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 $\gamma_{p,w}$. Next, we present a full characterization of extreme points of the unit ball in the sequence Lorentz space $\gamma_{1,w}$. We also establish a complete description with an isometry of the dual and predual spaces of the sequence Lorentz spaces $\gamma_{1,w}$ written in terms of the Marcinkiewicz spaces. Finally, we show a fundamental application of geometric structure of $\gamma_{1,w}$ to one-complemented subspaces of $\gamma_{1,w}$.