Stable points of unit ball in Orlicz spaces

Marek Wisła

Commentationes Mathematicae Universitatis Carolinae (1991)

  • Volume: 32, Issue: 3, page 501-515
  • ISSN: 0010-2628

Abstract

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The aim of this paper is to investigate stability of unit ball in Orlicz spaces, endowed with the Luxemburg norm, from the “local” point of view. Firstly, those points of the unit ball are characterized which are stable, i.e., at which the map z → { ( x , y ) : 1 2 ( x + y ) = z } is lower-semicontinuous. Then the main theorem is established: An Orlicz space L ϕ ( μ ) has stable unit ball if and only if either L ϕ ( μ ) is finite dimensional or it is isometric to L ∞ ( μ ) or ϕ satisfies the condition Δ r or Δ r 0 (appropriate to the measure μ and the function ϕ ) or c ( ϕ ) < ∞ , ϕ ( c ( ϕ ) ) < ∞ and μ ( T ) < ∞ . Finally, it is proved that the set of all stable points of norm one is dense in the unit sphere S ( L ϕ ( μ ) ) .

How to cite

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Wisła, Marek. "Stable points of unit ball in Orlicz spaces." Commentationes Mathematicae Universitatis Carolinae 32.3 (1991): 501-515. <http://eudml.org/doc/247274>.

@article{Wisła1991,
abstract = {The aim of this paper is to investigate stability of unit ball in Orlicz spaces, endowed with the Luxemburg norm, from the “local” point of view. Firstly, those points of the unit ball are characterized which are stable, i.e., at which the map $z\rightarrow \lbrace (x,y):\frac\{1\}\{2\}(x+y)=z\rbrace $ is lower-semicontinuous. Then the main theorem is established: An Orlicz space $L^\{\varphi \}(\mu )$ has stable unit ball if and only if either $L^\{\varphi \}(\mu )$ is finite dimensional or it is isometric to $L^\{\infty \}(\mu )$ or $\varphi $ satisfies the condition $\Delta _r$ or $\Delta _r^0$ (appropriate to the measure $\mu $ and the function $\varphi $) or $c(\varphi )<\infty , \varphi (c(\varphi ))<\infty $ and $\mu (T)<\infty $. Finally, it is proved that the set of all stable points of norm one is dense in the unit sphere $S(L^\{\varphi \}(\mu ))$.},
author = {Wisła, Marek},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {stable point; stable unit ball; extreme point; Orlicz space; extreme point; stability of the unit ball in Orlicz spaces, endowed with the Luxemburg norm; lower-semicontinuous; stable points of norm},
language = {eng},
number = {3},
pages = {501-515},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Stable points of unit ball in Orlicz spaces},
url = {http://eudml.org/doc/247274},
volume = {32},
year = {1991},
}

TY - JOUR
AU - Wisła, Marek
TI - Stable points of unit ball in Orlicz spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1991
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 32
IS - 3
SP - 501
EP - 515
AB - The aim of this paper is to investigate stability of unit ball in Orlicz spaces, endowed with the Luxemburg norm, from the “local” point of view. Firstly, those points of the unit ball are characterized which are stable, i.e., at which the map $z\rightarrow \lbrace (x,y):\frac{1}{2}(x+y)=z\rbrace $ is lower-semicontinuous. Then the main theorem is established: An Orlicz space $L^{\varphi }(\mu )$ has stable unit ball if and only if either $L^{\varphi }(\mu )$ is finite dimensional or it is isometric to $L^{\infty }(\mu )$ or $\varphi $ satisfies the condition $\Delta _r$ or $\Delta _r^0$ (appropriate to the measure $\mu $ and the function $\varphi $) or $c(\varphi )<\infty , \varphi (c(\varphi ))<\infty $ and $\mu (T)<\infty $. Finally, it is proved that the set of all stable points of norm one is dense in the unit sphere $S(L^{\varphi }(\mu ))$.
LA - eng
KW - stable point; stable unit ball; extreme point; Orlicz space; extreme point; stability of the unit ball in Orlicz spaces, endowed with the Luxemburg norm; lower-semicontinuous; stable points of norm
UR - http://eudml.org/doc/247274
ER -

References

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