## 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:

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