A search for linearity in the universe of topological vector spaces

Abstract

This thesis provides criteria (both negative and positive) that contribute to the existing literature and address open problems within the notions of (α, β)-lineability/spaceability, (α, β)-dense lineability, pointwise lineability, and [S]-lineability. In our exploration, we began by investigating the behavior of algebraic and topological structures present in the set of unbounded, continuous, and integrable functions on the interval [0, ∞). This investigation was initiated by Calderón-Moreno, Gerlach-Mena, and Prado-Bassas, where they demonstrated, among other results, that the set

A := f ∈ C [0, ∞) ∩ L1 [0, ∞) : lim sup x→∞ |f (x)| = ∞

is lineable. To better understand the dimensional relationships in this environment, we employed new techniques and gained additional insights into both the topological and algebraic structure of this set. Specifically, we proved its pointwise spaceability (and thus, spaceability). Additionally, we demonstrated that the set Lp[0, 1] \ S q∈(p,∞)Lq[0, 1], para p ∈ (0, ∞), although (1,c)-spaceable (see [21]), is not (א0,c)-spaceable. We established a general criterion for negative results concerning (α, β)-spaceability and verified that the set N D[0, 1] of nowhere differentiable functions cannot be (α, β)-spaceable for any infinite cardinal α. We also provided criteria for positive results, showing in particular that the sets l∞ \ F, where F ∈ {c, c0}, and Lp[0, 1] \ S q∈(p,∞)Lq[0, 1], for p ∈ (0, ∞), are

(α,c)-espaceable if and only if α is finite. We introduced the notion of (α, β)-dense lineability and provided a criterion to demonstrate in particular that the set Lp[0, 1] \ S q∈(p,∞)Lq[0, 1], for p ∈ (0, ∞) is (α, β)- dense lineable for every 0 ≤ α ≤ β and max {α, א0} ≤ β ≤ c. Our findings highlight that the geometry of the studied sets alone is insufficient and that the type of topology considered in each environment also plays a crucial role.