### A Bator's question on dual Banach spaces.

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The equivalence of the two following properties is proved for every Banach lattice $E$:1) $E$ is weakly sequentially complete.2) Every $\sigma ({E}^{*},E)$-Borel measurable linear functional on $E$ is $\sigma ({E}^{*},E)$-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 ${c}_{0}$-saturated, but whose dual is ${\ell}^{1}$-saturated.

The notion of the Orlicz space is generalized to spaces of Banach-space valued functions. A well-known generalization is based on $N$-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 ${\mathcal{L}}^{\infty}$, and representations of the dual space.