Deltell, Luis Oncina; (1997) Borel sets and σ-fragmentability of a Banach space. Masters thesis (M.Phil), UCL (University College London).

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Abstract

In this thesis we give a sufficient condition on a Banach space for it to have the same weak and norm Borel sets and to be a Borel subset of its bidual, when the latter is endowed with the weak* topology. We also deal with one-to-one maps between Banach spaces, say from X into Y, when Y has a countable cover by sets of small local diameter. Under these conditions we are able to characterize those maps which transfer that property to X. We use this kind of map to show that certain spaces have a countable cover by sets of small local diamter and to answer some questions on Co-sums of Banach spaces and on topological invariants for the weak topology as well as some questions related to C(K) spaces. We also study the inverses of some of these maps. Finally we construct injections of this type into co(Γ) for spaces with Projectional Resolutions of Identity.

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