Related papers: Trees and Branches in Banach Spaces

We say that a smooth normed space $X$ has a property (SL), if every mapping $f:X \to X$ preserving the semi-inner product on $X$ is linear. It is well known that every Hilbert space has the property (SL) and the same is true for every…

We show that there exist infinite-dimensional extremely non-complex Banach spaces, i.e. spaces $X$ such that the norm equality $\|Id + T^2\|=1 + \|T^2\|$ holds for every bounded linear operator $T:X\longrightarrow X$. This answers in the…

We introduce the point degree spectrum of a represented space as a substructure of the Medvedev degrees, which integrates the notion of Turing degrees, enumeration degrees, continuous degrees, and so on. The notion of point degree spectrum…

We show that any bounded operator $T$ on a separable, reflexive, infinite-dimensional Banach space $X$ admits a rank one perturbation which has an invariant subspace of infinite dimension and codimension. In the non-reflexive spaces, we…

When optimization theorists consider optimization problems in infinite dimensional spaces, they need to deal with closed convex subsets(usually cones) which mostly have empty interior. These subsets often prevent optimization theorists from…

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…

Given a Banach space $X$ and a real number $\alpha\ge 1$, we write: (1) $D(X)\le\alpha$ if, for any locally finite metric space $A$, all finite subsets of which admit bilipschitz embeddings into $X$ with distortions $\le C$, the space $A$…

We show that if an infinite-dimensional Banach space X has a symmetric basis then there exists a bounded, linear operator R : X --> X such that the set A = {x in X : ||R^n(x)|| --> infinity} is non-empty and nowhere dense in X. Moreover, if…

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…

The following dichotomy is established for a normalized weakly null sequence in a Banach space: Either every subsequence admits a convex block subsequence equivalent to the unit vector basis of c, the Banach space of null sequences under…

For every $\alpha<\omega_1$ we establish the existence of a separable Banach space whose Szlenk index is $\omega^{\alpha\omega+1}$ and which is universal for all separable Banach spaces whose Szlenk-index does not exceed…

This paper contains two improvements on a theorem of S. N. Bernstein for Banach spaces. We show that if $X$ is an arbitrary infinite-dimensional Banach space, $\{Y_n\}$ is a sequence of strictly nested subspaces of $ X$ and if $\{d_n\}$ is…

An example of an infinite dimensional and separable Banach space is given, that is not isomorphic to a subspace of l1 with no infinite equilateral sets.

These notes concern the nonlinear geometry of Banach spaces, asymptotic uniform smoothness and several Banach-Saks-like properties. We study the existence of certain concentration inequalities in asymptotically uniformly smooth Banach…

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 we extend to the quasi-reflexive setting the result of F. Baudier, N. Kalton and G. Lancien concerning the non-embeddability of the family of countably branching trees into reflexive Banach spaces whose Szlenk index and Szlenk…

In this paper, we develop a coarse analogue of treewidth. We prove that a graph $G$ admits a tree-decomposition in which each bag is contained in the union of a bounded number of balls of bounded radius, if and only if $G$ admits a…

We prove that if X is an infinite-dimensional Banach space with C^p smooth partitions of unity, then X and X\K are C^p diffeomorphic, for every weakly compact subset K of X.

We introduce a suitable notion of asymptotic smoothness on infinite dimensional Banach spaces, and we prove that, under some structural restrictions on the space, the convex envelope of an asymptotically smooth function is asymptotically…

Given a metric space $X$ of finite asymptotic dimension, we consider a quasi-isometric invariant of the space called dimension function. The space is said to have asymptotic Assouad-Nagata dimension less or equal $n$ if there is a linear…