Functional Analysis
- 2023
- Buch
- 2. Auflage
- Verfasst von
- S. Kesavan
- Buchreihe
- Texts and Readings in Mathematics
- Verlag
- Springer Nature Singapore
Über dieses Buch
Über dieses Buch
This second edition is thoroughly revised and includes several new examples and exercises. Proofs of many results have been rewritten for a greater clarity. While covering all the standard material expected of such a course, efforts have been made to illustrate the use of the topics to study differential equations and calculus of variations. The book includes a chapter on weak topologies and their applications. It also includes a chapter on the Lebesgue spaces, which discusses Sobolev spaces. The book includes a chapter on compact operators and their spectra, especially for compact self-adjoint operators on a Hilbert space. Each chapter has a large collection of exercises in the end, which give additional examples and counterexamples to the results given in the text. This book is suitable for a first course in functional analysis for graduate students who wish to pursue a career in the applications of mathematics.
Inhaltsverzeichnis
-
Frontmatter
-
Chapter 1. Preliminaries
S. KesavanAbstractFunctional analysis is the study of vector spaces endowed with topological structures (that are compatible with the linear structure of the space) and of (linear) mappings between such spaces. Throughout this book we will be working with vector spaces whose underlying field is the field of real numbers \(\mathbb {R}\) or the field of complex numbers \(\mathbb {C}\). -
Chapter 2. Normed Linear Spaces
S. KesavanAbstractIn order to do analysis on vector spaces, we need to endow these spaces with a topological structure which is compatible with the linear structure. This is made precise in the following definition. -
Chapter 3. Hahn-Banach Theorems
S. KesavanAbstractThe analytic form of the Hahn-Banach theorem concerns the extension of linear functionals defined on a subspace of a normed linear space to the entire space, preserving the norm of the functional. We will prove a slightly more general result in this direction. -
Chapter 4. Baire’s Theorem and Applications
S. KesavanAbstractBaire’s theorem is a result on complete metric spaces which will be used in this chapter to prove some very important results on Banach spaces. -
Chapter 5. Weak and Weak* Topologies
S. KesavanAbstractIn this chapter, we will study topologies on Banach spaces which are weaker (i.e., coarser) than the norm topology. -
Chapter 6. Spaces
S. KesavanAbstractThe Lebesgue spaces, also known as the \(L^p\) spaces, constitute a rich source of examples and counterexamples in functional analysis. They also form an important class of function spaces when studying the applications of mathematical analysis. In this chapter, we will study the important properties of these spaces from the functional analytic point of view. -
Chapter 7. Hilbert Spaces
S. KesavanAbstractHilbert spaces form a special class of Banach spaces with the geometric notion of orthogonality of vectors, or more generally, the notion of an angle between vectors, built into them. -
Chapter 8. Compact Operators
S. KesavanAbstractIn this chapter we will study a special class of linear transformations between Banach spaces which generalize several properties of linear transformations between finite dimensional spaces. -
Backmatter
- Titel
- Functional Analysis
- Verfasst von
-
S. Kesavan
- Copyright-Jahr
- 2023
- Verlag
- Springer Nature Singapore
- Electronic ISBN
- 978-981-19-7633-9
- DOI
- https://doi.org/10.1007/978-981-19-7633-9
Informationen zur Barrierefreiheit für dieses Buch folgen in Kürze. Wir arbeiten daran, sie so schnell wie möglich verfügbar zu machen. Vielen Dank für Ihre Geduld.
