Moduleability, algebraic structures, and nonlinear properties

We show that some pathological phenomena occur more often than one could expect, existing large algebraic structures (infinite dimensional vector spaces, algebras, positive cones or infinitely generated modules) enjoying certain special properties. In particular we construct infinite dimensional vector spaces of non-integrable, measurable functions, completing some recent results shown in Garcيa-Pacheco et al. (2009) [13], Garcيa-Pacheco and Seoane-Sepْlveda (2006) [15], Muٌoz-Fernلndez et al. (2008) [20]. We prove, as well, the existence of dense and not barrelled spaces of sequences every non-zero element of which has a finite number of zero coordinates (giving partial answers to a problem originally posed by R.M. Aron and V.I. Gurariy in 2003).