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| Preface | |
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| Linear Spaces and Operators | |
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| Introduction | |
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| Linear Spaces | |
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| Linear Operators | |
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| Passage from Finite- to Infinite-Dimensional Spaces | |
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| Exercises | |
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| Normed Linear Spaces: The Basics | |
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| Metric Spaces | |
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| Norms | |
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| Space of Bounded Functions | |
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| Bounded Linear Operators | |
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| Completeness | |
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| Comparison of Norms | |
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| Quotient Spaces | |
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| Finite-Dimensional Normed Linear Spaces | |
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| L[superscript p] Spaces | |
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| Direct Products and Sums | |
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| Schauder Bases | |
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| Fixed Points and Contraction Mappings | |
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| Exercises | |
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| Major Banach Space Theorems | |
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| Introduction | |
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| Baire Category Theorem | |
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| Open Mappings | |
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| Bounded Inverses | |
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| Closed Linear Operators | |
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| Uniform Boundedness Principle | |
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| Exercises | |
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| Hilbert Spaces | |
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| Introduction | |
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| Semi-Inner Products | |
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| Nearest Points and Convexity | |
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| Orthogonality | |
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| Linear Functionals on Hilbert Spaces | |
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| Linear Operators on Hilbert Spaces | |
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| Order Relation on Self-Adjoint Operators | |
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| Exercises | |
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| Hahn-Banach Theorem | |
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| Introduction | |
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| Basic Version of Hahn-Banach Theorem | |
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| Complex Version of Hahn-Banach Theorem | |
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| Application to Normed Linear Spaces | |
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| Geometric Versions of Hahn-Banach Theorem | |
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| Exercises | |
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| Duality | |
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| Examples of Dual Spaces | |
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| Adjoints | |
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| Double Duals and Reflexivity | |
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| Weak and Weak* Convergence | |
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| Exercises | |
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| Topological Linear Spaces | |
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| Review of General Topology | |
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| Topologies on Linear Spaces | |
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| Linear Functionals on Topological Linear Spaces | |
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| Weak Topology | |
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| Weak* Topology | |
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| Extreme Points and Krein-Milman Theorem | |
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| Operator Topologies | |
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| Exercises | |
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| The Spectrum | |
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| Introduction | |
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| Banach Algebras | |
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| General Properties of the Spectrum | |
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| Numerical Range | |
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| Spectrum of a Normal Operator | |
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| Functions of Operators | |
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| Brief Introduction to C*-Algebras | |
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| Exercises | |
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| Compact Operators | |
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| Introduction and Basic Definitions | |
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| Compactness Criteria in Metric Spaces | |
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| New Compact Operators from Old | |
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| Spectrum of a Compact Operator | |
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| Compact Self-Adjoint Operators on Hilbert Spaces | |
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| Invariant Subspaces | |
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| Exercises | |
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| Application to Integral and Differential Equations | |
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| Introduction | |
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| Integral Operators | |
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| Integral Equations | |
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| Second-Order Linear Differential Equations | |
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| Sturm-Liouville Problems | |
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| First-Order Differential Equations | |
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| Exercises | |
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| Spectral Theorem for Bounded, Self-Adjoint Operators | |
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| Introduction and Motivation | |
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| Spectral Decomposition | |
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| Extension of Functional Calculus | |
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| Multiplication Operators | |
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| Exercises | |
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| Zorn's Lemma | |
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| Stone-Weierstrass Theorem | |
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| Basic Theorem | |
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| Nonunital Algebras | |
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| Complex Algebras | |
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| Extended Real Numbers and Limit Points of Sequences | |
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| Extended Reals | |
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| Limit Points of Sequences | |
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| Measure and Integration | |
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| Introduction and Notation | |
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| Basic Properties of Measures | |
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| Properties of Measurable Functions | |
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| Integral of a Nonnegative Function | |
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| Integral of an Extended Real-Valued Function | |
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| Integral of a Complex-Valued Function | |
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| Construction of Lebesgue Measure on R | |
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| Completeness of Measures | |
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| Signed and Complex Measures | |
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| Radon-Nikodym Derivatives | |
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| Product Measures | |
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| Riesz Representation Theorem | |
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| Tychonoff's Theorem | |
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| Symbols | |
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| References | |
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| Index | |