We study the invertibility nonsmooth maps between infinite-dimensional Banach spaces. To this end, we introduce an analogue of the notion of pseudo-Jacobian matrix of Jeyakumar and Luc in this infinite-dimensional setting. Using this, we obtain several inversion results. In particular, we give a version of the classical Hadamard integral condition for global invertibility in this context.

Abstract. We study compactness and related topological properties in the space L 1 (m) of a Banach space valued measure m when the natural topologies associated to the convergence of the vector valued integrals are considered. The resulting topological spaces are shown to be angelic and the relationship of compactness and equi-integrability is explored. A natural norming subset of the dual unit ball of L 1 (m) appears in our discussion and we study when it is a boundary. The (almost) complete continuity of the integration operator is analyzed in relation with the positive Schur property of L 1 (m). The strong weakly compact generation of L 1 (m) is discussed as well.

We study strongly Asplund generated (SAG) and strongly conditionally weakly compactly generated (SCWCG) Banach spaces. These spaces are defined like the strongly weakly compactly generated (SWCG) Banach spaces of Schlüchtermann and Wheeler, but replacing weakly compact sets by Asplund sets and conditionally weakly compact sets, respectively. We show that every SAG space is SCWCG and that a Banach space is SWCG if and only if it is SAG/SCWCG and weakly sequentially complete. We also prove that the notions of SAG and SCWCG space coincide for Banach lattices. Some related results on Lebesgue-Bochner spaces are also given. We prove that if the norm of the Banach space X is weakly uniformly rotund (WUR) and μ is any probability measure, then L 1 (μ, X ) admits an equivalent norm which is WUR when restricted to any Asplund subspace of L 1 (μ, X ).

Abstract. We study the different ways in which a weakly compact set can generate a Banach lattice. Among other things, it is shown that in an order continuous Banach lattice X, the existence of a weakly compact set K ⊂ X such that X coincides with the band generated by K, implies that X is WCG. The general problemThe purpose of this note is to study Banach lattices which are generated in one way or another by a weakly compact set. Namely, we will explore the connection between the existence of a weakly compact set which generates a Banach lattice as a linear space, a lattice, an ideal or a band. Our motivation starts with the question of J. Diestel of whether every Banach lattice which is generated, as a lattice, by a weakly compact set must be weakly compactly generated (i.e., as a linear space).Recall that a Banach lattice is a Banach space endowed with additional order and lattice structures which behave well with respect to the norm and linear structure. This is in particular highlighted by the fact that x ≤ y whenever |x| ≤ |y|, or by the norm continuity of the lattice operations ∧ and ∨. However, for the weak topology, the relation with the order and lattice structures is more subtle, in particular it is not always true that the lattice operations are weakly continuous. In fact, on infinite dimensional Banach lattices the weak topology fails to be locally solid (see e.g. [1, Theorem 6.9]).A Banach space X is called weakly compactly generated (WCG) whenever there exists a weakly compact subset of X whose closed linear span coincides with X. This class of Banach spaces was first studied by Corson [12] and it was pushed further by the fundamental work of Amir and Lindenstrauss [5]. Nowadays, WCG spaces play a relevant role in non-separable Banach space theory. For complete information on WCG spaces, see [17,22,35].Weakly compact sets and weakly compact operators in Banach lattices have been the object of research by several authors (cf. [2,3,11,28]

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