Normal Structure and Weakly Normal Structure of Orlicz Sequence Spaces

• Published in: Transactions of the American Mathematical Society Vol. 285; no. 2; pp. 523 - 534
• Main Author: Landes, Thomas
• Format: Journal Article
• Language: English
• Published: American Mathematical Society 1984
• Subjects: Banach space; Diagonal lemma; Mathematical sequences; Perceptron convergence procedure; Scalars
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-1984-0752489-1

For a convex Orlicz function $\varphi: R_+ \rightarrow R_+ \cup\{\infty\}$ 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., $\lim \sup_{n \rightarrow \infty}||x_n - x||_\varphi < \lim \inf_{n \rightarrow \infty}|| x_n - y||_\varphi$ for all $y \neq x$. 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 $l_1 \Leftrightarrow \varphi'(0) = \lim_{t \rightarrow 0}\varphi(t)/t > 0$. (3) $\Leftrightarrow \varphi$ satisfy 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 $C(\varphi) > \frac{1}{2}$ or $\varphi'(0) > 0$. (5) $\Leftrightarrow \varphi$ satisfies the $\Delta_2$-condition at 0 and $C(\varphi) = 1$. The last equivalence contains a result of Lami-Dozo [10].