containing $\ell _1$, in particular a reflexive Banach space, admits $c_0$ as a unique asymptotic model, then it is Asymptotic $c_0$. These results provide a complete answer to a problem posed by Halbeisen and Odell [Isr. J. Math. 139 (2004), 253–291] and also complete a line of inquiry of the relation between specific asymptotic structures in Banach spaces, initiated in a previous paper by the first and fourth authors. For the definition of $\mathfrak {X}^{(p)}_{\text {awi}}$, we use saturation with asymptotically weakly incomparable constraints, a new method for defining a norm that remains small on a well-founded tree of vectors which penetrates any infinite dimensional closed subspace.

The purpose of this paper is to quantify the size of the Lebesgue constants $(\boldsymbol {L}_m)_{m=1}^{\infty }$ associated with the thresholding greedy algorithm in terms of a new generation of parameters that modulate accurately some features of a general basis. This fine tuning of constants allows us to provide an answer to the question raised by Temlyakov in 2011 to find a natural sequence of greedy-type parameters for arbitrary bases in Banach (or quasi-Banach) spaces which combined linearly with the sequence of unconditionality parameters $(\boldsymbol {k}_m)_{m=1}^{\infty }$ determines the growth of $(\boldsymbol {L}_m)_{m=1}^{\infty }$. Multiple theoretical applications and computational examples complement our study.

Magnitude is a numerical invariant of compact metric spaces, originally inspired by category theory and now known to be related to myriad other geometric quantities. Generalizing earlier results in $\ell _1^n$ and Euclidean space, we prove an upper bound for the magnitude of a convex body in a hypermetric normed space in terms of its Holmes–Thompson intrinsic volumes. As applications of this bound, we give short new proofs of Mahler’s conjecture in the case of a zonoid and Sudakov’s minoration inequality.

The classical Banach space $L_1(L_p)$ consists of measurable scalar functions f on the unit square for which

$$ \begin{align*}\|f\| = \int_0^1\Big(\int_0^1 |f(x,y)|^p dy\Big)^{1/p}dx < \infty.\end{align*} $$

We show that $L_1(L_p) (1 < p < \infty )$ is primary, meaning that whenever $L_1(L_p) = E\oplus F$, where E and F are closed subspaces of $L_1(L_p)$, then either E or F is isomorphic to $L_1(L_p)$. More generally, we show that $L_1(X)$ is primary for a large class of rearrangement-invariant Banach function spaces.

We present and thoroughly study natural Polish spaces of separable Banach spaces. These spaces are defined as spaces of norms, respectively pseudonorms, on the countable infinite-dimensional rational vector space. We provide an exhaustive comparison of these spaces with admissible topologies recently introduced by Godefroy and Saint-Raymond and show that Borel complexities differ little with respect to these two topological approaches.

We investigate generic properties in these spaces and compare them with those in admissible topologies, confirming the suspicion of Godefroy and Saint-Raymond that they depend on the choice of the admissible topology.

Any Lipschitz map $f : M \to N$ between two pointed metric spaces may be extended in a unique way to a bounded linear operator $\widehat {f} : \mathcal {F}(M) \to \mathcal {F}(N)$ between their corresponding Lipschitz-free spaces. In this paper, we give a necessary and sufficient condition for $\widehat {f}$ to be compact in terms of metric conditions on $f$. This extends a result by A. Jiménez-Vargas and M. Villegas-Vallecillos in the case of non-separable and unbounded metric spaces. After studying the behaviour of weakly convergent sequences made of finitely supported elements in Lipschitz-free spaces, we also deduce that $\widehat {f}$ is compact if and only if it is weakly compact.

Let $\{R_{k}\}_{k=1}^{\infty }$ be a sequence of expanding integer matrices in $M_{n}(\mathbb {Z})$, and let $\{D_{k}\}_{k=1}^{\infty }$ be a sequence of finite digit sets with integer vectors in ${\mathbb Z}^{n}$. In this paper, we prove that under certain conditions in terms of $(R_{k},D_{k})$ for $k\ge 1$, the Moran measure

$$ \begin{align*} \mu_{\{R_{k}\},\{D_{k}\}}:=\delta_{R_{1}^{-1}D_{1}}\ast\delta_{R_{1}^{-1}R_{2}^{-1}D_{2}}\ast\cdots \end{align*} $$

$(R,D)$

We prove that, given $2< p<\infty$, the Fourier coefficients of functions in $L_2(\mathbb {T}, |t|^{1-2/p}\,{\rm d}t)$ belong to $\ell _p$, and that, given $1< p<2$, the Fourier series of sequences in $\ell _p$ belong to $L_2(\mathbb {T}, \vert {t}\vert ^{2/p-1}\,{\rm d}t)$. Then, we apply these results to the study of conditional Schauder bases and conditional almost greedy bases in Banach spaces. Specifically, we prove that, for every $1< p<\infty$ and every $0\le \alpha <1$, there is a Schauder basis of $\ell _p$ whose conditionality constants grow as $(m^{\alpha })_{m=1}^{\infty }$, and there is an almost greedy basis of $\ell _p$ whose conditionality constants grow as $((\log m)^{\alpha })_{m=2}^{\infty }$.

The paper deals with the sets of numbers from [0,1] such that their binary representation is almost convergent. The aim of the study is to compute the Hausdorff dimensions of such sets. Previously, the results of this type were proved for a single summation method (e.g. Cesàro, Abel, Toeplitz). This study extends the results to a wide range of matrix summation methods.

We prove a general principle satisfied by weakly precompact sets of Lipschitz-free spaces. By this principle, certain infinite dimensional phenomena in Lipschitz-free spaces over general metric spaces may be reduced to the same phenomena in free spaces over their compact subsets. As easy consequences we derive several new and some known results. The main new results are: $\mathcal {F}(X)$ is weakly sequentially complete for every superreflexive Banach space $X$, and $\mathcal {F}(M)$ has the Schur property and the approximation property for every scattered complete metric space $M$.

Let $\mathcal {M}$ be a semifinite von Nemann algebra equipped with an increasing filtration $(\mathcal {M}_n)_{n\geq 1}$ of (semifinite) von Neumann subalgebras of $\mathcal {M}$. For $0<p <\infty $, let $\mathsf {h}_p^c(\mathcal {M})$ denote the noncommutative column conditioned martingale Hardy space and $\mathsf {bmo}^c(\mathcal {M})$ denote the column “little” martingale BMO space associated with the filtration $(\mathcal {M}_n)_{n\geq 1}$.

We prove the following real interpolation identity: if $0<p <\infty $ and $0<\theta <1$, then for $1/r=(1-\theta )/p$,

$$ \begin{align*} \big(\mathsf{h}_p^c(\mathcal{M}), \mathsf{bmo}^c(\mathcal{M})\big)_{\theta, r}=\mathsf{h}_{r}^c(\mathcal{M}), \end{align*} $$

For the case of complex interpolation, we obtain that if $0<p<q<\infty $ and $0<\theta <1$, then for $1/r =(1-\theta )/p +\theta /q$,

$$ \begin{align*} \big[\mathsf{h}_p^c(\mathcal{M}), \mathsf{h}_q^c(\mathcal{M})\big]_{\theta}=\mathsf{h}_{r}^c(\mathcal{M}) \end{align*} $$

These extend previously known results from $p\geq 1$ to the full range $0<p<\infty $. Other related spaces such as spaces of adapted sequences and Junge’s noncommutative conditioned $L_p$-spaces are also shown to form interpolation scale for the full range $0<p<\infty $ when either the real method or the complex method is used. Our method of proof is based on a new algebraic atomic decomposition for Orlicz space version of Junge’s noncommutative conditioned $L_p$-spaces.

We apply these results to derive various inequalities for martingales in noncommutative symmetric quasi-Banach spaces.

We show that every Lorentz sequence space $d(\textbf {w},p)$ admits a 1-complemented subspace Y distinct from $\ell _p$ and containing no isomorph of $d(\textbf {w},p)$. In the general case, this is only the second nontrivial complemented subspace in $d(\textbf {w},p)$ yet known. We also give an explicit representation of Y in the special case $\textbf {w}=(n^{-\theta })_{n=1}^\infty $ ($0<\theta <1$) as the $\ell _p$-sum of finite-dimensional copies of $d(\textbf {w},p)$. As an application, we find a sixth distinct element in the lattice of closed ideals of $\mathcal {L}(d(\textbf {w},p))$, of which only five were previously known in the general case.

We expand upon work from many hands on the decomposition of nuclear maps. Such maps can be characterised by their ability to be approximately written as the composition of maps to and from matrices. Under certain conditions (such as quasidiagonality), we can find a decomposition whose maps behave nicely, by preserving multiplication up to an arbitrary degree of accuracy and being constructed from order-zero maps (as in the definition of nuclear dimension). We investigate these conditions and relate them to a W*-analogue.