Function Spaces and Operators between them

This work aims to present functional analysis concepts for function spaces with non-single norm topologies,topological duals,and operators between them. It covers continuous,analytic,and smooth functions,sequence spaces,differentiation,integration,composition,multiplication,and partial differential operators. The novelty lies in connecting these topics in an accessible way for beginners and young researchers,with an emphasis on the connection between them. The book is intended to serve as an introduction to these areas and can be of interest to students and researchers in functional analysis and operator theory.

Format: Unspecified
Length: 269 pages
Publication date: 29 October 2023
Publisher: Springer International Publishing AG

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 Hörmander'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, differential and integral equations.

We begin by recalling some basic definitions and results from functional analysis, such as the definition of a Banach space, the norm and the topology, the concept of an isomorphism and the Hahn-Banach theorem
theorem. We then introduce the concept of a distribution, which is a generalization of the concept of a function to non-linear operators. Distributions.

We then discuss the theory of distributions, which is a fundamental tool in the study of functional analysis. Distributions are used to define functions on a Banach space, and they play an important role.

role in the study of partial differential equations. We introduce the concept of a distributional isomorphism, which is a generalization of the concept of an isomorphism to distributions. We then discuss the theory of distributions, which is a fundamental tool in the study of functional analysis. Distributions are used to define functions on a Banach space, and they play an important role in the study of partial differential equations. We introduce the concept of a distributional isomorphism, which is a generalization of the concept of an isomorphism to distributions. We then discuss the theory of distributions, which is a fundamental tool in the study of functional analysis. Distributions are used to define functions on a Banach space, and they play an important role in the study of partial differential equations. We introduce the concept of a distributional isomorphism, which is a generalization of the concept of an isomorphism to distributions. We then discuss the theory of distributions.

Weight: 659g
Dimension: 235 x 155 (mm)
ISBN-13: 9783031416019
Edition number: 1st ed. 2023