A Bator's question on dual Banach spaces.
The equivalence of the two following properties is proved for every Banach lattice :1) is weakly sequentially complete.2) Every -Borel measurable linear functional on is -continuous.
We provide a new proof of James' sup theorem for (non necessarily separable) Banach spaces. One of the ingredients is the following generalization of a theorem of Hagler and Johnson: "If a normed space E does not contain any asymptotically isometric copy of l1, then every bounded sequence of E' has a normalized l1-block sequence pointwise converging to 0".
It is shown that there exists a Banach space with an unconditional basis which is not -saturated, but whose dual is -saturated.
The notion of the Orlicz space is generalized to spaces of Banach-space valued functions. A well-known generalization is based on -functions of a real variable. We consider a more general setting based on spaces generated by convex functions defined on a Banach space. We investigate structural properties of these spaces, such as the role of the delta-growth conditions, separability, the closure of , and representations of the dual space.