# Functional Analysis

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Functional Analysis examines trends in functional analysis as a mathematical discipline and the ever-increasing role played by its techniques in applications. The theory of topological vector spaces is emphasized, along with the applications of functional analysis to applied analysis. Some topics of functional analysis connected with applications to mathematical economics and control theory are also discussed. Comprised of 18 chapters, this book begins with an introduction to the elements of the theory of topological spaces, the theory of metric spaces, and the theory of abstract measure spaces. Many results are stated without proofs. The discussion then turns to vector spaces, normed spaces, and linear operators and functionals. Subsequent chapters deal with the analytic representation of functionals; sequences of linear operators; the weak topology in a Banach space; and compact and adjoint operators. The last section focuses on functional equations, including the adjoint equation and functional equations of the second kind. This monograph is intended for students specializing in mathematical analysis and computational mathematics.

## Table of Contents

Preface to the Second Edition

From the Preface to the First Edition

Part I Linear Operators and Functionals

Chapter I. Topological and Metric Spaces

§ 1. General Information on Sets. Ordered Sets

§ 2. Topological Spaces

§ 3. Metric Spaces

§ 4. Completeness and Separability. Sets of the First and Second Categories

§ 5. Compactness in Metric Spaces

§ 6. Measure Spaces

Chapter II. Vector Spaces

§ 1. Basic Definitions

§ 2. Linear Operators and Functional

§ 3. Convex Sets and Seminorms

§ 4. The Hahn-Banach Theorem

Chapter III. Topological Vector Spaces

§ 1. General Definitions

§ 2. Locally Convex Spaces

§ 3. Duality

Chapter IV. Normed Spaces

§ 1. Basic Definitions and Simplest Properties of Normed Spaces

§ 2. Auxiliary Inequalities

§ 3. Normed Spaces of Measurable Functions and Sequences

§ 4. Other Normed Spaces of Functions

§ 5. Hilbert Space

Chapter V. Linear Operators and Functionals

§ 1. Spaces of Operators and Dual Spaces

§ 2. Some Functionals and Operators on Specific Spaces

§ 3. Linear Functionals and Operators on Hilbert Space

§ 4. Rings of Operators

§ 5. The Method of Successive Approximations

§ 6. The Ring of Operators on a Hilbert Space

§ 7. The Weak Topology and Reflexive Spaces

§ 8. Extensions of Linear Operators

Chapter VI. The Analytic Representation of Functionals

§ 1. Integral Representations for Functionals on Spaces of Measurable Functi

§ 2. The Spaces Lp(T,Σ,μ)

§ 3. A General Form for Linear Functionals on the Space C(K)

Chapter VII. Sequences of Linear Operators

§ 1. Basic Theorems

§ 2. Some Applications to the Theory of Functions

Chapter VIII. The Weak Topology in a Banach Space

§ 1. Weakly Bounded Sets

§ 2. Eberlein-Shmul'yan Theory

§ 3. Weak Convergence in Specific Spaces

§ 4. The Problem of Translocation of Mass and the Normed Space it Generates

Chapter IX. Compact and Adjoint Operators

§ 1. Compact Sets in Normed Spaces

§ 2. Compact Operators

§ 3. Adjoint Operators

§ 4. Compact Self-Adjoint Operators on Hilbert Space

§ 5. Integral Representations of Self-Adjoint Operators

Chapter X. Ordered Normed Spaces

§ 1. Vector Lattices

§ 2. Linear Operators and Functionals

§ 3. Normed Lattices

§ 4. KB-Spaces

§ 5. Convex Sets that are Closed with Respect to Convergence in Measure

Chapter XI. Integral Operators

§ 1. Integral Representations of Operators

§ 2. Operators on Sequence Spaces

§ 3. Integral Operators on Function Spaces

§ 4. Sobolev's Embedding Theorems

Part II Functional Equations

Chapter XII. The Adjoint Equation

§ 1. Theorems on Inverse Operators

§ 2. The Connection Between an Equation and its Adjoint

Chapter XIII. Functional Equations of the Second Kind

§ 1. Equations with Compact Kernels

§ 2. Complex Normed Spaces

§ 3. The Spectrum

§ 4. Resolvents

§ 5. The Fredholm Alternative

§ 6. Applications to Integral Equations

§ 7. Invariant Subspaces of Compact Operators. The Approximation Problem

Chapter XIV. A General Theory of Approximation Methods

§ 1. A General Theory for Equations of the Second Kind

§ 2. Equations Reducible to Equations of the Second Kind

§ 3. Applications to Infinite Systems of Equations

§ 4. Applications to Integral Equations

§ 5. Applications to Ordinary Differential Equations

§ 6. Applications to Boundary-Value Problems for Equations of Elliptic Type

Chapter XV. The Method of Steepest Descent

§ 1. The Solution of Linear Equations

§ 2. Determination of the Eigenvalues of Compact Operators

§ 3. Applications to Elliptic Differential Equations

§ 4. Minimization of Convex Differentiable Functionals

§ 5. Minimization of Convex Functionals on Finite-Dimensional Spaces

Chapter XVI. The Fixed-Point Principle

§ 1. The Caccioppoli-Banach Principle

§ 2. Auxiliary Propositions

§ 3. Schauder's Principle

§ 4. Applications of the Fixed-Point Principle

§ 5. Kakutani's Theorem

Chapter XVII. Differentiation of Non-Linear Operators

§ 1. The First Derivative

§ 2. Second Derivatives and Bilinear Operators

§ 3. Examples

§ 4. The Implicit Function Theorem

Chapter XVIII. Newton's Method

§ 1. Equations of the Form P(x) = 0

§ 2. Consequences of the Convergence Theorem for Newton's Method

§ 3. Applications of Newton's Method to Specific Functional Equations

§ 4. Newton's Method in Lattice-Normed Spaces

Monographs on Functional Analysis and Related Topics

References

Subject Index

Index of Notation

Index of Abbreviations

## Product details

- No. of pages: 604
- Language: English
- Copyright: © Pergamon 1982
- Published: May 18, 2014
- Imprint: Pergamon
- eBook ISBN: 9781483147741