Normal Structure and Weakly Normal Structure of Orlicz Sequence Spaces
Thomas Landes · Transactions of the American Mathematical Society · 1984
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., $\lim {\sup _{n \to \infty }}\parallel {x_n} - x{\parallel _{\varphi }} 0,0 0$. (3) $\Leftrightarrow \varphi$ satisfies the ${\Delta _2}$-condition at $0, \varphi$ is not linear at $0$ and $C(\varphi ) = \sup \{ \varphi (t) \frac {1}{2}$. (4) $\Leftrightarrow \varphi$ satisfies the ${\Delta _2}$-condition at $0$ and $C (\varphi ) > \frac {1}{2}\;{\rm {or}}\;\varphi ’(0) > 0$. (5) $\Leftrightarrow \;\varphi$ satisfies the ${\Delta _2}$-condition at $0$ and $C(\varphi ) = 1$. The last equivalence contains a result of Lami-Dozo [10].