| Title:
|
Stable points of unit ball in Orlicz spaces (English) |
| Author:
|
Wisła, Marek |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
32 |
| Issue:
|
3 |
| Year:
|
1991 |
| Pages:
|
501-515 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $z\rightarrow \{(x,y):\frac{1}{2}(x+y)=z\}$ 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 $S(L^{\varphi }(\mu ))$. (English) |
| Keyword:
|
stable point |
| Keyword:
|
stable unit ball |
| Keyword:
|
extreme point |
| Keyword:
|
Orlicz space |
| MSC:
|
46B20 |
| MSC:
|
46E30 |
| idZBL:
|
Zbl 0770.46013 |
| idMR:
|
MR1159798 |
| . |
| Date available:
|
2009-01-08T17:46:36Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/116986 |
| . |
| Reference:
|
[1] Clausing A., Papadopoulou S.: Stable convex sets and extreme operators.Math. Ann. 231 (1978), 193-200. MR 0467249 |
| Reference:
|
[2] Engelking R.: General Topology.Polish Scientific Publishers, Warsaw, 1977. Zbl 0684.54001, MR 0500780 |
| Reference:
|
[3] Grząślewicz R.: Finite dimensional Orlicz spaces.Bull. Acad. Polon. Sci.: Math. 33 (1985), 277-283. MR 0816376 |
| Reference:
|
[4] Lazar A.J.: Affine functions on simplexes and extreme operators.Israel J. Math. 5 (1967), 31-43. Zbl 0149.08703, MR 0211246 |
| Reference:
|
[5] Lima Å.: On continuous convex functions and split faces.Proc. London Math. Soc. 25 (1972), 27-40. Zbl 0236.46024, MR 0303243 |
| Reference:
|
[6] Luxemburg W.A.J.: Banach function spaces.Thesis, Delft, 1955. Zbl 0162.44701, MR 0072440 |
| Reference:
|
[7] Michael E.: Continuous selections I.Ann. of Math. (2) 63 (1956), 361-382. Zbl 0071.15902, MR 0077107 |
| Reference:
|
[8] Musielak J.: Orlicz spaces and modular spaces.Lecture Notes in Math. 1034, Springer Verlag, 1983. Zbl 0557.46020, MR 0724434 |
| Reference:
|
[9] O'Brien R.C.: On the openness of the barycentre map.Math. Ann. 223 (1976), 207-212. Zbl 0321.46004, MR 0420221 |
| Reference:
|
[10] Orlicz W.: Über eine gewisse Klasse von Räumen vom Typus B.Bull. Intern. Acad. Pol., série A, Kraków, 1932, 207-220. Zbl 0006.31503 |
| Reference:
|
[11] Papadopoulou S.: On the geometry of stable compact convex sets.Math. Ann. 229 (1977), 193-200. Zbl 0339.46001, MR 0450938 |
| Reference:
|
[12] Suarez-Granero A.: Stable unit balls in Orlicz spaces.Proc. Amer. Math. Soc. 109, 1 (1990), 97-104. Zbl 0722.46014, MR 1000154 |
| Reference:
|
[13] Vesterstrøm J.: On open maps, compact convex sets and operator algebras.J. London Math. Soc. 6 (1973), 289-297. MR 0315464 |
| Reference:
|
[14] Wisła M.: Extreme points and stable unit balls in Orlicz sequence spaces.Archiv der Math. 56 (1991), 482-490. MR 1100574 |
| Reference:
|
[15] Wisła M.: Stable unit balls in finite dimensional generalized Orlicz spaces.Proceedings of the Second Conference ``Function Spaces'', Poznań, 1989, Teubner Texte zur Mathematik, to appear. MR 1155158 |
| Reference:
|
[16] Wisła M.: Continuity of the identity embedding of Musielak-Orlizc sequence spaces.Proc. of the 14th Winter School on Abstract Analysis, Srní, 1986, Supp. ai Rendiconti del Circolo Mat. di Palermo 14 (1987), 427-437. MR 0920876 |
| . |
