We prove that every Banach space, not necessarily separable, can be isometrically embedded into a $mathcal L_{infty}$-space in a way that the corresponding quotient has the Radon-Nikodym and the Schur properties. As a consequence, we obtain $mathcal
L_infty$ spaces of arbitrary large densities with the Schur and the Radon-Nikodym properties. This extents the a classical result by J. Bourgain and G. Pisier.
A subset $A$ of a Banach space is called Banach-Saks when every sequence in $A$ has a Ces{`a}ro convergent subsequence. Our interest here focusses on the following problem: is the convex hull of a Banach-Saks set again Banach-Saks? By means of a comb
inatorial argument, we show that in general the answer is negative. However, sufficient conditions are given in order to obtain a positive result.
We prove that, unless assuming additional set theoretical axioms, there are no reflexive space without unconditional sequences of density the continuum. We give for every integer $n$ there are normalized weakly-null sequences of length $om_n$ without
unconditional subsequences. This together with a result of cite{Do-Lo-To} shows that $om_omega$ is the minimal cardinal $kappa$ that could possibly have the property that every weakly null $kappa$-sequence has an infinite unconditional basic subsequence . We also prove that for every cardinal number $ka$ which is smaller than the first $om$-Erdos cardinal there is a normalized weakly-null sequence without subsymmetric subsequences. Finally, we prove that mixed Tsirelson spaces of uncountable densities must always contain isomorphic copies of either $c_0$ or $ell_p$, with $pge 1$.
We study the problem of the existence of unconditional basic sequences in Banach spaces of high density. We show, in particular, the relative consistency with GCH of the statement that every Banach space of density $aleph_omega$ contains an unconditional basic sequence.
