### A characterization of the duals of some echelon Köthe spaces of Banach valued functions.

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For Banach-space-valued functions, the concepts of 𝒫-measurability, λ-measurability and m-measurability are defined, where 𝒫 is a δ-ring of subsets of a nonvoid set T, λ is a σ-subadditive submeasure on σ(𝒫) and m is an operator-valued measure on 𝒫. Various characterizations are given for 𝒫-measurable (resp. λ-measurable, m-measurable) vector functions on T. Using them and other auxiliary results proved here, the basic theorems of [6] are rigorously established.

In this paper, a precise projection decomposition in reflexive, smooth and strictly convex Orlicz-Bochner spaces is given by the representation of the duality mapping. As an application, a representation of the metric projection operator on a closed hyperplane is presented.

We prove unconditionality of general Franklin systems in ${L}^{p}\left(X\right)$, where X is a UMD space and where the general Franklin system corresponds to a quasi-dyadic, weakly regular sequence of knots.

In the present work we prove that, in the space of Pettis integrable functions, any subset that is decomposable and closed with respect to the topology induced by the so-called Alexiewicz norm $\left|\u2225\xb7\u2225\right|$ (where $\left|\u2225f\u2225\right|={sup}_{[a,b]\subset [0,1]}\parallel {\int}_{a}^{b}f\left(s\right)ds\parallel $) is convex. As a consequence, any such family of Pettis integrable functions is also weakly closed.

If $(\Omega ,\Sigma ,\mu )$ is a finite measure space and $X$ a Banach space, in this note we show that ${L}_{{w}^{*}}^{1}(\mu ,{X}^{*})$, the Banach space of all classes of weak* equivalent ${X}^{*}$-valued weak* measurable functions $f$ defined on $\Omega $ such that $\parallel f\left(\omega \right)\parallel \le g\left(\omega \right)$ a.e. for some $g\in {L}_{1}\left(\mu \right)$ equipped with its usual norm, contains a copy of ${c}_{0}$ if and only if ${X}^{*}$ contains a copy of ${c}_{0}$.

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.

We prove the following quasi-dichotomy involving the Banach spaces C(α,X) of all X-valued continuous functions defined on the interval [0,α] of ordinals and endowed with the supremum norm. Suppose that X and Y are arbitrary Banach spaces of finite cotype. Then at least one of the following statements is true. (1) There exists a finite ordinal n such that either C(n,X) contains a copy of Y, or C(n,Y) contains a copy of X. (2) For any infinite countable...