Related papers: Indiscernible Subspaces and Minimal Wide Types

We show that there exists a strong uniform embedding from any proper metric space into any Banach space without cotype. Then we prove a result concerning the Lipschitz embedding of locally finite subsets of $\mathcal{L}_{p}$-spaces. We use…

Although Berkovich spaces may fail to be metrizable when defined over too big a field, we prove that a large part of their topology can be recovered through sequences: for instance, limit points of subsets are actual limits of sequences and…

We consider a normalized basis in a Banach space with the following property: any normalized block sequence of the basis has a subsequence equivalent to the basis. We show that under uniformity or other natural assumptions, a basis with…

In the short note we prove that for every $0<p<1$, there exists an infinite dimensional closed linear subspace of $\mathcal{L}\left( \ell_{p};\ell_{p}\right) $ every nonzero element of which is non $(r,s)$-absolutely summing operator for…

We are considering typed hierarchies of total, continuous functionals using complete, separable metric spaces at the base types. We pay special attention to the so called Urysohn space constructed by P. Urysohn. One of the properties of the…

The paper elucidates the relationship between the density of a Banach space and possible sizes of well-separated subsets of its unit sphere. For example, it is proved that for a large enough space $X$, the unit sphere $S_X$ always contains…

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…

We prove that an infinitesimally Hilbertian CD(0,N) space containing a line splits as the product of $R$ and an infinitesimally Hilbertian CD(0,N-1) space. By `infinitesimally Hilbertian' we mean that the Sobolev space $W^{1,2}(X,d,m)$,…

A subset of a Banach space is called equilateral if the distances between any two of its distinct elements are the same. It is proved that there exist non-separable Banach spaces (in fact of density continuum) with no infinite equilateral…

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…

In this work infinitely divisible cylindrical probability measures on arbitrary Banach spaces are introduced. The class of infinitely divisible cylindrical probability measures is described in terms of their characteristics, a…

We find conditions on a function space $\bf{L}$ that ensure that it behaves as an $L_p$-space in the sense that any unconditional basis of a complemented subspace of $\bf{L}$ either is equivalent to the unit vector system of $\ell_2$ or has…

Generalizing a recent result on lineability of sets of non-injective linear operators, we prove, for quite general linear spaces $A$ of maps from an arbitraty set to a sequence space, that, for every $0 \neq f \in A$, the subset of $A$ of…

Let $1\leq p\leq q\leq\infty.$ Being motivated by the classical notions of limited, $p$-limited and coarse $p$-limited subsets of a Banach space, we introduce and study $(p,q)$-limited subsets and their equicontinuous versions and coarse…

We prove that a non ergodic Banach space must be near Hilbert. In particular, $\ell_p$ ($2<p<\infty$) is ergodic. This reinforces the conjecture that $\ell_2$ is the only non ergodic Banach space. As an application of our criterion for…

For a measure space $\Omega$ we extend the theory of Orlicz spaces generated by an even convex integrand $\varphi \colon \Omega \times X \to \left[ 0, \infty \right]$ to the case when the range Banach space $X$ is arbitrary. Besides…

We study Banach spaces whose group of isometries acts micro-transitively on the unit sphere. We introduce a weaker property, which one-complemented subspaces inherit, that we call uniform micro-semitransitivity. We prove a number of results…

We prove that a typical Lipschitz mapping between any two Banach spaces is non-differentiable at typical points of any given subset of its domain in the most extreme form. This is a new result even for Lipschitz mappings between Euclidean…

We continue the study of Uniformly Finitely Extensible Banach spaces (in short, UFO) initiated in Moreno-Plichko, \emph{On automorphic Banach spaces}, Israel J. Math. 169 (2009) 29--45 and Castillo-Plichko, \emph{Banach spaces in various…

We show that any separable stable Banach space can be represented as a group of isometries on a separable reflexive Banach space, which extends a result of S. Guerre and M. Levy. As a consequence, we can then represent homeomorphically its…