Lineabilidad y propiedades no lineales en el ámbito del Análisis RealLineability and nonlinear properties in the real setting

Abstract

The present thesis covers two fundamental topics: lineability and convolution of functions. These topics correspond to the two main sections of the thesis. The first, therefore, focuses on lineability, understood as the study of the type of structures (algebraic or topological) that can be found in the interior of certain sets which, a priori, do not possess any structure. More specifically, it is devoted to the study of the algebraic structure of sets of overjective functions. It concludes with the development of connections between different degrees of extreme overjectivity. From there, a second theoretical block is developed, which delves into the differentiability of the convolution of functions. We use as an introduction the Volterra and periodic functions to construct two differentiable functions whose convolution is not differentiable. The work ends with an extension of this second block by specifying the algebraic structure of the set of continuous and non-differentiable functions at any point, whose convolutions provide a continuous but non-differentiable function at any point.