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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Normal structure and weakly normal structure of Orlicz sequence spaces


Author: Thomas Landes
Journal: Trans. Amer. Math. Soc. 285 (1984), 523-534
MSC: Primary 46B20; Secondary 46A45, 47H10
MathSciNet review: 752489
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Abstract: For a convex Orlicz function $ \varphi :{{\bf {R}}_ + } \to {{\bf {R}}_ + } \cup \{ \infty \} $ and the associated Orlicz sequence space $ {l_\varphi }$, we consider the following five properties:

(1) $ {l_\varphi }$ has a subspace isometric to $ {l_1}$.

(2) $ {l_\varphi }$ is Schur.

(3) $ {l_\varphi }$ has normal structure.

(4) Every weakly compact subset of $ {l_\varphi }$ has normal structure.

(5) Every bounded sequence in $ {l_\varphi }$ has a subsequence $ ({x_n})$ which is pointwise and almost convergent to $ x \in {l_\varphi }$, i.e., $ \lim \,{\sup_{n \to \infty }}\parallel {x_n} - x{\parallel_{\varphi }} < \lim \inf _{n \to \infty }\parallel {x_n} - y{\parallel_\varphi }$ for all $ y \ne x$.

Our results are:

(1) $ \Leftrightarrow \;\varphi $ is either linear at $ 0\;(\varphi (s)/s = c > 0,0 < s \leqslant t)$ or does not satisfy the $ {\Delta_2}$-condition at 0.

(2) $ \Leftrightarrow \;{l_\varphi }$ is isomorphic to $ {l_1}\; \Leftrightarrow \;\varphi^{\prime}(0) = {\lim_{t \to 0}}\,\varphi \,(t)/t > 0$.

(3) $ \Leftrightarrow \varphi $ satisfies the $ {\Delta_2}$-condition at $ 0, \varphi $ is not linear at 0 and $ C(\varphi ) = \sup \,\{ \varphi \,(t) < 1\} > \frac{1}{2}$.

(4) $ \Leftrightarrow \,\varphi $ satisfies the $ {\Delta_2}$-condition at 0 and $ C\,(\varphi ) > \frac{1}{2}\;{\rm {or}}\;\varphi^{\prime}(0) > 0$.

(5) $ \Leftrightarrow \;\varphi $ satisfies the $ {\Delta_2}$-condition at 0 and $ C(\varphi ) = 1$.

The last equivalence contains a result of Lami-Dozo [10].


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0752489-1
PII: S 0002-9947(1984)0752489-1
Article copyright: © Copyright 1984 American Mathematical Society