Filtros en espacios de Banach

Abstract

Sequences play a fundamental role in mathematics because of their usefulness in the proofs of theorems and properties of topology, they are also fundamental in applied mathematics. Just to name a few examples, sequences are key in the characterization of continuous functions or in the characterization of compact subsets in metrizable spaces, they are used to prove the existence of solutions of certain equations through the Fixed Point Theorem or in iterative methods such as divide and conquer, and even in the asymptotic notation that allows estimating the efficiency of an algorithm. In the last 50 years or so, several mathematicians have made contributions on generalizations of this concept. Specifically, they have made generalizations of the classical concept of convergence by means of conjunctive notions. For example, Kostyrko, Šalát and Wilczyński use the notion of topological ideal, introduced by Kuratowski in 1933, to generate a convergence of sequences via ideals. Also well known is the generalization of convergence of sequences using the notion of filter, which were introduced by Cartan in 1937. It is not known for sure who introduced the notion of convergence using filters, what is certain is that it is already part of the folklore within topology and is used by many mathematicians to make generalizations of theories based on this concept. In this work a study of filters is made, examples are given, their main properties are stated and demonstrated. Use is made of Zorn's Lemma to guarantee the existence, under certain conditions, of ultrafilters (maximal filters), the collection of all ultrafilters over N is endowed with a topology, the topological space obtained turns out to be the Stone-Čech compactification of the natural numbers. Then, given a filter F, the notion of F-convergent succession over a topological space is studied. Punctually, Ferreira's article is broken down, in which the concept of convergence of sequences using free filters on the natural numbers is worked out. In addition, common notions of topology such as: point of adhesion or accumulation and the behavior of F-convergent sequences under continuous functions are characterized. As already mentioned, in 2000 Kostyrko, Šalát and Wilczyński generalize the notion of convergence by means of a dual structure to that of filters: ideals. In this paper, for an ideal I, the notion of I-convergence is introduced, properties and characterizations are studied, among other things. Over the same decade, the notions of I-Cauchy succession, weak I-convergence and weak I-convergence∗ are introduced. The notion of I-Cauchy was introduced in the year 2005, by Dems, in this work the relation between I-Cauchy sequences and I-convergent sequences is studied, even when it could be thought that these notions could lead to a Banach I-space, it is surprising to read the result provided by the authors where they characterize Banach spaces in terms of I-Cauchy sequences and I-convergent sequences, which provides an additional tool for the study of this type of spaces. In 2010 Pelihvan, Şençimen and Yaman work on the notions of weak I-convergence and I-weak∗ and establish properties of these similar to those satisfied by weakly convergent sequences and sequences of weakly*convergent operators. In this work, a study of notions of convergence from the point of view of filters is carried out, which represents a modest contribution to the literature since no references evidencing the existence of these have been found to date. Finally, asymptotic notations are understood as the fundamental tool to estimate the computational complexity of algorithms, that is, to study their growth rate. Considering the nature of asymptotic notations, it is possible to interpret them in terms of sequences and, therefore, to generalize them using filters. Thus, in this paper, a generalization of the asymptotic notations OF and oF is introduced, relations between these two notations, the properties they satisfy, as well as their relation with the previously defined and studied notions are established.