18.102 Intro to Functional Analysis
Welcome to 18.102: Introduction to functional analysis
- Syllabus: vector spaces, topological spaces, metric spaces, completeness, Baire category theorem, normed and Banach spaces, Hahn-Banach theorem, open mapping theorem, closed graph theorem, (some) abstract measure theory, Lebesgue measure, Lebesgue integration, L^p spaces, dual spaces and weak topologies, Banach-Alaoglu theorem, Hilbert spaces, compact operators, spectral theorem.
- Prerequisites: Linear algebra (18.06, 18.700, or 18.701) and real analysis (18.100A, 18.100B, 18.100P, 18.100Q) or permission from the lecturer.
- Homework: There will be 7 problem sets in total. Only the best 6 problem sets count towards your final grade. No late homework will be accepted except for cases approved by S3 (https://studentlife.mit.edu/s3).
- Grading: The final grade is the weighted average of the problem sets (30%, i.e. 5% for each problem set), the two midterms (30%, i.e. 15% each), and the final (40%).
- Office hours:
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- Christoph Kehle: Tuesday, 1.30 pm-2.30 pm, 2-277, kehle@mit.edu (no office hour on April 8!)
- Ryan Chen: Thursday 10 am - 11 am, 2-255, rcchen@mit.edu
- Ryan will have office hours from 1:45 PM to 2:45 PM on Friday March 21, 2025 (moved from Thursday March 20).
- Divya Shyamal: Thursday, 8 - 9 pm (moved from Monday, March 31), Zoom Links to an external site., dshyamal@mit.edu
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- Exams: There will be two midterms on 3 March and 9 April and one final on 21 May.
- Lecture notes Download Lecture notes
- Problem sets:
- Problem set 1 Download Problem set 1 (due 2/14)
- Problem set 2 Download Problem set 2 (due 2/24)
- Problem set 3 Download Problem set 3 (due 3/10)
- Problem set 4 Download Problem set 4 (due 4/4)
- Problem set 5 Download Problem set 5 (due 4/14)
- Solutions:
- Midterms:
| Lecture number | Date | Notes | Contents | Pset number | Pset due (at 11.59pm) |
| 1 | 2/3/2025 | Zorn's lemma, topological spaces | |||
| 2 | 2/5/2025 | Continuity, compactness, Tychnoff's theorem | |||
| 3 | 2/10/2025 | Metric spaces, completeness, completion | |||
| 4 | 2/12/2025 | l^p, C(K) spaces and their completeness | 1 | 2/14/2025 (Friday) | |
| 5 | 2/18/2025 | moved from 2/17/2025 | Baire category theorem and applications, normed spaces, Banach spaces | ||
| 6 | 2/19/2025 | Riesz' lemma, linear operators | |||
| 7 | 2/24/2025 | Hahn--Banach theorem, Banach Steinhaus theorem | 2 | 2/24/2025 | |
| 8 | 2/26/2025 | Open mapping theorem, closed graph theorem, inverse mapping theorem | |||
| 9 | 3/3/2025 | midterm, in class | |||
| 10 | 3/5/2025 |
Measure spaces, outer measures |
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| 11 | 3/10/2025 | Carathéodory extension theorem and Lebesgue measure | 3 | 3/13/2025 | |
| 12 | 3/12/2025 | Approximation theorems for Lebesgue measures, Hausdorff measure | |||
| 13 | 3/17/2025 | Measurable functions and integration, | |||
| 14 | 3/19/2025 |
Convergence theorems, L^p spaces |
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| Spring break | 3/24/2025 | ||||
| Spring break | 3/26/2025 | ||||
| 15 | 3/31/2025 |
Completeness of L^p spaces, Dual spaces, weak topologies, Locally convex spaces |
4 | 3/31/2025 | |
| 16 | 4/2/2025 |
Banach-Alaoglu theorem, Riesz representation for L^p spaces I |
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| 17 | 4/7/2025 | Riesz representation for L^p spaces II, Inner product spaces, Hilbert spaces | |||
| 18 | 4/9/2025 | midterm, in class | |||
| 19 | 4/14/2025 |
Projection theorem, Bessel's inequality, Parseval's identity |
5 | 4/14/2025 | |
| 20 | 4/16/2025 |
Riesz representation theorem for Hilbert space, Hilbert-Adjoint Operator |
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| Patriots' day | 4/21/2025 | no lecture | |||
| Drop date | 4/22/2025 | ||||
| 21 | 4/23/2025 |
Spectral theory on Banach spaces |
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| 22 | 4/28/2025 |
Compact operators |
6 | 4/28/2025 | |
| 23 | 4/30/2025 | Fredholm alternative | |||
| 24 | 5/5/2025 | Spectral theory for bounded self-adjoint operators | |||
| 25 | 5/7/2025 | Fourier series and L^2 (0,2pi) | |||
| 26 | 5/12/2025 | Review of the course, exam preparation | 7 |
5/09/2025 (Friday) |
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| Final exam | 21 May | in person |
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Course Summary:
| Date | Details | Due |
|---|---|---|
| Fri Feb 14, 2025 | Assignment Problem set 1 | due by 11:59pm |
| Mon Feb 24, 2025 | Assignment Problem set 2 | due by 11:59pm |
| Mon Mar 10, 2025 | Assignment Problem Set 3 | due by 11:59pm |
| Mon Mar 31, 2025 | Assignment Problem set 4 | due by 11:59pm |