Copies of the classnotes are on the internet in PDF format as given below. The "Proofs of Theorems" files were prepared in Beamer. The "Printout of Proofs" are printable PDF files of the Beamer slides without the pauses.

1. Linear Spaces and Operators.

• 1.1. Introduction. Section 1.1 notes
• 1.2. Linear Spaces. Section 1.2 notes
• 1.3. Linear Operators. Section 1.3 notes
• Supplement. Proofs of Theorems in Section 1.3. Beamer file of Section 1.3 proofs
• Supplement. Printout of the Proofs of Theorems in Section 1.3. Print file of Section 1.3 proofs
• 1.4. Passage from Finite- to Infinite-Dimensional Spaces. Section 1.4 notes
• Study Guide 1.

2. Normed Linear Spaces: The Basics.

• 2.1. Metric Spaces. Section 2.1 notes
• 2.2. Norms. Section 2.2 notes
• Supplement. Proofs of Theorems in Section 2.2. Beamer file of Section 2.2 proofs
• Supplement. Printout of the Proofs of Theorems in Section 2.2. Print file of Section 2.2 proofs
• 2.3. Space of Bounded Functions. Section 2.3 notes
• Supplement. Proofs of Theorems in Section 2.3. Beamer file of Section 2.3 proofs
• Supplement. Printout of the Proofs of Theorems in Section 2.3. Print file of Section 2.3 proofs
• 2.4. Bounded Linear Operators. Section 2.4 notes
• Supplement. Proofs of Theorems in Section 2.4. Beamer file of Section 2.4 proofs
• Supplement. Printout of the Proofs of Theorems in Section 2.4. Print file of Section 2.4 proofs
• 2.5. Completeness. Section 2.5 notes
• Supplement. Proofs of Theorems in Section 2.5. Beamer file of Section 2.5 proofs
• Supplement. Printout of the Proofs of Theorems in Section 2.5. Print file of Section 2.5 proofs
• 2.6. Comparison of Norms. Section 2.6 notes
• Supplement. Proofs of Theorems in Section 2.6. Beamer file of Section 2.6 proofs
• Supplement. Printout of the Proofs of Theorems in Section 2.6. Print file of Section 2.6 proofs
• 2.7. Quotient Spaces. Section 2.7 notes
• Supplement. Proofs of Theorems in Section 2.7. Beamer file of Section 2.7 proofs
• Supplement. Printout of the Proofs of Theorems in Section 2.7. Print file of Section 2.7 proofs
• 2.8. Finite-Dimensional Normed Linear Spaces. Section 2.8 notes
• Supplement. Proofs of Theorems in Section 2.8. Beamer file of Section 2.8 proofs
• Supplement. Printout of the Proofs of Theorems in Section 2.8. Print file of Section 2.8 proofs
• 2.9. L^p Spaces. Section 2.9 notes
• 2.10. Direct Sums and Products. Section 2.10 notes
• 2.11. Schauder Bases. Section 2.11 notes
• Supplement. Groups, Fields, and Vector Spaces (Section 5.1 from Real Analysis with an Introduction to Wavelets and Applications - There is a detailed discussion of Hamel and Schauder bases). Groups, Fields, and Vector Spaces notes (Includes a proof that every vector space has a Hamel basis and that any two Hamel bases for a given vector space have the same cardinality.)
• Supplement. Proofs of Theorems in Section 5.1 of Real Analysis with an Introduction to Wavelets and Applications. Beamer file of Groups, Fields, and Vector Spaces proofs
• Supplement. Printout of the Proofs of Theorems in Section 5.1 of Real Analysis with an Introduction to Wavelets and Applications. Print file of Groups, Fields, and Vector Spaces proofs
• 2.12. Fixed Points and Contraction Mappings. Section 2.12 notes
• Supplement. Proofs of Theorems in Section 2.12. Beamer file of Section 2.12 proofs
• Supplement. Printout of the Proofs of Theorems in Section 2.12. Print file of Section 2.12 proofs
• Study Guide 2.

3. Major Banach Space Theorems.

• 3.1. Introduction. Section 3.1 notes
• 3.2. Baire Category Theorem. Section 3.2 notes
• Supplement. Proofs of Theorems in Section 3.2. Beamer file of Section 3.2 proofs
• Supplement. Printout of the Proofs of Theorems in Section 3.2. Print file of Section 3.2 proofs
• 3.3. Open Mappings. Section 3.3 notes
• Supplement. Proofs of Theorems in Section 3.3. Beamer file of Section 3.3 proofs
• Supplement. Printout of the Proofs of Theorems in Section 3.3. Print file of Section 3.3 proofs
• 3.4. Bounded Inverses. 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
• 3.5. Closed Linear Operators. 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
• 3.6. Uniform Boundedness Principle. 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
• Study Guide 3.

4. Hilbert Spaces.

• 4.1. Introduction. Section 4.1 notes
• 4.2. Semi-Inner Products. Section 4.2 notes
• Supplement. Proofs of Theorems in Section 4.2. Beamer file of Section 4.2 proofs
• Supplement. Printout of the Proofs of Theorems in Section 4.2. Print file of Section 4.2 proofs
• Supplement. Inner Product Spaces (Section 5.2 from Real Analysis with an Introduction to Wavelets and Applications). Inner Product Spaces notes
• Supplement. Proofs of Theorems in Section 5.2 of Real Analysis with an Introduction to Wavelets and Applications. Beamer file of Inner Product Spaces proofs.
• Supplement. Printout of Theorems in Section 5.2 of Real Analysis with an Introduction to Wavelets and Applications. Print file of Inner Product Spaces proofs
• 4.3. Nearest Points and Convexity. Section 4.3 notes
• Supplement. Proofs of Theorems in Section 4.3. Beamer file of Section 4.3 proofs
• Supplement. Printout of the Proofs of Theorems in Section 4.3. Print file of Section 4.3 proofs
• 4.4. Orthogonality. 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
• Supplement. The Space L² (Section 5.3 from Real Analysis with an Introduction to Wavelets and Applications). The Space L² notes
• Supplement. Projections and Hilbert Space Isomorphisms (Section 5.4 from Real Analysis with an Introduction to Wavelets and Applications). Projections and Hilbert Space Isomorphisms notes
• Supplement. Proofs of Theorems in Section 5.4 of Real Analysis with an Introduction to Wavelets and Applications. Beamer file of Projections and Hilbert Space Isomorphisms proofs
• Supplement. Printout of the Proofs of Theorems in Section 5.4 of Real Analysis with an Introduction to Wavelets and Applications. Print file of Projections and Hilbert Space Isomorphisms proofs
• 4.5. Linear Functionals on Hilbert Spaces. Section 4.5 notes
• Supplement. A Proof of Theorem 4.22. Proof of Theorem 4.22 notes
• 4.6. Linear Operators on Hilbert Spaces. 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
• 4.7. Order Relation on Self-Adjoint Operators. Section 4.7 notes
• Supplement. Proofs of Theorems in Section 4.7. Beamer file of Section 4.7 proofs
• Supplement. Printout of the Proofs of Theorems in Section 4.7. Print file of Section 4.7 proofs
• Study Guide 4.

5. Hahn-Banach Theorem.

• 5.1. Introduction. Section 5.1 notes
• 5.2. Basic Version of Hahn-Banach Theorem. Section 5.2 notes
• Supplement. Proofs of Theorems in Section 5.2. Beamer file of Section 5.2 proofs
• Supplement. Printout of the Proofs of Theorems in Section 5.2. Print file of Section 5.2 proofs
• 5.3. Complex Version of Hahn-Banach Theorem. Section 5.3 notes
• Supplement. Proofs of Theorems in Section 5.3. Beamer file of Section 5.3 proofs
• Supplement. Printout of the Proofs of Theorems in Section 5.3. Print file of Section 5.3 proofs
• 5.4. Application to Normed Linear Spaces. Section 5.4 notes
• Supplement. Proofs of Theorems in Section 5.4. Beamer file of Section 5.4 proofs
• Supplement. Printout of the Proofs of Theorems in Section 5.4. Print file of Section 5.4 proofs
• 5.5. Geometric Versions of Hahn-Banach Theorem. Section 5.5 notes
• Supplement. Proofs of Theorems in Section 5.5. Beamer file of Section 5.5 proofs
• Supplement. Printout of the Proofs of Theorems in Section 5.5. Print file of Section 5.5 proofs
• Study Guide 5.

6. Duality.

• 6.1. Examples of Dual Spaces. Section 6.1 notes
• Supplement. Proofs of Theorems in Section 6.1. Beamer file of Section 6.1 proofs
• Supplement. Printout of the Proofs of Theorems in Section 6.1. Print file of Section 6.1 proofs
• 6.2. Adjoints. Section 6.2 notes
• Supplement. Proofs of Theorems in Section 6.2. Beamer file of Section 6.2 proofs
• Supplement. Printout of the Proofs of Theorems in Section 6.2. Print file of Section 6.2 proofs
• 6.3. Double Duals and Reflexivity. Section 6.3 notes
• Supplement. Proofs of Theorems in Section 6.3. Beamer file of Section 6.3 proofs
• Supplement. Printout of the Proofs of Theorems in Section 6.3. Print file of Section 6.3 proofs
• 6.4. Weak and Weak* Convergence. Section 6.4 notes
• Supplement. Proofs of Theorems in Section 6.4. Beamer file of Section 6.4 proofs
• Supplement. Printout of the Proofs of Theorems in Section 6.4. Print file of Section 6.4 proofs
• Study Guide 6.

7. Topological Linear Spaces.

• 7.1. Review of General Topology.
• 7.2. Topologies in Linear Spaces.
• 7.3. Linear Functionals on Topological Linear Spaces.
• 7.4. Weak Topology.
• 7.5. Weak* Topology.
• 7.6. Extreme Points and Krein-Milman Theorem.
• 7.7. Operator Topologies.
• Study Guide 7.

8. The Spectrum.

• 8.1. Introduction. Section 8.1 notes
• 8.2. Banach Algebras. Section 8.2 notes
• Supplement. Proofs of Theorems in Section 8.2. Beamer file of Section 8.2 proofs
• Supplement. Printout of the Proofs of Theorems in Section 8.2. Print file of Section 8.2 proofs
• 8.3. General Properties of the Spectrum. Section 8.3 notes
• Supplement. Proofs of Theorems in Section 8.3. Beamer file of Section 8.3 proofs
• Supplement. Printout of the Proofs of Theorems in Section 8.3. Print file of Section 8.3 proofs
• 8.4. Numerical Range. Section 8.4 notes
• Supplement. Proofs of Theorems in Section 8.4. Beamer file of Section 8.4 proofs
• Supplement. Printout of the Proofs of Theorems in Section 8.4. Print file of Section 8.4 proofs
• 8.5. Spectrum of a Normal Operator. Section 8.5 notes
• Supplement. Proofs of Theorems in Section 8.5. Beamer file of Section 8.5 proofs
• Supplement. Printout of the Proofs of Theorems in Section 8.5. Print file of Section 8.5 proofs
• 8.6. Functions of Operators. Section 8.6 notes
• Supplement. Proofs of Theorems in Section 8.6. Beamer file of Section 8.6 proofs
• Supplement. Printout of the Proofs of Theorems in Section 8.6. Print file of Section 8.6 proofs
• 8.7. Brief Introduction to C*-Algebras. Section 8.7 notes
• Study Guide 8.

9. Compact Operators.

• 9.1. Introduction and Basic Definitions. Section 9.1 notes
• Supplement. Proofs of Theorems in Section 9.1. Beamer file of Section 9.1 proofs
• Supplement. Printout of the Proofs of Theorems in Section 9.1. Print file of Section 9.1 proofs
• 9.2. Compactness Criteria in Metric Spaces. Section 9.2 notes
• Supplement. Proofs of Theorems in Section 9.2. Beamer file of Section 9.2 proofs
• Supplement. Printout of the Proofs of Theorems in Section 9.2. Print file of Section 9.2 proofs
• 9.3. New Compact Operators from Old. Section 9.3 notes
• Supplement. Proofs of Theorems in Section 9.3. Beamer file of Section 9.3 proofs
• Supplement. Printout of the Proofs of Theorems in Section 9.3. Print file of Section 9.3 proofs
• 9.4. Spectrum of a Compact Operator. Section 9.4 notes
• Supplement. Proofs of Theorems in Section 9.4. Beamer file of Section 9.4 proofs
• Supplement. Printout of the Proofs of Theorems in Section 9.4. Print file of Section 9.4 proofs
• 9.5. Compact Self-Adjoint Operators on Hilbert Spaces. Section 9.5 notes
• Supplement. Proofs of Theorems in Section 9.5. Beamer file of Section 9.5 proofs
• Supplement. Printout of the Proofs of Theorems in Section 9.5. Print file of Section 9.5 proofs
• 9.6. Invariant Subspaces. Section 9.6 notes
• Supplement. Proofs of Theorems in Section 9.6. Beamer file of Section 9.6 proofs
• Supplement. Printout of the Proofs of Theorems in Section 9.6. Print file of Section 9.6 proofs
• Study Guide 9.

Return to Bob Gardner's home page