Functional Analysis, Sobolev Spaces and Partial Differential Equations

verfasst von: Haim Brezis

Verlag: Springer New York

Buchreihe : Universitext

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

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Über dieses Buch

This textbook is a completely revised, updated, and expanded English edition of the important Analyse fonctionnelle (1983). In addition, it contains a wealth of problems and exercises (with solutions) to guide the reader. Uniquely, this book presents in a coherent, concise and unified way the main results from functional analysis together with the main results from the theory of partial differential equations (PDEs). Although there are many books on functional analysis and many on PDEs, this is the first to cover both of these closely connected topics. Since the French book was first published, it has been translated into Spanish, Italian, Japanese, Korean, Romanian, Greek and Chinese. The English edition makes a welcome addition to this list.

Inhaltsverzeichnis

Frontmatter

Chapter 1. The Hahn–Banach Theorems. Introduction to the Theory of Conjugate Convex Functions

Abstract

Let E be a vector space over \(\mathbb{R}.\) We recall that a functional is a function defined on E, or on some subspace of E, with values in \(\mathbb{R}.\) The main result of this section concerns the extension of a linear functional defined on a linear subspace of E by a linear functional defined on all of E.

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Chapter 2. The Uniform Boundedness Principle and the Closed Graph Theorem

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Chapter 3. Weak Topologies. Reflexive Spaces. Separable Spaces. Uniform Convexity

Abstract

We begin this chapter by recalling a well-known concept in topology. Suppose X is a set (without any structure) and \({({Y_i})_{i \in I}}\) is a collection of topological spaces.

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Chapter 4. Lp Spaces

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Chapter 5. Hilbert Spaces

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Chapter 6. Compact Operators. Spectral Decomposition of Self-Adjoint Compact Operators

Abstract

Throughout this chapter, and unless otherwise specified, E and F denote two Banach spaces.

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Chapter 7. The Hille–Yosida Theorem

Abstract

Throughout this chapter H denotes a Hilbert space.

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Chapter 8. Sobolev Spaces and the Variational Formulation of Boundary Value Problems in One Dimension

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Chapter 9. Sobolev Spaces and the Variational Formulation of Elliptic Boundary Value Problems in N Dimensions

Abstract

Let \(\Omega \subset \mathbb{R}^N.\) be an open set and let \(P\in\mathbb{R}\) with \(1 \le p \le \infty .\)

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Chapter 10. Evolution Problems: The Heat Equation and the Wave Equation

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Chapter 11. Miscellaneous Complements

Abstract

This chapter contains various complements that have not been incorporated in the main body of the book in order to keep the presentation more compact. They are connected to Chapters 1–7. Some of the proofs are very sketchy. Several proofs have been omitted, and the interested reader is invited to consult the references.

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Backmatter

Titel: Functional Analysis, Sobolev Spaces and Partial Differential Equations
verfasst von: Haim Brezis
Verlag: Springer New York
Electronic ISBN: 978-0-387-70914-7
Print ISBN: 978-0-387-70913-0
DOI: https://doi.org/10.1007/978-0-387-70914-7