Some remarks on the geometry of Lorentz spaces
type: Item
instance of: scholarly article
title: Some remarks on the geometry of Lorentz spaces
zbMATH Open document ID: 1253.46019
author: Pawel Foralewski
published in: Indagationes Mathematicae. New Series
publication date: 2012-08-23T00:00:00Z
review text: The author considers several properties of Lorentz spaces \(\Lambda _{1, w}\). Section 2 deals with the Kadec-Klee property with respect to local convergence in measure and several related notions. It is worth mentioning that Kadec-Klee properties with respect to local (global) convergence in measure in Calderón-Lozanovskiĭ spaces have recently been studied in [\textit{P. Kolwicz}, ``Kadec-Klee properties of Calderón-Lozanovskiĭ function spaces'', J. Funct. Spaces Appl. 2012, Article ID 314068, 21 p. (2012; Zbl 1244.46012)] and [\textit{P. Kolwicz}, ``Kadec-Klee properties of Calderón-Lozanovskiĭ sequence spaces'', Collect. Math. 63 , 45--58 (2012)]. In Section 3 the author characterizes non-square Lorentz spaces \(\Lambda _{1, w}\) and he also shows that these spaces are not locally uniformly non-square. In the last section extreme points of Lorentz spaces \(\Lambda _{1, w}\) are considered and described in full. Moreover, it is proved that these spaces have no strong U-points (recall that each strong U-points is an extreme point). The results in Sections 3 and 4 require quite delicate proofs. The author applies methods concerning decreasing rearrangements and symmetric spaces which may be of independent interest.
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46B42
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 6070488
zbMATH Keywords: Lorentz function space
zbMATH Keywords: separability
zbMATH Keywords: Kadec-Klee property
zbMATH Keywords: strict monotonicity
zbMATH Keywords: lower locally uniform monotonicity
zbMATH Keywords: upper locally uniform monotonicity
zbMATH Keywords: non-squareness
zbMATH Keywords: local uniform non-squareness
zbMATH Keywords: extreme point
zbMATH Keywords: strong U-point
zbMATH Keywords: rotundity
reviewed by: Paweł Kolwicz
MaRDI profile type: Publication
full work available at URL: j.indag.2012.02.002
OpenAlex ID: W1607483482
cites work: Uniform convexity in Lorentz sequence spaces
cites work: Q4039801
cites work: Q5331549
cites work: Equidistributed random variables in \(L_{p,q}\)
cites work: On the geometry of the unit spheres of the Lorentz spaces Lw,1
cites work: On the isometries of the Lorentz function spaces
cites work: Characterizations of Kadec-Klee properties in symmetric spaces of measurable functions
cites work: On some local geometry of Orlicz sequence spaces equipped with the Luxemburg norm
cites work: Non-squareness properties of Orlicz spaces equipped with the \(p\)-Amemiya norm
cites work: Q4386398
cites work: Are generalized Lorentz “spaces” really spaces?
cites work: Q3329946
cites work: Q4679060
cites work: Rotundity structure of local nature for generalized Orlicz-Lorentz function spaces
cites work: Some fundamental geometric and topological properties of generalized Orlicz-Lorentz function spaces
cites work: Local rotundity structure of Cesàro--Orlicz sequence spaces
cites work: Q5296440
cites work: Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings
cites work: Function Spaces
cites work: Uniform convexity in function spaces
cites work: Uniformly non-$l_{n}^{(1)}$ Orlicz spaces with Luxemburg norm
cites work: Monotonicity Properties of Lorentz Spaces
cites work: Monotonicity properties of Musielak-Orlicz spaces and dominated best approximation in Banach lattices
cites work: Strongly extreme points in Köthe-Bochner spaces
cites work: Uniformly non-square Banach spaces
cites work: Super-reflexive spaces with bases
cites work: Some Remarks on Orlicz‐Lorentz Spaces
cites work: On isometric copies of \(\ell _{\infty }\) and James constants in Cesáro-Orlicz sequence spaces
cites work: Order convexity and concavity of Lorentz spaces Λp,w, 0<p<∞
cites work: Convexity and concavity constants in Lorentz and Marcinkiewicz spaces
cites work: Q3943431
cites work: On James and Jordan-von Neumann constants of Lorentz sequence spaces
cites work: Q5697168
cites work: Q3955929
cites work: Q4136031
cites work: Q4188284
cites work: Some new functional spaces
cites work: On the theory of spaces \(\Lambda\)
cites work: On the James constant and \(B\)-convexity of Cesàro and Cesàro--Orlicz sequence spaces
cites work: Q4016778
cites work: Q5529945
DOI: 10.1016/J.INDAG.2012.02.002
Recommended article: Q5296440
Recommended article: On the geometry of the unit spheres of the Lorentz spaces Lw,1
Recommended article: Q5690137
Recommended article: On the geometry of some Calderón-Lozanovskij interpolation spaces
Recommended article: Q3975732
Recommended article: Sequence Lorentz spaces and their geometric structure
Recommended article: Q4858492
Recommended article: Q4881083
Recommended article: Q3620733
Recommended article: Kadec-Klee properties of Calderón-Lozanovskiĭ sequence spaces