Related papers: On spreading sequences and asymptotic structures

In this paper, we study the coarse Lipschitz geometry of Banach spaces with several asymptotic properties. Specifically, we look at asymptotically uniformly smoothness and convexity, and several distinct Banach-Saks-like properties. Among…

In this note the result by A. Swift concerning the embeddability of countably branching bundle graphs into Banach spaces is extended from the context of reflexive spaces with an unconditional asymptotic structure to the context of dual…

In this paper structure of infinite dimensional Banach spaces is studied by using an asymptotic approach based on stabilization at infinity of finite dimensional subspaces which appear everywhere far away. This leads to notions of…

Let $(x_n)$ be a sequence in a Banach space $X$ which does not converge in norm, and let $E$ be an isomorphically precisely norming set for $X$ such that \[ \sum_n |x^*(x_{n+1}-x_n)|< \infty, \; \forall x^* \in E. \qquad (*) \] Then there…

It is shown that a separable Banach space $X$ can be given an equivalent norm $|\!|\!|\cdot |\!|\!|$ with the following properties:\quad If $(x_n)\subseteq X$ is relatively weakly compact and $\lim_{m\to\infty} \lim_{n\to\infty}\break…

It is shown that every conditional spreading sequence can be decomposed into two well behaved parts, one being unconditional and the other being convex block homogeneous, i.e. equivalent to its convex block sequences. This decomposition is…

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. D. Freeman,…

A well-known application of the Ramsey Theorem in the Banach Space Theory is the proof of the fact that every normalized basic sequence has a subsequence which generates a spreading model (the Brunel-Sucheston Theorem). Based on this…

We provide a generalization of two results of Knaust and Odell from \cite{KO2} and \cite{KO}. We prove that if $X$ is a Banach space and $(g_n)_{n=1}^\infty$ is a right dominant Schauder basis such that every normalized, weakly null…

In this paper we survey known results of characterizations of reflexive Banach spaces, which are based on convergence of usual and generalized arithmetic mean (or Ces\`aro sum), weakly compact subsets, affine sets in a Banach space or its…

We present a reflexive Banach space $\mathfrak{X}_{_{^\text{usm}}}$ which is Hereditarily Indecomposable and satisfies the following properties. In every subspace $Y$ of $\mathfrak{X}_{_{^\text{usm}}}$ there exists a weakly null normalized…

A subsequence principle is obtained, characterizing Banach spaces containing $c_0$, in the spirit of the author's 1974 characterization of Banach spaces containing $\ell^1$. Definition: A sequence $(b_j)$ in a Banach space is called {\it…

The following strengthening of the Elton-Odell theorem on the existence of a $(1+\epsilon)-$separated sequences in the unit sphere $S_X$ of an infinite dimensional Banach space $X$ is proved: There exists an infinite subset $S\subseteq S_X$…

This paper brings new results on the FPP in Banach spaces $X$ with a Schauder basis. We first deal with the problem of whether there is a Banach space isomorphic to $\co$ having the FPP. We show that the answer is negative if $X$ contains a…

Two examples of asymptotic $\ell_{1}$ Banach spaces are given. The first, $X_{u}$, has an unconditional basis and is arbitrarily distortable. The second, $X$, does not contain any unconditional basic sequence. Both are spaces of the type of…

We study the boundedness of averaging projections associated with symmetric Schauder bases in quasi-Banach spaces. Although this property is standard in the Banach setting, it is far from clear in the absence of local convexity and, indeed,…

We prove a structural property of the class of unconditionally saturated separable Banach spaces. We show, in particular, that for every analytic set $\aaa$, in the Effros-Borel space of subspaces of $C[0,1]$, of unconditionally saturated…

A Banach space contains either a minimal subspace or a continuum of incomparable subspaces. General structure results for analytic equivalence relations are applied in the context of Banach spaces to show that if $E_0$ does not reduce to…

It is known that there exists a Banach space $X$ with a Schauder basis $(e_i)_{i=1}^{\infty}$ which does not admit $\ell_p$ as the model space obtained by a finite chain of sequences such that each element is a spreading model of a block…

Let $X$ be a Banach space with separable dual. It is proved that for every $\varepsilon\in (0,1)$, $X$ embeds isometrically into a Banach space $W$ with a shrinking basis $(w_n)$ which is $(1+ \varepsilon)$-monotone. Moreover, if $X$ has…