Integration, geometry and topology in Banach spaces

Laburpena

The thesis is composed of three chapters and two appendices. In the first chapter we study the Riemann integral of vector-valued functions. Lebesgue's Criterion of Riemann integrability fails in general for functions taking values in a Banach space. We say that a Banach space X has the Lebesgue property (resp. weak Lebesgue property) or LP (resp. WLP) if every Riemann integrable function from [0,1] to X is norm-continuous (resp. weak-continuous) a.e. All classical infinite-dimensional Banach spaces except l_1 do not have the LP. Nevertheless, every Banach space with separable dual has the WLP. The main aim of this chapter is to study the LP and the WLP and to obtain new characterizations and examples in order to understand better these properties. The main results of this chapter are: -The James tree space does not have the WLP. -The WLP is stable under l_1-sums. -C(K)* has the WLP whenever K is a compact space in the class MS. -We answer a question of M.A. Sofi concerning the existence of weak*-continuous nonRiemann integrable functions. In the second chapter we study properties which relate families of Banach spaces to their weak*-compact sets. In particular, a compact space is said to be Radon-Nikodym (resp. weakly Radon-Nikodym) if it is homeomorphic to a weak*-compact subset of a dual Banach space with the Radon-Nikodym property (resp. with the weak Radon-Nikodym property). This line of research focuses on studying topological properties of (weakly) Radon-Nikodym compact spaces and their relation with other classes of compact spaces. In this chapter we show the following results: -The continuous image of a Radon-Nikodym compact space may not be weakly Radon-Nikodym. In particular, this result answers a question of E. Glasner and M. Megrelishvili. -Every zero-dimensional continuous image of a weakly Radon-Nikodym compact space is weakly Radon-Nikodym. -Talagrand's compact is not a continuous image of a weakly Radon-Nikodym compact space. -The class of Rosenthal compact spaces and the class of weakly Radon-Nikodym compact spaces are uncomparable. The third chapter studies different sequential properties in topological spaces. In particular, we study sequential compactness and sequentiality in dual Banach spaces with the weak*-topology. There is no characterization of Banach spaces with weak*-sequentially compact dual ball and none of the classes presently under study offers a viable candidate for such a characterization. J. Hagler, E. Odell and R. Haydon provided examples of Banach spaces containing no copy of l_1 but such that their dual ball is not weak*-sequentially compact. Motivated by these examples, R. Haydon asked whether every infinite weakly Radon-Nikodym compact space has a nontrivial convergent sequence. The main goals of this line of research are to study Banach spaces with weak*-sequentially compact dual ball or weak*-sequential dual ball and to answer Haydon's question. Furthermore, we study different convex versions of these properties. The main results in this line are the following: -We obtain sufficient conditions for a Banach space to have weak*-sequentially compact dual ball. -We provide a partial answer to Haydon's question. -We answer a question of A. Plichko about the existence of Banach spaces with weak*-sequential dual ball nonFréchet-Urysohn. In particular, we prove that Johnson-Lindenstrauss space JL_2 and every Banach space C(K) with K an scattered compact space of countable height have weak*-sequential dual ball.