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Lecture 1. Basics of Functional Analysis. Metric Spaces.
Examples of Metric Spaces Separable Spaces Maps of Metric Spaces Properties of Complete Metric Spaces

Lecture 2. Metric Spaces. Normed Spaces. Seminorms and Polynormed Spaces. Banach Spaces.
Wrapping Up the Previous Lecture: Properties of Complete Metric Spaces Normed Spaces An Example of a Linear Metric Space with a Shift-Invariant Metric That Is Not a Normed Space Seminorms and Polynormed Spaces Banach Spaces Self-Study Exercises

Lecture 3. Euclidean and Hilbert Spaces.
Proof of the Uniqueness of the Completion Why Banach Spaces Are not Good Enough Euclidean and Hilbert Spaces Properties of Dot Product Orthogonal Systems in Euclidean and Hilbert Spaces

Lecture 4. Separable Hilbert Spaces. Bases in Hilbert Spaces.
Further Development of the Previous Lecture: Existence of an Orthonormal Basis in Separable Hilbert Space Applications to Quantum Mechanics and Isometric Isomorphisms of Separable Hilbert Spaces Discussion of Self-Study Problems Typical Examples of Hilbert Spaces Exercises Exercises: Typical Examples of Bases in Hilbert Spaces Self-Study Exercises

Lecture 5. Compact and Precompact Sets in Metric Spaces.
Compactness Criteria Example: Closed Unit Ball in ℓ₂ is Not Compact Corollary: Unit Closed Ball is not Compact in Infinite-Dimensional Space Hausdorff Criterion for Precompactness Criteria for Precompactness in Specific Normed Spaces

Lecture 6. Compact and Precompact Sets: Exercises.
Proof of the Arzelà–Ascoli Theorem Theorem on Precompact Sets in Lₚ (Without Proof) Discussion of Self-Study Exercises from the Previous Lecture Exercises on Precompactness Self-Study Exercises

Lecture 7. Linear Operators and Functionals in Normed Spaces.
Bounded Operators Examples: Finding Norms of Operators B(X, Y) is Banach if Y is Banach Linear Functionals and Adjoint Spaces

Lecture 8. Linear Operators and Functionals in Normed Spaces: Exercises.
Discussion of Self-Study Exercises from the Previous Lecture Exercises on Bounded Operators and Functionals Self-Study Exercises

Lecture 9. The Hahn–Banach Theorem and the Corollaries.
The Hahn–Banach Theorem Corollaries of the Hahn–Banach Theorem Reflexive Spaces Adjoint Space to C[a,b]

Lecture 10. (C[a,b])*. Norms of Functionals.
Discussion of Self-Study Problems from the Previous Lecture Adjoint Space to C[a,b] Example of Finding the Norm of a Functional Self-Study Problems

Lecture 11. Hilbert Space Duality. Modes of Convergence.
Riesz Representation Theorem Reproducing Kernel Hilbert Space Modes of Convergence in Normed Spaces Modes of Convergence in B(X,Y)

Lecture 12. Reproducing Kernels and Weak Convergence: Exercises.
Discussion of Self-Study Problems from the Previous Lecture Exercises on Reproducing Kernels and Weak Convergence Self-Study Exercises

Lecture 13. Adjoint, Self-Adjoint, and Normal Operators. Hellinger–Toeplitz Theorem.
(Banach) Adjoint Operators Hilbert Adjoint Operators Self-Adjoint Operators Examples and Properties of Normal Operators Quadratic Form Associated to an Operator Boundedness and Weak Boundedness of Sets in Normed Spaces Hellinger–Toeplitz Theorem

Lecture 14. Adjoint Operators: Exercises.
Discussion of Self-Study Problems from the Previous Lecture Exercises on Adjoint Operators Self-Study Exercises

Lecture 15. Compact Operators. Inverse Operator.
Properties of Compact Operators Example: Integral Operators in C[a,b] and L₂[a.b] Inverse Operator Example: Inverse Operator to the Left-Shift Operator Two-Sided Inverse Operator is Linear

Lecture 16. Exercises on Compact and Inverse Operators.
Discussion of the Self-Study Problems from the Previous Lecture Exercises on Compact Operators Relation Between Notions of Compact and Adjoint Operators Exercises on Inverse Operators Self-Study Exercises

Lecture 17. Spectrum of a Bounded Operator. Classification of Points in the Spectrum.
Banach Bounded Inverse Theorem Properties of the Spectrum Spectrum of the Adjoint Operator Spectrum of a Normal Operator Spectrum of a Self-Adjoint Operator Spectral Radius

Lecture 18. Exercises on Spectra of Operators.
Discussion of the Self-Study Problems from the Previous Lecture Spectra of Similar Operators Self-Study Exercises

Lecture 19. The Hilbert–Schmidt Theorem.
Weyl Sequences Application of Weyl Sequences to Study the Spectrum of Multiplication Operator The Hilbert–Schmidt Theorem: Auxiliary Propositions The Hilbert–Schmidt Theorem Example: a Compact Operator in ℓ₂

Lecture 20. Applications of the Hilbert–Schmidt Theorem.
Discussion of Self-Study Exercises from the Previous Lecture Exercises: Applications of the Hilbert–Schmidt Theorem Schatten–von Neumann Classes and Nuclear Operators Self-Study Exercises

Lecture 21. Fredholm Theory.
Fredholm Theory: Introduction Auxiliary Lemmas Fredholm Solvability Conditions The Fredholm Alternative The Third Fredholm Theorem History of the Fredholm Theory Corollaries: Spectra of Compact Operators in Banach Spaces

Lecture 22. Fredholm Theory: Exercises.
Localization of Eigenvalues of a Compact Operator Discussion of the Self-Study Problems from the Previous Lecture Fredholm Theory: Exercises Self-Study Exercises

Lecture 23. Unbounded Operators: Introduction.
Volterra Operators Examples of Volterra Operators Unbounded Operators: Introduction Closed Operators Example of a Nonclosed Operator Closable Operators The Adjoint of an Unbounded Operator Theorem: A* is Closed Closability of a Densely Defined Operator

Lecture 24. Symmetric Operators.
Discussion of the Self-Study Problems from the Previous Lecture Properties of Relatively Bounded Operators The Set of Regular Points Self-Adjointness of a Closed Symmetric Operator with n₊(A)=n₋(A)=0 Examples: Non-Self-Adjoint Symmetric Operators and Self-Adjoint Extensions Self-Study Exercises

Lecture 25. Isometric Operators and the Cayley Transform. Self-Adjoint Extensions of Symmetric Operators.
Regular Points and Deficiency Indices of an Isometric Operator Unitary Operators Cayley Transform Domain of the Adjoint of a Symmetric Operator Description of Self-Adjoint Extensions of a Symmetric Operator in Terms of Extensions of the Corresponding Isometry

Lecture 26. Functional Calculus.
Spectral Theorem: Introduction Spectral Mapping Theorem Continuous Functional Calculus Borel Functional Calculus Example: Projection-Valued Measure Projection-Valued Measure for a Finite-Dimensional Operator and a Multiplication Operator Self-Study Exercises

Lecture 27. Spectral Theorem for Self-Adjoint Operators. Fourier Transform in L₁.
Cyclic Vectors Spectral Theorem for Bounded Self-Adjoint Operators with Simple Spectrum Spectral Theorem for Bounded Self-Adjoint Operators Fourier Transform Properties of the Fourier Transform in L₁

Lecture 28. Fourier Transform in L₁, S, and L₂.
Properties of the Fourier Transform in L₁: Further Discussion The Fourier Transform in Schwartz Space The Fourier Transform in L₂ Unitarity of the Fourier Transform in L₂ Spectrum of the Fourier Transform in L₂ Example: Calculation of the Fourier Transform

Lecture 29. Test Functions and Distributions.

Lecture 30. Convolution.