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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)

Citations:

Zbl 0473.47043
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References:

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