Applied Functional Analysis 3rd Edition John 1498761151 9781498761154

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Product details:

• ISBN-10: 1498761151
• ISBN-13: 9781498761154
• Author: John Tinsley Oden

Applied Functional Analysis, Third Edition provides a solid mathematical foundation for the subject. It motivates students to study functional analysis by providing many contemporary applications and examples drawn from mechanics and science. This well-received textbook starts with a thorough introduction to modern mathematics before continuing with detailed coverage of linear algebra, Lebesque measure and integration theory, plus topology with metric spaces. The final two chapters provides readers with an in-depth look at the theory of Banach and Hilbert spaces before concluding with a brief introduction to Spectral Theory. The Third Edition is more accessible and promotes interest and motivation among students to prepare them for studying the mathematical aspects of numerical analysis and the mathematical theory of finite elements.

Table of contents:

1. Preliminaries
1.1 Sets and Preliminary Notations, Number Sets
1.2 Level One Logic
1.3 Algebra of Sets
1.4 Level Two Logic
1.5 Infinite Unions and Intersections
1.6 Cartesian Products, Relations
1.7 Partial Orderings
1.8 Equivalence Relations, Equivalence Classes, Partitions
1.9 Fundamental Definitions
1.10 Compositions, Inverse Functions
1.11 Fundamental Notions
1.12 Ordering of Cardinal Numbers
1.13 Operations, Abstract Systems, Isomorphisms
1.14 Examples of Abstract Systems
1.15 The Real Number System
1.16 Open and Closed Sets
1.17 Sequences
1.18 Limits and Continuity
1.19 Derivatives and Integrals of Functions of One Variable
1.20 Multidimensional Calculus
2. Linear Algebra
2.1 Concept of a Vector Space
2.2 Subspaces
2.3 Equivalence Relations and Quotient Spaces
2.4 Linear Dependence and Independence, Hamel Basis, Dimension
2.5 Linear Transformations—The Fundamental Facts
2.6 Isomorphic Vector Spaces
2.7 More about Linear Transformations
2.8 Linear Transformations and Matrices
2.9 Solvability of Linear Equations
2.10 The Algebraic Dual Space, Dual Basis
2.11 Transpose of a Linear Transformation
2.12 Tensor Products, Covariant and Contravariant Tensors
2.13 Elements of Multilinear Algebra
2.14 Scalar (Inner) Product, Representation Theorem in Finite-Dimensional Spaces
2.15 Basis and Cobasis, Adjoint of a Transformation, Contra- and Covariant Components of Tensors
3. Lebesgue Measure and Integration
3.1 Elementary Abstract Measure Theory
3.2 Construction of Lebesgue Measure in
3.3 The Fundamental Characterization of Lebesgue Measure
3.4 Measurable and Borel Functions
3.5 Lebesgue Integral of Nonnegative Functions
3.6 Fubini’s Theorem for Nonnegative Functions
3.7 Lebesgue Integral of Arbitrary Functions
3.8 Lebesgue Approximation Sums, Riemann Integrals
3.9 Hölder and Minkowskilder inequality
4. Topological and Metric Spaces
4.1 Topological Structure—Basic Notions
4.2 Topological Subspaces and Product Topologies
4.3 Continuity and Compactness
4.4 Sequences
4.5 Topological Equivalence. Homeomorphism
4.6 Metric and Normed Spaces, Examples
4.7 Topological Properties of Metric Spaces
4.8 Completeness and Completion of Metric Spaces
4.9 Compactness in Metric Spaces
4.10 Contraction Mappings and Fixed Points
5. Banach Spaces
5.1 Topological Vector Spaces—An Introduction
5.2 Locally Convex Topological Vector Spaces
5.3 Space of Test Functions
5.4 The Hahn–Banach Theorem
5.5 Extensions and Corollaries
5.6 Fundamental Properties of Linear Bounded Operators
5.7 The Space of Continuous Linear Operators
5.8 Uniform Boundedness and Banach–Steinhaus Theorems
5.9 The Open Mapping Theorem
5.10 Closed Operators, Closed Graph Theorem
5.11 Example of a Closed Operator
5.12 Examples of Dual Spaces, Representation Theorem for Topological Duals of Spaces
5.13 Bidual, Reflexive Spaces
5.14 Weak Topologies, Weak Sequential Compactness
5.15 Compact (Completely Continuous) Operators
5.16 Topological Transpose Operators, Orthogonal Complements
5.17 Solvability of Linear Equations in Banach Spaces, the Closed Range Theorem
5.18 Generalization for Closed Operators
5.19 Closed Range Theorem for Closed Operators
5.20 Examples
5.21 Equations with Completely Continuous Kernels. Fredholm Alternative
6. Hilbert Spaces
6.1 Inner Product and Hilbert Spaces
6.2 Orthogonality and Orthogonal Projections
6.3 Orthonormal Bases and Fourier Series
6.4 Riesz Representation Theorem
6.5 The Adjoint of a Linear Operator
6.6 Variational Boundary-Value Problems
6.6.1 Classical Calculus of Variations
6.6.2 Abstract Variational Problems
6.6.3 Examples of Variational Formulations
6.6.4 Other Examples
6.6.5 The Case with No Essential Boundary Conditions
6.7 Generalized Green’s Formulas for Operators on Hilbert Spaces
6.8 Resolvent Set and Spectrum
6.9 Spectra of Continuous Operators. Fundamental Properties
6.10 Spectral Theory for Compact Operators
6.11 Spectral Theory for Self-Adjoint Operators

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