Positivity 2: 311–337, 1998.
© 1998 Kluwer Academic Publishers. Printed in the Netherlands.
Geometric Properties of Symmetric Spaces with
Applications to Orlicz–Lorentz Spaces
, HENRYK HUDZIK
, ANNA KAMI
Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, E-08071 Barcelona,
Spain E-mail: firstname.lastname@example.org;
Faculty of Mathematics and Computer Sciences, Adam
Mickiewicz University, Pozna´n, Poland E–mail: email@example.com and
Department of Mathematical Sciences, The University of Memphis,
Memphis TN 38152, U.S.A. E–mail: firstname.lastname@example.org
(Received: 27 February 1998; Accepted: 6 April 1998)
Abstract. We deal with the basic convexity properties –rotundity, and uniform, local uniform and
full rotundity — for symmetric spaces. A characterization of Orlicz–Lorentz spaces with the Kadec–
Klee property for pointwise convergence is given. These results are applied to obtain criteria of
convexity properties for Orlicz–Lorentz sequence spaces, and some new proofs of the sufﬁciency
part of criteria for rotundity and uniform rotundity for Orlicz–Lorentz function spaces.
Mathematics Subject Classiﬁcation (1991): Primary 46E30, 46B20. Secondary: 46B42
Key words: rotundity, uniform convexity, symmetric space, Orlicz-Lorentz space
Let E denote a Banach function lattice over a given measure space with a mea-
sure µ whichisassumedtobeσ–ﬁnite and nonatomic, or the counting measure
if = N andthenwealsosaythatE is a Banach sequence lattice. This means
that E is a Banach space which is a linear subspace of L
(µ), the space
of all (equivalence classes of) measurable functions on , with the following two
(i) x ∈ E and x
y, whenever x ∈ L
, y ∈ E and |x|
(ii) There exists x ∈ E such that x(ω) > 0almosteverywhereon.
If w is a positive measurable function on , E(w) will represent the weighted
Banach function space of all x ∈ L
such that x
This work was done while the three last mentioned authors were visiting the C.R.M. in
Supported by DGICYT Grant PB94-0897 and by Suport a Grups de Recerca 1997SGR 00185.
Supported by KBN Grant 2PO3A 03110.
Supported by KBN Grant 2PO3A 05009.