Linear and algebraic structures in function sequence spaces

Abstract

Historically, many mathematicians of all ages have been attracted and fascinated by the existence of large algebraic structures that satisfy certain properties that, a priori, contradict the mathematical intuition. The aim of the present Dissertation is the study of the lineability of certain families of sequences of functions with very specific properties. The Dissertation is divided in 6 chapters, where Chapters 1, 2 and 3 focus on introducing the basic notation and main terminology of the theory of Lineability and modes of convergence that will be used along this Dissertation. In Chapter 4 we begin with the study of the algebraic size of two families of sequences of functions with different modes of convergence in the closed unit interval [0, 1]: convergence in measure but pointwise almost everywhere and pointwise but not uniform convergence. In Chapter 5 we focus our attention on the setting of (Lebesgue) integrable functions. We start with sequences of integrable functions with different modes of convergence in comparison to the L1-convergence, and finish the chapter with the algebraic size of the family of unbounded, continuous and integrable functions on [0, +∞) and sequences of them. Finally, in Chapter 6 we turn into the setting of series of functions, obtaining positive results about the linear and algebraic size of the family of sequences of functions whose series converges absolutely and uniformly but does not verify the hypothesis of the Weierstrass M-test.