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Let $X$ be a sequence space and denote by $Z(X)$ the subset of $X$ formed by sequences having only a finite number of zero coordinates. We study algebraic properties of $Z(X)$ and show (among other results) that (for $p in [1,infty]$) $Z(ell_p)$ does not contain infinite dimensional closed subspaces. This solves an open question originally posed by R. M. Aron and V. I. Gurariy in 2003 on the linear structure of $Z(ell_infty)$. In addition to this, we also give a thorough analysis of the existing algebraic structures within the set $X setminus Z(X)$ and its algebraic genericity.
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.
We study Banach spaces X with a strongly asymptotic l_p basis (any disjointly supported finite set of vectors far enough out with respect to the basis behaves like l_p) which are minimal (X embeds into every infinite dimensional subspace). In particular such spaces embed into l_p.
We study density requirements on a given Banach space that guarantee the existence of subsymmetric basic sequences by extending Tsirelsons well-known space to larger index sets. We prove that for every cardinal $kappa$ smaller than the first Mahlo ca
We introduce the concept of {em maximal lineability cardinal number}, $mL(M)$, of a subset $M$ of a topological vector space and study its relation to the cardinal numbers known as: additivity $A(M)$, homogeneous lineability $HL(M)$, and lineability
We study the generic behavior of the method of successive approximations for set-valued mappings in Banach spaces. We consider, in particular, the case of those set-valued mappings which are defined by pairs of nonexpansive mappings and give a positi