The aim of this work is to present, in a unified and reasonably self-contained way, certain aspects of functional analysis which are needed to treat function spaces whose topology is not derived from a single norm, their topological duals and operators between those spaces. We treat spaces of continuous, analytic and smooth functions as well as sequence spaces. Operators of differentiation, integration, composition, multiplication and partial differential operators between those spaces are studied. A brief introduction to Laurent Schwartz’s theory of distributions and to Lars H�rmander’s approach to linear partial differential operators is presented.

The novelty of our approach lies mainly on two facts. First of all, we show all these topics together in an accessible way, stressing the connection between them. Second, we keep it always at a level that is accessible to beginners and young researchers. Moreover, parts of the book might be of interest for researchers in functional analysis and operator theory. Our aim is not to build and describe a whole, complete theory, but to serve as an introduction to some aspects that we believe are interesting. We wish to guide any reader that wishes to enter in some of these topics in their first steps. Our hope is that they learn interesting aspects of functional analysis and become interested to broaden their knowledge about function and sequence spaces and operators between them.

The text is addressed to students at a master level, or even undergraduate at the last semesters, since only knowledge on real and complex analysis is assumed. We have intended to be as self-contained as possible, and wherever an external citation is needed, we try to be as precise as we can. Our aim is to be an introduction to topics in, or connected with, different aspects of functional analysis. Many of them are in some sense classical, but we tried to show a unified direct approach; some others are new. This is why parts of these lectures might be of some interest even for researchers in related areas of functional analysis or operator theory. There is a full chapter about transitive and mean ergodic operators on locally convex spaces. This material is new in book form. It is a novel approach and can be of interest for researchers in the area.

About the Author

Jos� Bonet was born in Valencia in June 1955. He was interested in Mathematics since his childhood. He graduated and obtained his Doctoral degree in Mathematics in the Universitat de Val�ncia. He became full professor at the Universitat Polit�cnica de Val�ncia in 1987. He is full member of the Real Academia de Ciencias de Espa�a since 2008. He published many articles and taught courses at master level on functional analysis and operator theory. He likes reading novels, cinema and travelling.

David Jornet was born in Ontinyent (Valencia) in July 1976. He has been interested in mathematics since he was a child, partly motivated by his father, who is also a mathematician. He graduated from the Universitat de Val�ncia and obtained his doctorate from the Universitat Polit�cnica de Val�ncia. He is a tenured professor at the Universitat Polit�cnica de Val�ncia since 2011. Since 2014, he has taught master’s level classes in function spaces and operator theory. He likes sports, reading and travelling.

Pablo Sevilla-Peris was born in Valencia in 1975. Since primary school, he enjoyed learning and trying to understand mathematics. He took his degree at the Universitat de Val�ncia, where he also did his PhD (with stays in Ireland and Germany). Immediately after he joined the Universitat Polit�cnica de Val�ncia, where he has taught several courses on operator theory, functions spaces and related topics at master’s level. He is full professor since 2021. He plays (amateur) bassoon at the Societat Musical d’Alboraia.

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