"Hilbert Spaces and Applications" Webpage

Hilbert Spaces and Applications Class Notes
Introduction to Hilbert Spaces with Applications, 1st Edition, Lokenath Debnath and Piotr Mikusinski (Academic Press, 1990) Debnath and Mikusinski's Introduction to Hilbert Spaces book

Debnath and Mikusinski's Introduction to Hilbert Spaces book

Copies of the class notes are on the internet in PDF format as given below. These notes have not been classroom tested and may contain typographical errors. The notes are based on a section of Applied Math 1 (MATH 5610) I taught fall 1998. This material ultimately evolved into the current ETSU class "Fundamentals of Funtional Analysis" (MATH 5740); notes for Fundamentals of Functional Analysis are online here. The "Proofs of Theorems" files were prepared in Beamer. The "Printout of Proofs" are printable PDF files of the Beamer slides without the pauses.

Syllabus for the Applied Math 1 class. Syllabus

Chapter 1. Normed Vector Spaces.

Section 1.1. Introduction.
Section 1.2. Vector Spaces. Section 1.2 notes

Supplement. Proofs of Theorems in Section 1.2. Beamer file of Section 1.2 proofs
Supplement. Printout of the Proofs of Theorems in Section 1.2. Print file of Section 1.2 proofs

Section 1.3. Linear Independence, Basis, Dimension. Section 1.3 notes
Section 1.4. Normed Spaces. Section 1.4 notes

Supplement. Proofs of Theorems in Section 1.4. Beamer file of Section 1.4 proofs
Supplement. Printout of the Proofs of Theorems in Section 1.4. Print file of Section 1.4 proofs

Section 1.5. Banach Spaces. Section 1.5 notes

Supplement. Proofs of Theorems in Section 1.5. Beamer file of Section 1.5 proofs
Supplement. Printout of the Proofs of Theorems in Section 1.5. Print file of Section 1.5 proofs

Section 1.6. Linear Mappings. Section 1.6 notes

Supplement. Proofs of Theorems in Section 1.6. Beamer file of Section 1.6 proofs
Supplement. Printout of the Proofs of Theorems in Section 1.6. Print file of Section 1.6 proofs

Section 1.7. Completion of Normed Spaces. Section 1.7 notes
Section 1.8. Contraction Mappings and the Fixed Point Theorem.
Supplement. Vector Spaces: Introduction, Examples, Dimension and Isomorphism. Vector Spaces notes (From the B.U.T.T.H.E.A.D. Seminars of the 1990s.)
Study Guide 1.

Chapter 3. Hilbert Spaces and Orthonormal Systems.

Section 3.1. Introduction. Section 3.1 notes
Section 3.2. Inner Product Spaces. Section 3.2 notes
Section 3.3. Examples of Inner Product Spaces. Section 3.3 notes
Section 3.4. Norm in an Inner Product Space. Section 3.4 notes

Supplement. Proofs of Theorems in Section 3.4. Beamer file of Section 3.4 proofs
Supplement. Printout of the Proofs of Theorems in Section 3.4. Print file of Section 3.4 proofs

Section 3.5. Hilbert Spaces - Definition and Examples. Section 3.5 notes

Supplement. Proofs of Theorems in Section 3.5. Beamer file of Section 3.5 proofs
Supplement. Printout of the Proofs of Theorems in Section 3.5. Print file of Section 3.5 proofs

Section 3.6. Strong and Weak Convergence. Section 3.6 notes

Supplement. Proofs of Theorems in Section 3.6. Beamer file of Section 3.6 proofs
Supplement. Printout of the Proofs of Theorems in Section 3.6. Print file of Section 3.6 proofs

Section 3.7. Orthogonal and Orthonormal Systems. Section 3.7 notes

Supplement. Proofs of Theorems in Section 3.7. Beamer file of Section 3.7 proofs
Supplement. Printout of the Proofs of Theorems in Section 3.7. Print file of Section 3.7 proofs

Section 3.8. Properties of Orthonormal Systems. Section 3.8 notes

Supplement. Proofs of Theorems in Section 3.8. Beamer file of Section 3.8 proofs
Supplement. Printout of the Proofs of Theorems in Section 3.8. Print file of Section 3.8 proofs

Section 3.9. Trigonometric Fourier Series.
Section 3.10. Orthogonal Complements and Projection Theorem.
Section 3.11. Linear Functionals and the Riesz Representation Theorem. Section 3.11 notes

Supplement. Proofs of Theorems in Section 3.11. Beamer file of Section 3.11 proofs
Supplement. Printout of the Proofs of Theorems in Section 3.11. Print file of Section 3.11 proofs

Section 3.12. Separable Hilbert Spaces. Section 3.12 notes

Supplement. Proofs of Theorems in Section 3.12. Beamer file of Section 3.12 proofs
Supplement. Printout of the Proofs of Theorems in Section 3.12. Print file of Section 3.12 proofs

Supplement. Vector Spaces and Hilbert Spaces: Norms, Completeness, Inner Products and Hilbert Spaces.Vector Spaces and Hilbert Spaces notes (From the B.U.T.T.H.E.A.D. Seminars of the 1990s.)
Study Guide 3.

Chapter 4. Linear Operators on Hilbert Spaces.

Section 4.1. Introduction.
Section 4.2. Examples of Operators. Section 4.2 notes
Section 4.3. Bilinear Functionals and Quadratic Forms.
Section 4.4. Adjoint and Self Adjoint Operators. Section 4.4 notes

Supplement. Proofs of Theorems in Section 4.4. Beamer file of Section 4.4 proofs
Supplement. Printout of the Proofs of Theorems in Section 4.4. Print file of Section 4.4 proofs

Section 4.5. Invertible, Normal, Isometric, and Unitary Operators. Section 4.5 notes

Supplement. Proofs of Theorems in Section 4.5. Beamer file of Section 4.5 proofs
Supplement. Printout of the Proofs of Theorems in Section 4.5. Print file of Section 4.5 proofs

Section 4.6. Positive Operators. Section 4.6 notes

Supplement. Proofs of Theorems in Section 4.6. Beamer file of Section 4.6 proofs
Supplement. Printout of the Proofs of Theorems in Section 4.6. Print file of Section 4.6 proofs

Section 4.7. Projection Operators.Section 4.7 notes
Section 4.8. Compact Operators. Section 4.8 notes
Section 4.9. Eigenvalues and Eigenvectors. Section 4.9 notes

Supplement. Proofs of Theorems in Section 4.9. Beamer file of Section 4.9 proofs
Supplement. Printout of the Proofs of Theorems in Section 4.9. Print file of Section 4.9 proofs

Section 4.10. Spectral Decomposition. Section 4.10 notes
Section 4.11. The Fourier Transform. Section 4.11 notes

Supplement. Proofs of Theorems in Section 4.11. Beamer file of Section 4.11 proofs
Supplement. Printout of the Proofs of Theorems in Section 4.11. Print file of Section 4.11 proofs

Supplement. 5.11. Applications of Fourier Transforms. Applications to ODEs and Integral Equations. Section 5.11 notes
Section 4.12. Unbounded Operators.
Supplement. Linear Operators on Hilbert Spaces: Operators, Norms, Self Adjoint Operators. Linear Operators on Hilbert Spaces notes (From the B.U.T.T.H.E.A.D. Seminars of the 1990s.)
Study Guide 4.

Chapter 7. Mathematical Foundations of Quantum Mechanics.

Section 7.1. Introduction. Section 7.1 notes
Section 7.2. Basic Concepts and Equations of Classical Mechanics. Section 7.2 notes
Section 7.3. Basic Concepts and Postulates of Quantum Mechanics. Section 7.3 notes
Section 7.4. The Heisenberg Uncertainty Principle.
Section 7.5. The Schrödinger Equations of Motion.
Section 7.6. The Schrödinger Picture.
Section 7.7. The Heisenberg Picture and the Heisenberg Equation of Motion.
Section 7.8. The Interaction Picture.
Section 7.9. The Linear Harmonic Oscillator.
Section 7.10. Angular Momentum Operators.
Study Guide 7.

Additional Chapters.

Chapter 2. The Lebesgue Integral.
Chapter 5. Applications to Integral and Differential Equations.
Chapter 6. Generalized Functions and Partial Differential Equations?
Chapter 8. Optimization Problems and Other Miscellaneous Applications.

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