Related papers: Maps with the Radon-Nikod\'ym property
Non-convex functions that yet satisfy a condition of uniform convexity for non-close points can arise in discrete constructions. We prove that this sort of discrete uniform convexity is inherited by the convex envelope, which is the key to…
Very recently, motivated by the result of Bhatia and \v{S}emrl which characterizes the Birkhoff-James orthogonality of operators on a finite dimensional Hilbert space in terms of norm attaining points, the Bhatia-\v{S}emrl property was…
We consider semidifferentiable (possibly nonsmooth) maps, acting on a subset of a Banach space, that are nonexpansive either in the norm of the space or in the Hilbert's or Thompson's metric inherited from a convex cone. We show that the…
Some necessary and sufficient conditions are found for Banach function lattices to have the Radon-Nikod\'ym property. Consequently it is shown that an Orlicz space $L_\varphi$ over a non-atomic $\sigma$-finite measure space $(\Omega,…
We prove that a continuous image of a Radon-Nikod\'ym compact space of weight less than b is Radon-Nikod\'ym compact. As a Banach space counterpart, subspaces of Asplund generated Banach spaces of density character less than b are Asplund…
In this paper, two main results concerning uniformly continuous retractions are proved. First, an $\alpha$-H\"older retraction from any separable Banach space onto a compact convex subset whose closed linear span is the whole space is…
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…
We show that the duals of Banach algebras of scalar-valued bounded holomorphic functions on the open unit ball $B_E$ of a Banach space $E$ lack weak$^*$-strongly exposed points. Consequently, we obtain that some Banach algebras of…
We study smoothness and strict convexity of (the bidual) of Banach spaces in the presence of diameter 2 properties. We prove that the strong diameter 2 property prevents the bidual from being strictly convex and being smooth, and we…
The purpose of this note is to show that, if $\mcB$ is a uniformly convex Banach, then the dual space $\mcB'$ has a "Hilbert space representation" (defined in the paper), that makes $\mcB$ much closer to a Hilbert space then previously…
We show that every metric space with bounded geometry uniformly embeds into an explicit reflexive Banach space (a direct sum of l^p spaces). In the case of discrete groups we show the analogue of a-T-menability. That is, we construct a…
The main results of the paper: {\bf (1)} The dual Banach space $X^*$ contains a linear subspace $A\subset X^*$ such that the set $A^{(1)}$ of all limits of weak$^*$ convergent bounded nets in $A$ is a proper norm-dense subset of $X^*$ if…
For a fixed $K\gg 1$ and $n\in\mathbb{N}$, $n\gg 1$, we study metric spaces which admit embeddings with distortion $\le K$ into each $n$-dimensional Banach space. Classical examples include spaces embeddable into $\log n$-dimensional…
In this survey we collect some recent results obtained by the authors and collaborators concerning the fine structure of functions of bounded deformation (BD). These maps are $\mathrm{L}^1$-functions with the property that the symmetric…
We introduce a new topological property called (*) and the corresponding class of topological spaces, which includes spaces with $G_\delta$-diagonals and Gruenhage spaces. Using (*), we characterise those Banach spaces which admit…
We study universal approximation of continuous functionals on compact subsets of products of Hilbert spaces. We prove that any such functional can be uniformly approximated by models that first take finitely many continuous linear…
In [2] we characterized in terms of a quadratic growth condition various metric regularity properties of the subdifferential of a lower semicontinuous convex function acting in a Hilbert space. Motivated by some recent results in [16] where…
The Radon-Nikodym formalism is used to study the structure of the set of positive maps from $\mathcal{B}(\mathcal{H})$ into itself, where $\mathcal{H}$ is a finite dimensional Hilbert space. In particular, this formalism was employed to…
We show that Sobolev maps with values in a dual Banach space can be characterized in terms of weak derivatives in a weak* sense. Since every metric space embeds isometrically into a dual Banach space, this implies a characterization of…
We introduce a family of Banach spaces of measures, each containing the set of measures with density of bounded variation. These spaces are suitable for the study of weighted transfer operators of piecewise-smooth maps of the interval where…