Functional Analysis

Table of Contents
Preface
Chapter I. Preliminaries
1.1. Scope of the Chapter
1.2. Sets
1.3. Set Operations
1.4. Cartesian Product. Relations
1.5. Functions
1.6. Inverse Functions
1.7. Partial Ordering
1.8. Equivalence Relation
1.9. Operations on Sets
1.10. Cardinality of Sets
1.11. Abstract Mathematical Systems
1.12. Various Abtract Systems
Exercises
Chapter II. Linear Vector Spaces
2.1. Scope of the Chapter
2.2. Linear Vector Spaces
2.3. Subspaces
2.4. Linear Independence and Dependence
2.5. Basis and Dimension
2.6. Tensor Product of Linear Spaces
2.7. Linear Transformations
2.8. Matrix Representations of Linear Transformations
2.9. Equivalent and Similar Linear Transformations
2.10. Linear Functionals. Algebraic Dual
2.11. Linear Equations
2.12. Eigenvalues and Eigenvectors
Exercises
Chapter III. Introduction to Real Analysis
3.1. Scope of the Chapter
3.2. Properties of Sets of Real Numbers
3.3. Compactness
3.4. Sequences
3.5. Limit and Continuity in Functions
3.6. Differentiation and Integration
3.7. Measure of a Set Lebesgue Integral
Exercises
Chapter IV. Topological Spaces
4.1. Scope of the Chapter
4.2. Topological Structure
4.3. Bases and Subbases
4.4. Some Topological Concepts
4.5. Numerical Functions
4.6. Topological Vector Spaces
Exercises
Chapter V. Metric Spaces
5.1. Scope of the Chapter
5.2. The Metric and the Metric Topology
5.3. Various Metric Spaces
5.4. Topological Properties of Metric Spaces
5.5. Completeness of Metric Spaces
5.6. Contraction Mappings
5.7. Compact Metric Spaces
5.8. Approximation
5.9. The Space of Fractals
Exercises
Chapter VI. Normes Spaces
6.1. Scope of the Chapter
6.2. Normed Spaces
6.3. Semi-Norms
6.4. Series of Vectors
6.5. Bounded Linear Operators
6.6. Equivalent Normed Spaces
6.7. Bounded Below Operators
6.8. Continuous Linear Functionals
6.9. Topological Dual
6.10. Strong and Weak Topologies
6.11. Compact Operators
6.12. Closed Operators
6.13. Conjugate Operators
6.14. Classification of Continuous Linear Operators
Exercises
Chapter VII. Inner Product Spaces
7.1. Scope of the Chapter
7.2. Inner Product Spaces
7.3. Orthogonal Subspaces
7.4. Orthonormal Sets and Fourier Series
7.5. Duals of Hilbert Spaces
7.6. Linear Operators in Hilbert Spaces
7.7. Forms and Variational Equations
Exercises
Chapter VIII. Spectral Theory of Linear Operators
8.1. Scope of the Chapter
8.2. The Resolvent Set and the Spectrum
8.3. The Resolvent Operator
8.4. The Spectrum of a Bounded Operator
8.5. The Spectrum of a Compact Operator
8.6. Functions. of Operators
8.7. Spectral Theory in Hilbert Spaces
Exercises
Chapter IX. Differentiation of Operators
9.1. Scope of the Chapter
9.2. Gâteaux and Fréchet Derivatives
9.3. Higher Order Fréchet Derivatives
9.4. Integration of Operators
9.5. The Method of Newton
9.6. The Method of Steepest Descent
9.7. The Implicit Function Theorem
Exercises
References
A, B, C, D, G
H, J, K, L, M, N, O, P, R
R, S, T, V, W, Y, Z
Index of Symbols
Name Index
Subject Index