Técnicas en análisis lineal (y no lineal) y aplicaciones

Abstract

La presente tesis est a centrada en dos temas principales: el primero abarca el primer cap tulo y el segundo se divide entre los cap tulos dos y tres. En el primer cap tulo estudio un problema que apareci o como tal hace relativamente poco tiempo (aunque ya en la segunda mitad del pasado siglo se publicaron una serie de resultados que, con la terminolog a adecuada, estar an englobados dentro de esta teor a). Nos interesaremos en la b usqueda de estructuras algebraicas (como espacios vectoriales, algebras, espacios de Banach) contenidas en subconjuntos de funciones cuyos elementos (con la posible excepci on del elemento nulo) veri can ciertas propiedades anti-intuitivas (propiedades de dif cil visualizaci on). Ello nos puede conducir a la idea de c omo la intuci on puede enga~narnos, y sugerir que, aunque se haya dedicado una ingente cantidad de esfuerzo y tiempo para encontrar un unico ejemplo que veri que tales propiedades, y dicho trabajo pueda dar la idea de que no existen muchos m as espec menes de similares caracter sticas, de hecho existen ejemplares su cientes como para construir espacios \grandes" cuyos elementos (salvo el cero) satisfacen las mismas propiedades. M as espec camente, decimos que un subconjunto de un espacio vectorial topol ogico es -lineable (dado un numero cardinal ) si podemos garantizar la existencia de un espacio vectorial de dimensi on contenido en el conjunto (uni on el elemento cero, en caso de que cero no forme parte del conjunto de partida). Si el espacio vectorial es cerrado, nos referiremos a este conjunto como - espaciable (y la propiedad que trataremos ser a la de -espaciabilidad) y si la estructura en cuesti on es un algebra de Banach, entonces diremos que el conjunto es ( ; )-algebrable (donde aqu es la cardinalidad de un conjunto minimal de generadores del algebra)...

This thesis will be divided into two topics: the rst one will cover the rst chapter and will deal with a problem that took form little time ago (even though already in the second half of the past century there would be some results). We will be interested on nding algebraic structures (vector spaces, algebras, Banach spaces) contained in subsets of functions whose elements ful ll some anti-intuitive property, union the zero function. Thereby, we can have an idea of how the intuition may mislead us, and hint that, even though we may think that because of having to spend a huge e ort in nding one example of such elements we may not nd many more, in fact there are enough to consider huge spaces all whose elements except from the zero element satisfy the same property. More speci cally, we de ne a subset of a topological vector space to be {u100000}lineable (for a cardinal number ) if we can nd a vector space of dimension contained in the set (union the zero element, in case it is not included). If the vector space is closed, then we will be talking about {u100000}spaceability (and we will say that the set is {u100000}spaceable), and if the structure included is a Banach algebra then we will de ne the set to be ( ; ){u100000}algebrable (where here would be the cardinality of a minimal set of generators of the algebra). If no cardinal number is de ned, then we will assume the structure to be in nite dimensional. This trend was developed as an independent theory in the end of the last Century, in [5], and since its appearance it has resulted in a fruitful eld of study, as the amount of results show (see for example [4], [7], [12], [24], [26] or [54], a very recent and detailed paper giving an exhausting overview of the results published until 2014 can be found in [16]). The sets that will be considered here when studying those anti-intuitive properties will deal with functions de ned over the real line, more concretely results that lie beneath the de nition of di erentiability (for example the relationship between bounds of the di erential and the Lipschitzianity of the function). In particular, we will revisit the famous example given by Weierstrass. There will also be some sections dedicated to the analyticity of real functions and its relation with the in nite di erentiability...