A sufficient and necessary condition for weak convergence of sequences in a class of Banach sequence lattices is obtained. As a direct application, a complete criterion of a weak convergence of sequences in l infinity is formulated.
In this paper, we give necessary and sufficient conditions for a point in a Musielak-Orlicz sequence space equipped with the Orlicz norm to be an H-point. We give necessary and sufficient conditions for a Musielak-Orlicz sequence space equipped with the Orlicz norm to have the Kadec-Klee property, the uniform Kadec-Klee property and to be nearly uniformly convex. We show that a Musielak-Orlicz sequence space equipped with the Orlicz norm has the fixed point property if and only if it is reflexive....
It is known that a Banach lattice with order continuous norm contains a copy of if and only if it contains a lattice copy of . The purpose of this note is to present a more direct proof of this useful fact, which extends a similar theorem due to R.C. James for Banach spaces with unconditional bases, and complements the - and -cases considered by Lozanovskii, Mekler and Meyer-Nieberg.
Let (S, ∑, m) be any atomless finite measure space, and X any Banach space containing a copy of . Then the Bochner space is uncomplemented in ccabv(∑,m;X), the Banach space of all m-continuous vector measures that are of bounded variation and have a relatively compact range; and ccabv(∑,m;X) is uncomplemented in cabv(∑,m;X). It is conjectured that this should generalize to all Banach spaces X without the Radon-Nikodym property.
A characterization of property of an arbitrary Banach space is given. Next it is proved that the Orlicz-Bochner sequence space has the property if and only if both spaces and have it also. In particular the Lebesgue-Bochner sequence space has the property iff has the property . As a corollary we also obtain a theorem proved directly in [5] which states that in Orlicz sequence spaces equipped with the Luxemburg norm the property , nearly uniform convexity, the drop property and...