Preface Notation and Terminology 0.1 Notation 0.2 Terminology Part 1 EXERCISES ET AL Chapter 1 Normed Vector Spaces.Banach Spaces 1.1 Basics 1.1.1 Definitions and Examples 1.1.2 Operations on Banach Spaces 1.1.3 Convex Sets 1.1.4 LP-Spaces 1.2 True or False: Questions 1.3 Exercises with Solutions 1.4 Tests 1.5 More Exercises Chapter 2 Bounded Linear Operators on Banach Spaces 2.1 Basics 2.1.1 Basic Definitions 2.1.2 Dual Spaces 2.1.3 The Fourier Transform 2.1.4 Invertibility 2.1.5 Series in Banach Spaces 2.1.6 Neumann Series 2.1.7 Operator-Valued Analytic Functions 2.1.8 The Four Pillars of Functional Analysis 2.2 True or False: Questions 2.3 Exercises with Solutions 2.4 Tests 2.5 More Exercises Chapter 3 Hilbert Spaces 3.1 Basics 3.1.1 Definitions and Examples 3.1.2 Modes of Convergence of Sequences 3.1.3 Orthogonality 3.1.4 The Closest Point Property and the Projection Theorem 3.1.5 Orthonormal Sets and Basy ana tne rro 3.1.6 Orthogonal Polynomials 3.1.7 Linear Operators on a Hilbert Space 3.1.8 Operator Topologies 3.2 True or False: Questions 3.3 Exercises with Solutions 3.4 Tests 3.5 More Exercises Chapter 4 Classes of Linear Operators on Hilbert Spaces 99 4.1 Basics 4.1.1 Operator Theory Basics 4.1.2 Cartesian Form 4.1.3 Operator Matrices 4.1.4 A Word on the Invariant Subspace Problem 4.2 True or False: Questions
4.3 Exercises with Solutions 4.4 Tests 4.5 More Exercises Chapter 5 Positive Operators.Square Root 5.1 Basics 5.1.1 Positive Operators 5.1.2 Invertibility Revisited 5.1.3 Square Root of Positive Operators 5.2 True or False: Questions 5.3 Exercises with Solutions 5.4 Tests 5.5 More Exercises Chapter 6 Absolute Value.Polar Decomposition of an Operator 6.1 Basics 6.1.1 Definitions and Properties 6.1.2 The Absolute Value and the Product 6.1.3 The Absolute Value and the Sum 6.1.4 The Absolute Value and Other Inequalities 6.1.5 Polar Decomposition 6.2 True or False: Questions 6.3 Exercises with Solutions 6.4 Tests 6.5 More Exercises Chapter 7 Spectrum of an Operator 7.1 Basics 7.1.1 Definitions and Properties 7.1.2 The Resolvent Function 7.1.3 Subsets of the Spectrum 7.1.4 Spectrum of the Product of Operators 7.1.5 On The Operator Equation AX - XB = C 7.1.6 Polynomial Calcul-yuawva A 7.2 True or False: Questions 7.3 Exercises with Solutions 7.4 Tests 7.5 More Exercises Chapter 8 Spectral Radius Numerical Range 8.1 Basics 8.1.1 Spectral Radius 8.1.2 Numerical Range 8.2 True or False: Questions 8.3 Exercises with Solutions 8.4 Tests 8.5 More Exercises Chapter 9 Compact Operators 9.1 Basics 9.1.1 Compact Operators 9.1.2 Hilbert-Schmidt Operators 9.1.3 Spectrum and Compact Operators 9.2 True or False: Questions 9.3 Exercises with Solutions
9.4 Tests 9.5 More Exercises Chapter 10 Closed Operators 10.1 Basics 10.1.1 Spectrum 10.1.2 Closedness of Products and Sums 10.1.3 A Word on Distributional Derivatives 10.2 True or False: Questions 10.3 Exercises with Solutions 10.4 Tests 10.5 More Exercises Chapter 11 Functional Calculi 11.1 Basics 11.1.1 Functional Calculus for Self-adjoint Operators 11.1.2 Functional Calculus for Normal Operators 11.1.3 A Little Digression 11.1.4 The Fuglede-Putnam Theorem Revisited 11.1.5 The Beck-Putnam Theorem 11.2 True or False: Questions 11.3 Exercises with Solutions 11.4 Tests 11.5 More Exercises Chapter 12 Hyponormal Operators 12.1 Basics 12.2 True or False: Questions 12.3 Exercises with Solutions 12.4 Tests 12.5 More Exercises Chapter 13 Similarities of Operators 13.1 Important Theorems 13.1.1 The Berberian Theorem 13.1.2 The Williams Theorem 13.1.3 The Embry Theorem 13.2 Exercises with Solutions 13.3 Tests 13.4 More Exercises Part 2 SOLUTIONS Chapter 1 Normed Vector Spaces Banach Spaces 1.2 True or False: Answers 1.3 Solutions to Exercises 1.4 Hints/Answers to Tests Chapter 2 Bounded Linear Operators on Banach Spaces 2.2 True or False: Answers 2.3 Solutions to Exercises 2.4 Hints/Answers to Tests Chapter 3 Hilbert Spaces 3.2 True or False: Answers 3.3 Solutions to Exercises 3.4 Hints/Answers to Tests Chapter 4 Classes of Linear Operators on Hilbert Spaces 377
4.2 True or False: Answers 4.3 Solutions to Exercises 4.4 Hints/Answers to Tests Chapter 5 Positive Operators Square Root 5.2 True or False: Answers 5.3 Solutions to Exercises 5.4 Hints/Answers to Tests Chapter 6 Absolute Value Polar Decomposition of an Operator 6.2 True or False: Answers 6.3 Solutions to Exercises 6.4 Hints/Answers to Tests Chapter 7 Spectrum of an Operator 7.2 True or False: Answers 7.3 Solutions to Exercises 7.4 Hints/Answers to Tests Chapter 8 Spectral Radius Numerical Range 8.2 True or False: Answers 8.3 Solutions to Exercises 8.4 Hints/Answers to Tests Chapter 9 Compact Operators 9.2 True or False: Answers 9.3 Solutions to Exercises 9.4 Hints/Answers to Tests Chapter 10 Closed Operators 10.2 True or False: Answers 10.3 Solutions to Exercises 10.4 Hints/Answers to Tests Chapter 11 Functional Calculi 11.2 True or False: Answers 11.3 Solutions to Exercises 11.4 Hints/Answers to Tests Chapter 12 Hyponormal Operators 12.2 True or False: Answers 12.3 Solutions to Exercises 12.4 Hints/Answers to Tests Chapter 13 Similarities of Operators 13.2 Solutions to Exercises 13.3 Hints/Answers to Tests Bibliography Index 編輯手記
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