### Absolutely summing operators between Banach spaces of finite cotype.

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The spaces L¹(m) of all m-integrable (resp. $L{\xb9}_{w}\left(m\right)$ of all scalarly m-integrable) functions for a vector measure m, taking values in a complex locally convex Hausdorff space X (briefly, lcHs), are themselves lcHs for the mean convergence topology. Additionally, $L{\xb9}_{w}\left(m\right)$ is always a complex vector lattice; this is not necessarily so for L¹(m). To identify precisely when L¹(m) is also a complex vector lattice is one of our central aims. Whenever X is sequentially complete, then this is the case. If, additionally,...

We study compactness and related topological properties in the space L¹(m) of a Banach space valued measure m when the natural topologies associated to convergence of 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 L¹(m) appears in our discussion and we study when it is a boundary. The (almost) complete continuity of the integration...

We obtain theorems of metrization and quasi-metrization for several topologies of weak* type on the unit ball of the dual of any separable quasi-normed cone. This is done with the help of an appropriate version of the Alaoglu theorem which is also obtained here.

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