Maciej Ciesielski

Hardy–Littlewood–Pólya relation in the best dominated approximation in symmetric spaces

Abstract We investigate a correspondence between strict K -monotonicity, K -order continuity and the best dominated approximation problems with respect to the Hardy–Littlewood–Polya relation ≺ . Namely, we study, in terms of an LKM point and a UKM point, a necessary condition for uniqueness of the best dominated approximation under the relation ≺ in a symmetric space E . Next, we characterize a relation between a point of K -order continuity and an existence of a best dominated approximant with respect to ≺ . Finally, we present a compete criteria, written in a notion of K -order continuity, under which a closed and K -bounded above subset of a symmetric space E is proximinal.

Sequence Lorentz Spaces and Their Geometric Structure

This article is dedicated to geometric structure of the Lorentz and Marcinkiewicz spaces in case of the pure atomic measure. We study complete criteria for order continuity, the Fatou property, strict monotonicity, and strict convexity in the sequence Lorentz spaces

Monotonicity and rotundity of Lorentz spaces Γp,w

gamma _{p,w}.

Local approach to Kadec–Klee properties in symmetric function spaces

γp,w. Next, we present a full characterization of extreme points of the unit ball in the sequence Lorentz space

On geometric structure of symmetric spaces

gamma _{1,w}.

Best approximation properties in spaces of measurable functions

γ1,w. We also establish a complete description up to isometry of the dual and predual spaces of the sequence Lorentz spaces