Class meets Monday & Wednesday, 75 minutes per session.
Text: Hillen, Elements of Applied Functional Analysis (June 2023)
Important Dates¶
First day of classes: Thursday, January 29
Presidents’ Day (no class): Monday, February 16
Thursday, February 19: Monday schedule followed
Spring recess: March 15–22
Patriots’ Day (no class): Monday, April 20
Friday, April 24: Monday schedule followed
Last day of classes: Friday, May 8
Schedule¶
| # | Date | Day | Topic | Hillen | HW |
|---|---|---|---|---|---|
| 1 | Feb 2 | Mon | Completeness - Cauchy sequences, Banach spaces, complete | §2.1, §2.3 | |
| 2 | Feb 4 | Wed | Density & Separability - Approximation, mollifiers, Weierstrass | §2.1.3 | |
| 3 | Feb 9 | Mon | Compactness - Sequential compactness, Arzelà-Ascoli, failure in -dim | §2.4 | HW 1 due |
| 4 | Feb 11 | Wed | Linear Operators - Bounded operators, operator norm | §3.1–3.2 | |
| 5 | Feb 18 | Wed | Baire Category Theorem | §3.3 | |
| 6 | Feb 19 | Thu | Banach-Steinhaus (Uniform Boundedness) | §3.3 | Mon schedule |
| 7 | Feb 23 | Mon | Open Mapping & Closed Graph Theorems | §3.3 | HW 2 due |
| 8 | Feb 25 | Wed | Compact operators and spectral theory | §3.4, §8.1–8.4 | |
| 9 | Mar 2 | Mon | Duality - Dual spaces, Zorn’s lemma | §4.1 | |
| 10 | Mar 4 | Wed | Hahn-Banach Theorem (with proof) | §4.1 | HW 3 due |
| 11 | Mar 9 | Mon | Dual of Hilbert spaces, Riesz representation | §4.2 | |
| 12 | Mar 11 | Wed | Reflexive spaces, weak convergence | §4.3–4.4 | |
| Mar 15-22 | SPRING BREAK | ||||
| 13 | Mar 23 | Mon | Weak-* convergence, Banach-Alaoglu | §4.5 | HW 4 due |
| 14 | Mar 25 | Wed | Sobolev Spaces - Distributions, weak derivatives | §5.1 | |
| 15 | Mar 30 | Mon | Sobolev spaces, embeddings | §5.2–5.3 | |
| 16 | Apr 1 | Wed | Fixed Points & Lax-Milgram - Banach fixed point, Lax-Milgram | §6.1, §6.4 | HW 5 due |
| 17 | Apr 6 | Mon | Calculus of Variations - Euler-Lagrange equations | §7.1–7.2 | |
| 18 | Apr 8 | Wed | Existence of minimizers, direct method | §7.7 | |
| 19 | Apr 13 | Mon | Semigroups - C₀-semigroups, generators | §9.1–9.2 | HW 6 due |
| 20 | Apr 15 | Wed | Hille-Yosida Theorem | §9.4 | |
| 21 | Apr 22 | Wed | Applications to PDEs | §9.6 | |
| 22 | Apr 24 | Fri | Review / Buffer | Mon schedule | |
| 23 | Apr 27 | Mon | Student Lectures | HW 7 due | |
| 24 | Apr 29 | Wed | Student Lectures | ||
| 25 | May 4 | Mon | Student Lectures | ||
| 26 | May 6 | Wed | Student Lectures | ||
| May 11-15 | Student Lectures (finals week) |
Topic Summary¶
| Topic | Classes | Hillen | Content |
|---|---|---|---|
| Foundations | 3 | Ch 2 | Completeness, density/separability, compactness |
| Linear Operators | 5 | Ch 3, 8 | Bounded ops, Baire, Banach-Steinhaus, Open Mapping, compact ops + spectral theory |
| Duality | 5 | Ch 4 | Hahn-Banach (with proof), Riesz, weak convergence, Banach-Alaoglu |
| Sobolev Spaces | 2 | Ch 5 | Distributions, weak derivatives, embeddings |
| Fixed Points & Lax-Milgram | 1 | Ch 6 | Banach fixed point, Lax-Milgram |
| Calculus of Variations | 2 | Ch 7 | Euler-Lagrange, direct method |
| Semigroups | 3 | Ch 9 | C₀-semigroups, Hille-Yosida, PDE applications |
| Student Lectures | 4+ | Student presentations |
Assessment¶
Homework (7 assignments): Weekly problem sets
Student Lectures: Each student presents 1-2 lectures on a topic (scheduled during finals week and potentially earlier)
Optional take-home: TBD
Possible Student Lecture Topics¶
Neural networks and universal approximation (Barron spaces)
Bifurcation theory
Reaction-diffusion equations
Schrödinger equation and quantum mechanics
Wave equation
Additional fixed point theorems (Leray-Schauder)
Trace theorems and boundary values
Analytic semigroups