## Normal structure and weakly normal structure of Orlicz sequence spaces.(English)Zbl 0594.46010

Summary: For a convex Orlicz function $$\phi:{\mathbb{R}}_+\to {\mathbb{R}}_+\cup \{\infty \}$$ and the associated Orlicz sequence space $$\ell_{\phi}$$, we consider the following five properties:
(1) $$\ell_{\phi}$$ has a subspace isometric to $$\ell_ 1.$$
(2) $$\ell_{\phi}$$ is Schur.
(3) $$\ell_{\phi}$$ has normal structure.
(4) Every weakly compact subset of $$\ell_{\phi}$$ has normal structure.
(5) Every bounded sequence in $$\ell_{\phi}$$ has a subsequence $$(x_ n)$$ which is pointwise and almost convergent to $$x\in \ell_{\phi}$$, i.e., $$\limsup_{n\to \infty}\| x_ n-x\|_{\phi}<\lim \inf_{n\to \infty}\| x_ n-y\|_{\phi \quad}$$ for all $$y\neq x.$$
Our results are:
(1) $$\Leftrightarrow$$ $$\phi$$ is either linear at $$0(\phi (s)/s=c>0$$, $$0<s\leq t)$$ or does not satisfy the $$\Delta_ 2$$-condition at 0.
(2) $$\Leftrightarrow$$ $$\ell_{\phi}$$ is isomorphic to $$\ell_ 1\Leftrightarrow \phi '(0)=\lim_{t\to 0}\phi (t)/t>0.$$
(3) $$\Leftrightarrow$$ $$\phi$$ satisfies the $$\Delta_ 2$$-condition at 0, $$\phi$$ is not linear at 0 and $$C(\phi)=\sup (\phi (t)<1)>.$$
(4) $$\Leftrightarrow$$ $$\phi$$ satisfies the $$\Delta_ 2$$-condition at 0 and $$C(\phi)>$$ or $$\phi '(0)>0.$$
(5) $$\Leftrightarrow$$ $$\phi$$ satisfies the $$\Delta_ 2$$-condition at 0 and $$C(\phi)=1.$$
The last equivalence contains a result of E. Lami-Dozo [Lect. Notes Math. 886, 199-207 (1981; Zbl 0473.47043)].

### MSC:

 46B20 Geometry and structure of normed linear spaces 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 46B25 Classical Banach spaces in the general theory 46A45 Sequence spaces (including Köthe sequence spaces)

### Keywords:

convex Orlicz function; Orlicz sequence; normal structure

Zbl 0473.47043
Full Text:

### References:

  J.-B. Baillon and R. Schöneberg, Asymptotic normal structure and fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 81 (1981), no. 2, 257 – 264. · Zbl 0465.47038  M. S. Brodskiĭ and D. P. Mil$$^{\prime}$$man, On the center of a convex set, Doklady Akad. Nauk SSSR (N.S.) 59 (1948), 837 – 840 (Russian).  Felix E. Browder, Nonlinear mappings of nonexpansive and accretive type in Banach spaces., Bull. Amer. Math. Soc. 73 (1967), 875 – 882. · Zbl 0176.45302  Michael Edelstein, The construction of an asymptotic center with a fixed-point property, Bull. Amer. Math. Soc. 78 (1972), 206 – 208. · Zbl 0231.47029  Kazimierz Goebel, On a fixed point theorem for multivalued nonexpansive mappings, Ann. Univ. Mariae Curie-Skłodowska Sect. A 29 (1975), 69 – 72 (1977) (English, with Polish and Russian summaries). · Zbl 0365.47032  Dietrich Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251 – 258 (German). · Zbl 0127.08005  J.-P. Gossez and E. Lami Dozo, Structure normale et base de Schauder, Acad. Roy. Belg. Bull. Cl. Sci. (5) 55 (1969), 673 – 681 (French, with English summary). · Zbl 0192.46903  W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004 – 1006. · Zbl 0141.32402  E. Lami Dozo, Centres asymptotiques dans certains \?-espaces, Boll. Un. Mat. Ital. B (5) 17 (1980), no. 2, 740 – 747 (French, with Italian summary). · Zbl 0456.47049  E. Lami Dozo, Asymptotic centers in particular spaces, Lecture Notes in Math., vol. 886, Springer, Berlin-New York, 1981, pp. 199 – 207. · Zbl 0473.47043  Thomas Landes, A characterization of totally normal structure, Arch. Math. (Basel) 37 (1981), no. 3, 248 – 255. · Zbl 0452.46005  Thomas Landes, Permanence properties of normal structure, Pacific J. Math. 110 (1984), no. 1, 125 – 143. · Zbl 0534.46015  Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Springer-Verlag, Berlin-New York, 1977. Sequence spaces; Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92. · Zbl 0362.46013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.