Functional Analysis

Preface. 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 Reaction. 1.9. Operations on Sets. 1.10. Cardinality of Sets. 1.11. Abstract Mathematical Systems. 1.12. Various Abstract Systems. Exercises. 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. 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. Lesbegue Integral. Exercises. 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. 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. Compactness of Metric Spaces. 5.6. Contraction Mappings. 5.7. Compact Metric Spaces. 5.8. Approximation. 5.9. The Space of Fractals. Exercises. VI: Normed 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. 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. 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. IX: Differentiation of Operators. 9.1. Scope of the Chapter. 9.2. Gateaux and Frechet Derivatives. 9.3. Higher Order Frechet 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. Index of Symbols. Name Index.