ABSTRACT
Chapter 1. After some preliminaries dealing with adequate
families of sets, we formulate and prove some equivalences, all of
them implying that the family defines a Gul'ko compactum. We
provide a characterization of Gul'ko compacta in terms of
pairings.
Chapter 2. Deals with the class of non-separable weakly
Lindelöf determined Banach spaces and their relatives. We give a
characterization of weakly Lindelöf determined Banach spaces by
mean of the existence of a full projectional generator on it. We
study some remarks on biorthogonal systems in Banach spaces. We
prove by the technology of PRI's, an extension of a
result due to Argyros and Mercourakis.
Chapter 3. For $(c_0(\Gamma),\|\cdot\|_\infty)$, with
$\Gamma\subset\mathbb{R}$, we provide an equivalent norm on $c_0(\Gamma)$ that
is strictly convex.
Chapter 4. We consider a characterization of subspaces of
the class of weakly compactly generated Banach spaces in terms of
a covering property of the closed unit ball, by means of
$\epsilon$-weakly compact sets. We replace this concept by a more
precise one that we call $\epsilon$-weakly
self-compactness, this concept allows a better description.
Chapter 5. We give intrinsic necessary and sufficient
conditions for a Banach space $X$ to be generated by $c_0(\Gamma)$
or $\ell_p(\Gamma)$ for $p\in(1,+\infty)$. As a byproduct we give
a new proof of a result of Rosenthal on fixing copies from
$c_0(\Gamma)$ into Banach spaces.