Functional Analysis Class Notes
From A First Course In Functional Analysis, S. David Promislow
with Supplemental Notes on Hilbert Spaces from
Real Analysis with an Introduction to Wavelets and Applications, D. Hong, J. Wang, and R. Gardner  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. PDF.
• 1.2. Linear Spaces. PDF.
• 1.3. Linear Operators. PDF.
• 1.4. Passage from Finite- to Infinite-Dimensional Spaces. PDF.
• Study Guide 1.

2. Normed Linear Spaces: The Basics.

• 2.1. Metric Spaces. PDF.
• 2.2. Norms. PDF.
• Supplement. Proofs of Theorems in Section 2.2. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 2.2. PDF.
• 2.3. Space of Bounded Functions. PDF.
• 2.4. Bounded Linear Operators. PDF.
• Supplement. Proofs of Theorems in Section 2.4. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 2.4. PDF.
• 2.5. Completeness. PDF.
• Supplement. Proofs of Theorems in Section 2.5. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 2.5. PDF.
• 2.6. Comparison of Norms. PDF.
• Supplement. Proofs of Theorems in Section 2.6. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 2.6. PDF.
• 2.7. Quotient Spaces. PDF.
• Supplement. Proofs of Theorems in Section 2.7. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 2.7. PDF.
• 2.8. Finite-Dimensional Normed Linear Spaces. PDF.
• Supplement. Proofs of Theorems in Section 2.8. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 2.8. PDF.
• 2.9. Lp Spaces. PDF.
• 2.10. Direct Sums and Products. PDF.
• 2.11. Schauder Bases. PDF.
• 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). PDF. (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. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 5.1 of Real Analysis with an Introduction to Wavelets and Applications. PDF.
• 2.12. Fixed Points and Contraction Mappings. PDF.
• Supplement. Proofs of Theorems in Section 2.12. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 2.12. PDF.
• Study Guide 2.

3. Major Banach Space Theorems.

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

4. Hilbert Spaces.

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

5. Hahn-Banach Theorem.

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

6. Duality.

• 6.1. Examples of Dual Spaces. PDF.
• Supplement. Proofs of Theorems in Section 6.1. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 6.1. PDF.
• Supplement. Proofs of Theorems in Section 6.2. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 6.2. PDF.
• 6.3. Double Duals and Reflexivity. PDF.
• Supplement. Proofs of Theorems in Section 6.3. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 6.3. PDF.
• 6.4. Weak and Weak* Convergence. PDF.
• Supplement. Proofs of Theorems in Section 6.4. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 6.4. PDF.
• 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. PDF.
• 8.2. Banach Algebras. PDF.
• Supplement. Proofs of Theorems in Section 8.2. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 8.2. PDF.
• 8.3. General Properties of the Spectrum. PDF.
• Supplement. Proofs of Theorems in Section 8.3. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 8.3. PDF.
• 8.4. Numerical Range. PDF.
• Supplement. Proofs of Theorems in Section 8.4. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 8.4. PDF.
• 8.5. Spectrum of a Normal Operator. PDF.
• Supplement. Proofs of Theorems in Section 8.5. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 8.5. PDF.
• 8.6. Functions of Operators. PDF.
• Supplement. Proofs of Theorems in Section 8.6. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 8.6. PDF.
• 8.7. Brief Introduction to C*-Algebras. PDF.
• Study Guide 8.

9. Compact Operators.

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