# Funciones de tipo no acotado y topologías en espacios de aplicaciones analíticas

### Impacto

López-Salazar Codes, Jerónimo (2013) Funciones de tipo no acotado y topologías en espacios de aplicaciones analíticas. [Thesis]

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## Abstract

The space H(U) of all holomorphic functions on an open subset U of the complex plane has always been a classic example in the theory of locally convex spaces. Endowed with the compact-open topology, H(U) is metrizable, complete, barrelled, bornological, etc. However, if U is an open subset of an infinite dimensional space, the compact-open topology is not strong enough. Because of this, several researchers introduced other topologies on H(U). This thesis is mainly devoted to study the properties of the topology \tau_{\delta}, which was defined by Coeuré and Nachbin in 1970.The first problem that we consider is the metrizability of H(U). It is proved that if U is an open subset of a metrizable space of infinite dimension, then H(U) is not metrizable. This result generalizes previous theorems due to Alexander, Ansemil and Ponte. We next study the representation of H(U) as an inductive limit of Fréchet spaces. In 2009, Ansemil, Aron and Ponte proved that the limit which defines to \tau_{\delta} on H(E) is not countable when E is an infinite dimensional Banach space with a Schauder basis. In this thesis, we prove the analogous result for H(U), where U is an open subset of an infinite dimensional normed space. This thesis is also devoted to study the lineability of several collections of holomorphic functions. In Chapter 3, we prove the lineability of the set of entire functions of unbounded type. In addition, in Chapter 7, we show that if E is a separable Banach space and D is the open unit disc in the complex plane, then the set of holomorphic mappings f:D→E such that f(D) is dense in E is lineable and dense. Thus we generalize previous results due to Aron, Globevnik and Rudin.

Item Type: Thesis Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Departamento de Análisis Matemático, leída el 09-01-2013 DirectorsMartínez Ansemil, José MaríaPonte Miramontes, Socorro Función holomorfa, espacio localmente convexo, conjunto lineable Holomorphic function, locally convex space, lineable set Sciences > Mathematics > Mathematical analysis 20062 22 Feb 2013 09:21 26 Feb 2013 18:47