We use a special tiling for the hyperbolic d-space $\mathbb {H}^d$ for $d=2,3,4$ to construct an (almost) explicit isomorphism between the Lipschitz-free space $\mathcal {F}(\mathbb {H}^d)$ and $\mathcal {F}(P)\oplus \mathcal {F}(\mathcal {N})$, where P is a polytope in $\mathbb {R}^d$ and $\mathcal {N}$ a net in $\mathbb {H}^d$ coming from the tiling. This implies that the spaces $\mathcal {F}(\mathbb {H}^d)$ and $\mathcal {F}(\mathbb {R}^d)\oplus \mathcal {F}(\mathcal {M})$ are isomorphic for every net $\mathcal {M}$ in $\mathbb {H}^d$. In particular, we obtain that, for $d=2,3,4$, $\mathcal {F}(\mathbb {H}^d)$ has a Schauder basis. Moreover, using a similar method, we also give an explicit isomorphism between $\mathrm {Lip}(\mathbb {H}^d)$ and $\mathrm {Lip}(\mathbb {R}^d)$.

We consider two questions on the geometry of Lipschitz-free $p$-spaces $\mathcal {F}_p$, where $0< p\leq 1$, over subsets of finite-dimensional vector spaces. We solve an open problem and show that if $(\mathcal {M}, \rho )$ is an infinite doubling metric space (e.g. an infinite subset of an Euclidean space), then $\mathcal {F}_p (\mathcal {M}, \rho ^\alpha )\simeq \ell _p$ for every $\alpha \in (0,\,1)$ and $0< p\leq 1$. An upper bound on the Banach–Mazur distance between the spaces $\mathcal {F}_p ([0, 1]^d, |\cdot |^\alpha )$ and $\ell _p$ is given. Moreover, we tackle a question due to Albiac et al. [4] and expound the role of $p$, $d$ for the Lipschitz constant of a canonical, locally coordinatewise affine retraction from $(K, |\cdot |_1)$, where $K=\cup _{Q\in \mathcal {R}} Q$ is a union of a collection $\emptyset \neq \mathcal {R} \subseteq \{ Rw + R[0,\,1]^d: w\in \mathbb {Z}^d\}$ of cubes in $\mathbb {R}^d$ with side length $R>0$, into the Lipschitz-free $p$-space $\mathcal {F}_p (V, |\cdot |_1)$ over their vertices.

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 $\ell _1$.

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.

Galego and Samuel showed that if K, L are metrizable, compact, Hausdorff spaces, then $C(K)\widehat{\otimes}_\pi C(L)$ is c0-saturated if and only if it is subprojective if and only if K and L are both scattered. We remove the hypothesis of metrizability from their result and extend it from the case of the twofold projective tensor product to the general n-fold projective tensor product to show that for any $n\in\mathbb{N}$ and compact, Hausdorff spaces K1, …, Kn, $\widehat{\otimes}_{\pi, i=1}^n C(K_i)$ is c0-saturated if and only if it is subprojective if and only if each Ki is scattered.

For 0 ≤ ξ ≤ ω1, we define the notion of ξ-weakly precompact and ξ-weakly compact sets in Banach spaces and prove that a set is ξ-weakly precompact if and only if its weak closure is ξ-weakly compact. We prove a quantified version of Grothendieck’s compactness principle and the characterisation of Schur spaces obtained in [7] and [9]. For 0 ≤ ξ ≤ ω1, we prove that a Banach space X has the ξ-Schur property if and only if every ξ-weakly compact set is contained in the closed, convex hull of a weakly null (equivalently, norm null) sequence. The ξ = 0 and ξ= ω1 cases of this theorem are the theorems of Grothendieck and [7], [9], respectively.

We prove that every separable Banach space containing an isomorphic copy of $\ell _{1}$ can be equivalently renormed so that the new bidual norm is octahedral. This answers, in the separable case, a question in Godefroy [Metric characterization of first Baire class linear forms and octahedral norms, Studia Math. 95 (1989), 1–15]. As a direct consequence, we obtain that every dual Banach space, with a separable predual and failing to be strongly regular, can be equivalently renormed with a dual norm to satisfy the strong diameter two property.

We present a construction that enables one to find Banach spaces $X$ whose sets $\operatorname{NA}(X)$ of norm attaining functionals do not contain two-dimensional subspaces and such that, consequently, $X$ does not contain proximinal subspaces of finite codimension greater than one, extending the results recently provided by Read [Banach spaces with no proximinal subspaces of codimension 2, Israel J. Math. (to appear)] and Rmoutil [Norm-attaining functionals need not contain 2-dimensional subspaces, J. Funct. Anal. 272 (2017), 918–928]. Roughly speaking, we construct an equivalent renorming with the requested properties for every Banach space $X$ where the set $\operatorname{NA}(X)$ for the original norm is not “too large”. The construction can be applied to every Banach space containing $c_{0}$ and having a countable system of norming functionals, in particular, to separable Banach spaces containing $c_{0}$. We also provide some geometric properties of the norms we have constructed.

Based on a construction method introduced by Bourgain and Delbaen, we give a general definition of a Bourgain–Delbaen space and prove that every infinite-dimensional separable ${\mathcal{L}}_{\infty }$ -space is isomorphic to such a space. Furthermore, we provide an example of a ${\mathcal{L}}_{\infty }$ and asymptotic $c_{0}$ space not containing $c_{0}$ .

We investigate questions of maximal symmetry in Banach spaces and the structure of certain bounded non-unitarisable groups on Hilbert space. In particular, we provide structural information about bounded groups with an essentially unique invariant complemented subspace. This is subsequently combined with rigidity results for the unitary representation of $\text{Aut}(T)$ on $\ell _{2}(T)$, where $T$ is the countably infinite regular tree, to describe the possible bounded subgroups of $\text{GL}({\mathcal{H}})$ extending a well-known non-unitarisable representation of $\mathbb{F}_{\infty }$.

As a related result, we also show that a transitive norm on a separable Banach space must be strictly convex.

In this paper we investigate the following functional inequality

$ \begin{eqnarray*} \| f(x-y-z) - f(x-y+z) + f(y) +f(z)\| \leq \|f(x+y-z) - f(x)\| \end{eqnarray*}$

Assuming that (Ω, Σ) is a measurable space and (X) is a Banach space we provide a quite general sufficient condition on (X) for bvca(Σ, X) (the Banach space of all X-valued countably additive measures of bounded variation equipped with the variation norm) to contain a copy of c0 if and only if X does. Some well-known results on this topic are straightforward consequences of our main theorem.

Let $X$ and $Y$ be Banach spaces isomorphic to complemented subspaces of each other with supplements $A$ and $B$. In 1996, W. T. Gowers solved the Schroeder–Bernstein (or Cantor–Bernstein) problem for Banach spaces by showing that $X$ is not necessarily isomorphic to $Y$. In this paper, we obtain a necessary and sufficient condition on the sextuples $\left( p,\,q,\,r,\,s,\,u,\,v \right)$ in $\mathbb{N}$ with $p\,+\,q\,\ge \,1$, $r+s\ge 1$ and $u,\,v\,\in \,{{\mathbb{N}}^{*}}$, to provide that $X$ is isomorphic to $Y$, whenever these spaces satisfy the following decomposition scheme

$${{A}^{u}}\,\sim \,{{X}^{p}}\,\oplus \,{{Y}^{q}},\,{{B}^{v}}\,\sim \,{{X}^{r}}\,\oplus \,{{Y}^{s}}.$$

Namely, $\Phi \,=\,\left( p\,-\,u \right)\left( s\,-\,v \right)-\left( q\,+\,u \right)\left( r\,+\,v \right)$ is different from zero and $\Phi $ divides $p\,+\,q$ and $r\,+\,s$. These sextuples are called Cantor–Bernstein sextuples for Banach spaces. The simplest case $\left( 1,0,0,1,1,1 \right)$ indicates the well-known Pełczyński's decomposition method in Banach space. On the other hand, by interchanging some Banach spaces in the above decomposition scheme, refinements of the Schroeder– Bernstein problem become evident.

A characterization of the Banach spaces of type $C\left( K \right)$ that admit an equivalent locally uniformly rotund norm is obtained, and a method to apply it to concrete spaces is developed. As an application the existence of such renorming is deduced when $K$ is a Namioka–Phelps compact or for some particular class of Rosenthal compacta, results which were originally proved with ad hoc methods.

An example is found of a nonreflexive Banach space X such that the union of {0} and the set of non-norm-attaining functionals on X contains no two-dimensional subspace.

It is shown that if a Banach space is saturated with infinite dimensional subspaces in which all “special” $n$-tuples of vectors are equivalent with constants independent of $n$-tuples and of $n$, then the space contains asymptotic-${{l}_{p}}$ subspaces for some $1\,\le \,p\,\le \,\infty $. This extends a result by Figiel, Frankiewicz, Komorowski and Ryll-Nardzewski.

We present partial positive results supporting a conjecture that admitting an equivalent Lipschitz (or uniformly) weak* Kadec–Klee norm is a three space property.

We construct a continuum of mutually non-isomorphic separable Banach spaces which are complemented in each other. Consequently, the Schroeder–Bernstein Index of any of these spaces is ${{2}^{{\aleph_{0}}}}$. Our construction is based on a Banach space introduced by W. T. Gowers and B. Maurey in 1997. We also use classical descriptive set theory methods, as in some work of the first author and C. Rosendal, to improve some results of P. G. Casazza and of N. J. Kalton on the Schroeder–Bernstein Property for spaces with an unconditional finite-dimensional Schauder decomposition.

A hereditarily indecomposable asymptotic $\ell_2$ Banach space is constructed. The existence of such a space answers a question of B. Maurey and verifies a conjecture of W. T. Gowers.

the analytic concepts of martingale type $p$ and cotype $q$ of a banach space have an intimate relation with the geometric concepts of $p$-concavity and $q$-convexity of the space under consideration, as shown by pisier. in particular, for a banach space $x$, having martingale type $p$ for some $p{>}1$ implies that $x$ has martingale cotype $q$ for some $q{<}\infty$.

the generalisation of these concepts to linear operators was studied by the author, and it turns out that the duality above only holds in a weaker form. an example is constructed showing that this duality result is best possible.

so-called random martingale unconditionality estimates, introduced by garling as a decoupling of the unconditional martingale differences (umd) inequality, are also examined.

it is shown that the random martingale unconditionality constant of $l_\infty^{2^n}$ for martingales of length $n$ asymptotically behaves like $n$. this improves previous estimates by geiss, who needed martingales of length $2^n$ to show this asymptotic. at the same time the order in the paper is the best that can be expected.