We survey a combinatorial framework for studying subsequences of a given sequence in a Banach space, with particular emphasis on weakly-null sequences. We base our presentation on the crucial notion of barrier introduced long time ago by Nash-Williams. In fact, one of the purposes of this survey is to isolate the importance of studying mappings defined on barriers as a crucial step towards solving a given problem that involves sequences in Banach spaces. We focus our study on various forms of ?partial...

Three sets occurring in functional analysis are shown to be of class PCA (also called ${\Sigma }_2^1$) and to be exactly of that class. The definition of each set is close to the usual objects of modern analysis, but some subtlety causes the sets to have a greater complexity than expected. Recent work in a similar direction is in [1, 2, 10, 11, 12].

Bref survol du théorème de non-plongement de J. Cheeger et B. Kleiner pour le groupe d’Heisenberg dans $L^1$.

The aim of this paper is to extend the framework of the spectral method for proving property (T) to the class of reflexive Banach spaces and present a condition implying that every affine isometric action of a given group $G$ on a reflexive Banach space $X$ has a fixed point. This last property is a strong version of Kazhdan’s property (T) and is equivalent to the fact that $H^1(G,\pi )=0$ for every isometric representation $\pi$ of $G$ on $X$. The condition is expressed in terms of $p$-Poincaré constants and we provide examples...

We study classes of operators represented as a pointwise absolutely convergent series of simpler ones, starting with rank 1 operators. In this short note we address the question, how far the repetition of this procedure can lead.

We show that, given a Banach space X, the Lipschitz-free space over X, denoted by ℱ(X), is isomorphic to ${\left({\sum }_{n=1}^{\infty }ℱ\left(X\right)\right)}_{\ell ₁}$. Some applications are presented, including a nonlinear version of Pełczyński’s decomposition method for Lipschitz-free spaces and the identification up to isomorphism between ℱ(ℝⁿ) and the Lipschitz-free space over any compact metric space which is locally bi-Lipschitz embeddable into ℝⁿ and which contains a subset that is Lipschitz equivalent to the unit ball of ℝⁿ. We also show that ℱ(M)...

The concept of lushness, introduced recently, is a Banach space property, which ensures that the space has numerical index 1. We prove that for Asplund spaces lushness is actually equivalent to having numerical index 1. We prove that every separable Banach space containing an isomorphic copy of c₀ can be renormed equivalently to be lush, and thus to have numerical index 1. The rest of the paper is devoted to the study of lushness just as a property of Banach spaces. We prove that lushness is separably...

Let (Ω,∑,μ) be a purely non-atomic measure space, and let 1 < p < ∞. If $L^{p,\infty }(\Omega ,\sum ,\mu )$ is isomorphic, as a Banach space, to $L^{p,\infty }({\Omega }^{\text{'}},{\sum }^{\text{'}},{\mu }^{\text{'}})$ for some purely atomic measure space (Ω’,∑’,μ’), then there is a measurable partition $\Omega ={\Omega }_1\cup {\Omega }_2$ such that $({\Omega }_1,\Sigma \cap {\Omega }_1,\mu {|}_{\Sigma \cap {\Omega }_1})$ is countably generated and σ-finite, and that μ(σ) = 0 or ∞ for every measurable $\sigma \subseteq {\Omega }_2$. In particular, $L^{p,\infty }(\Omega ,\sum ,\mu )$ is isomorphic to ${\ell }^{p,\infty }$.