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2013 | Buch

Distance Kapitel lesen Erstes Kapitel lesen

Functional Analysis

Fundamentals and Applications

verfasst von: Michel Willem

Verlag: Springer New York

Buchreihe : Cornerstones

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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SUCHEN

Über dieses Buch

The goal of this work is to present the principles of functional analysis in a clear and concise way. The first three chapters of Functional Analysis: Fundamentals and Applications describe the general notions of distance, integral and norm, as well as their relations. The three chapters that follow deal with fundamental examples: Lebesgue spaces, dual spaces and Sobolev spaces. Two subsequent chapters develop applications to capacity theory and elliptic problems. In particular, the isoperimetric inequality and the Pólya-Szegő and Faber-Krahn inequalities are proved by purely functional methods. The epilogue contains a sketch of the history of functional analysis, in relation with integration and differentiation. Starting from elementary analysis and introducing relevant recent research, this work is an excellent resource for students in mathematics and applied mathematics.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Distance
Abstract
In Sect. 1.1, we recall without proof some basic properties of the real numbers. A metric space is a set on which the distance between two elements is defined. In Sect. 1.2, we introduce some basic notions, such as completeness and compactness. In Sect. 1.3, we define continuous and uniformly continuous mappings between metric spaces. Section 1.4 is devoted to simple and uniform convergence of sequences of functions.
Michel Willem
Chapter 2. The Integral
Abstract
Fontenelle
In Sect. 2.1, we recall the definition of the Cauchy integral of a continuous function with compact support. Four basic properties satisfied by the Cauchy integral are used in Sect. 2.2 to define the elementary integral and to construct the abstract Lebesgue integral. Fubini’s theorem and the change of variables formula are proved in Sects. 2.3 and 2.4.
Michel Willem
Chapter 3. Norms
Abstract
We introduce in this chapter the basic notions of functional analysis. Normed and Banach spaces are defined in Sect. 3.1. We study in Sect. 3.2 continuous linear mappings between normed spaces. Pre-Hilbert and Hilbert spaces are defined in Sect. 3.3. In Sect. 3.4, a proof of the spectral theorem for a compact symmetric operator is given.
Michel Willem
Chapter 4. Lebesgue Spaces
Abstract
In Sect.4.1, we define the basic notions of convexity, and we prove the Hahn–Banach theorem and a general convexity inequality. The Lebesgue spaces defined in Sect.4.2 are spaces of integrable functions with integral norms. In order to define complete spaces, we use the Lebesgue integral. Sections 4.3 and 4.4 are devoted to regularization and compactness in L p (Ω).
Michel Willem
Chapter 5. Duality
Abstract
The dual X ∗ of a normed space X is the space of continuous linear functionals on X. We define the weak convergence in X ∗ in Sect.5.1. We characterize X ∗ when X is a uniformly convex smooth Banach space (Sect.5.2), a Hilbert space (Sect.5.3), and a Lebesgue space (Sect.5.4).
Michel Willem
Chapter 6. Sobolev Spaces
Abstract
A locally integrable function has a weak derivative of order α when its derivative of order α in the sense of distributions is represented by a locally integrable function. Sobolev spaces are spaces of differentiable functions with integral norms. In order to define complete spaces, we use weak derivatives. The Sobolev embedding theorem is the most important result of this chapter.
Michel Willem
Chapter 7. Capacity
Abstract
We define the notion of abstract capacity in the sense of Choquet in Sect. 7.1 and the variational capacity of degree p in Sect. 7.2. The space of functions of bounded variation defined in Sect. 7.3 contains the Sobolev space W 1, 1(Ω) and the characteristic functions of bounded sets of finite perimeter. The relation between the perimeter and the variational capacity of degree 1 is given in Sect. 7.4.
Michel Willem
Chapter 8. Elliptic Problems
Abstract
The Laplacian, defined by
$$\displaystyle{ \Delta u = \mathrm{div\ }\nabla u = \frac{{\partial }^{2}u} {\partial x_{1}^{2}} +\ldots + \frac{{\partial }^{2}u} {\partial x_{ N}^{2}}, }$$
is related to the mean of functions.
Michel Willem
Chapter 9. Appendix: Topics in Calculus
Abstract
In this appendix, for the convenience of the reader we recall some topics in calculus. We begin with the formula for changing variables in multiple integrals.
Michel Willem
Chapter 10. Epilogue: Historical Notes on Functional Analysis
Abstract
In a concise description of mathematical methods, Henri Lebesgue underlined the importance of definitions and axioms (see [47]).
Michel Willem
Backmatter
Metadaten
Titel
Functional Analysis
verfasst von
Michel Willem
Copyright-Jahr
2013
Verlag
Springer New York
Electronic ISBN
978-1-4614-7004-5
Print ISBN
978-1-4614-7003-8
DOI
https://doi.org/10.1007/978-1-4614-7004-5

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