# Normal structure of Lorentz-Orlicz spaces

Annales Polonici Mathematici (1997)

- Volume: 67, Issue: 2, page 147-168
- ISSN: 0066-2216

## Access Full Article

top## Abstract

top## How to cite

topPei-Kee Lin, and Huiying Sun. "Normal structure of Lorentz-Orlicz spaces." Annales Polonici Mathematici 67.2 (1997): 147-168. <http://eudml.org/doc/270441>.

@article{Pei1997,

abstract = {Let ϕ: ℝ → ℝ₊ ∪ 0 be an even convex continuous function with ϕ(0) = 0 and ϕ(u) > 0 for all u > 0 and let w be a weight function. u₀ and v₀ are defined by
u₀ = supu: ϕ is linear on (0,u), v₀=supv: w is constant on (0,v)
(where sup∅ = 0). We prove the following theorem.
Theorem. Suppose that $Λ_\{ϕ,w\}(0,∞)$ (respectively, $Λ_\{ϕ,w\}(0,1)$) is an order continuous Lorentz-Orlicz space.
(1) $Λ_\{ϕ,w\}$ has normal structure if and only if u₀ = 0 (respectively, $∫_^\{v₀\} ϕ(u₀) · w < 2 and u₀ <∞).
$(2) $Λ_\{ϕ,w\}$ has weakly normal structure if and only if $∫_0^\{v₀\} ϕ(u₀)· w < 2$.},

author = {Pei-Kee Lin, Huiying Sun},

journal = {Annales Polonici Mathematici},

keywords = {Lorentz-Orlicz space; normal sturcture; order continuous; Young function; normal structure; weight function; order continuous Lorentz-Orlicz space},

language = {eng},

number = {2},

pages = {147-168},

title = {Normal structure of Lorentz-Orlicz spaces},

url = {http://eudml.org/doc/270441},

volume = {67},

year = {1997},

}

TY - JOUR

AU - Pei-Kee Lin

AU - Huiying Sun

TI - Normal structure of Lorentz-Orlicz spaces

JO - Annales Polonici Mathematici

PY - 1997

VL - 67

IS - 2

SP - 147

EP - 168

AB - Let ϕ: ℝ → ℝ₊ ∪ 0 be an even convex continuous function with ϕ(0) = 0 and ϕ(u) > 0 for all u > 0 and let w be a weight function. u₀ and v₀ are defined by
u₀ = supu: ϕ is linear on (0,u), v₀=supv: w is constant on (0,v)
(where sup∅ = 0). We prove the following theorem.
Theorem. Suppose that $Λ_{ϕ,w}(0,∞)$ (respectively, $Λ_{ϕ,w}(0,1)$) is an order continuous Lorentz-Orlicz space.
(1) $Λ_{ϕ,w}$ has normal structure if and only if u₀ = 0 (respectively, $∫_^{v₀} ϕ(u₀) · w < 2 and u₀ <∞).
$(2) $Λ_{ϕ,w}$ has weakly normal structure if and only if $∫_0^{v₀} ϕ(u₀)· w < 2$.

LA - eng

KW - Lorentz-Orlicz space; normal sturcture; order continuous; Young function; normal structure; weight function; order continuous Lorentz-Orlicz space

UR - http://eudml.org/doc/270441

ER -

## References

top- [1] N. L. Carothers, S. J. Dilworth, C. J. Lennard and D. A. Trautman, A fixed point property for the Lorentz space ${L}_{p,1}\left(\mu \right)$, Indiana Univ. Math. J. 40 (1991), 345-352. Zbl0736.47029
- [2] N. L. Carothers, R. Haydon and P.-K. Lin, On the isometries of the Lorentz function spaces, Israel J. Math. 84 (1993), 265-287. Zbl0799.46024
- [3] S. Chen, Geometry of Orlicz spaces, Dissertationes Math. 356 (1996).
- [4] J. Diestel, Sequences and Series in Banach Spaces, Springer, 1984.
- [5] S. J. Dilworth and Y.-P. Hsu, The uniform Kadec-Klee property for the Lorentz space ${L}_{w,1}$, J. Austral. Math. Soc. Ser. A 60 (1996), 7-17. Zbl0852.46030
- [6] D. V. van Dulst and V. D. de Valk, (KK)-properties, normal structure and fixed points of nonexpansive mappings in Orlicz sequence spaces, Canad. J. Math. 38 (1986), 728-750. Zbl0615.46016
- [7] A. Kamińska, Some remarks on Orlicz-Lorentz spaces, Math. Nachr. 147 (1990), 29-38. Zbl0742.46013
- [8] A. Kamińska, P.-K. Lin and H. Y. Sun, Uniformly normal structure of Orlicz-Lorentz spaces, in: Interaction between Functional Analysis, Harmonic Analysis, and Probability, N. Kalton, E. Saab and S. Montgomery-Smith (eds.), Lecture Notes in Pure and Appl. Math. 175, Dekker, New York, 1996, 229-238. Zbl0856.46015
- [9] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006. Zbl0141.32402
- [10] T. Landes, Permanence properties of normal structure, Pacific J. Math. 110 (1984), 125-143. Zbl0534.46015
- [11] T. Landes, Normal structure and weakly normal structure of Orlicz sequence spaces, Trans. Amer. Math. Soc. 285 (1984), 523-534. Zbl0594.46010
- [12] P.-K. Lin and H. Y. Sun, Some geometric properties of Lorentz-Orlicz spaces, Arch. Math. (Basel) 64 (1995), 500-511. Zbl0823.46019
- [13] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Springer, 1979. Zbl0403.46022
- [14] M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker, 1991.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.