Contents

• 0 Motivating Example: Fourier Series
  • 0.1  Fourier series: basic notions
    • 0.1.1 2π-periodic functions
    • 0.1.2 Integrating the complex exponential function
  • 0.2 The vibrating string
    • 0.2.1 Separation of variables
    • 0.2.2 Principle of Superposition
  • 0.3 Historic: Joseph Fourier
• 1 Basics of Metric Spaces
  • 1.1 Metric Spaces
    • 1.1.1 Metric spaces: definition and examples
    • 1.1.2 Open and closed sets
    • 1.1.3 Convergence and continuity
  • 1.2 Useful properties of metric spaces
    • 1.2.1 Cauchy sequences and completeness
    • 1.2.2 Compactness
• 2 Basics of Linear Spaces
  • 2.1 Banach spaces (basic definitions only)
  • 2.2 Hilbert spaces
  • 2.3 Subspaces
  • 2.4 Linear spans
• 3 Orthogonality
  • 3.1 Orthogonal System in Hilbert Space
  • 3.2 Bessel’s inequality
  • 3.3 The Riesz–Fischer theorem
  • 3.4 Construction of Orthonormal Sequences
  • 3.5 Orthogonal complements
• 4 Duality of Linear Spaces
  • 4.1 Dual space of a normed space
  • 4.2 Self-duality of Hilbert space
• 5 Fourier Analysis
  • 5.1 Fourier series
  • 5.2 Fejér’s theorem
  • 5.3 Parseval’s formula
  • 5.4 Some Application of Fourier Series
• 6 Operators
  • 6.1 Linear operators
  • 6.2 Orthoprojections
  • 6.3 B(H) as a Banach space (and even algebra)
  • 6.4 Adjoints
  • 6.5 Hermitian, unitary and normal operators
• 7 Spectral Theory
  • 7.1 The spectrum of an operator on a Hilbert space
  • 7.2 The spectral radius formula
  • 7.3 Spectrum of Special Operators
• 8 Compactness
  • 8.1 Compact operators
  • 8.2 Hilbert–Schmidt operators
• 9 Compact normal operators
  • 9.1 Spectrum of normal operators
  • 9.2 Compact normal operators
• 10 Integral equations
• 11 Banach and Normed Spaces
  • 11.1 Normed spaces
  • 11.2 Bounded linear operators
  • 11.3 Dual Spaces
  • 11.4 Hahn–Banach Theorem
  • 11.5 C(X) Spaces
• 12 Measure Theory
  • 12.1 Basic Measure Theory
  • 12.2 Extension of Measures
  • 12.3 Complex-Valued Measures and Charges
  • 12.4 Constructing Measures, Products
• 13 Integration
  • 13.1 Measurable functions
  • 13.2 Lebesgue Integral
  • 13.3 Properties of the Lebesgue Integral
  • 13.4 Integration on Product Measures
  • 13.5 Absolute Continuity of Measures
• 14 Functional Spaces
  • 14.1 Integrable Functions
  • 14.2 Dense Subspaces in L_p
  • 14.3 Continuous functions
  • 14.4 Riesz Representation Theorem
• 15 Fourier Transform
  • 15.1 Convolutions on Commutative Groups
  • 15.2 Characters of Commutative Groups
  • 15.3 Fourier Transform on Commutative Groups
  • 15.4 The Schwartz space of smooth rapidly decreasing functions
  • 15.5 Fourier Integral
• 16 Advances of Metric Spaces
  • 16.1 The Stone–Weierstrass Theorem
  • 16.2 Contraction mappings and fixed point theorems
    • 16.2.1 The Banach fixed point theorem
    • 16.2.2 Applications of fixed point theory: The Picard-Lindelöf Theorem
    • 16.2.3 Applications of fixed point theory: Inverse and Implicit Function Theorems
  • 16.3 The Baire Category Theorem and Applications
    • 16.3.1 The Baire’s Categories
    • 16.3.2 Banach–Steinhaus Uniform Boundedness Principle
    • 16.3.3 The open mapping theorem
    • 16.3.4 The closed graph theorem
  • 16.4 Semi-norms and locally convex topological vector spaces
• A Tutorial Problems
  • A.1 Tutorial problems I
  • A.2 Tutorial problems II
  • A.3 Tutorial Problems III
  • A.4 Tutorial Problems IV
  • A.5 Tutorial Problems V
  • A.6 Tutorial Problems VI
  • A.7 Tutorial Problems VII
• B Solutions of Tutorial Problems
  • B.1 Solution of Tuitorial Problem I
  • B.2 Solutions of Tutorial Problems II
  • B.3 Solutions of Tutorial Problems III
  • B.4 Solutions of Tutorial Problems IV
  • B.5 Solutions of Tutorial Problems V
  • B.6 Solutions of Tutorial Problems VI
  • B.7 Solutions of Tutorial Problems VII
• C Course in the Nutshell
  • C.1 Some useful results and formulae (1)
  • C.2 Some useful results and formulae (2)
• D Supplementary Sections
  • D.1 Reminder from Complex Analysis