MATH 525B, Real Analysis, Spring 2027

Course: MATH 525B, Real Analysis, Spring 2027
Prerequisite: MATH 525A
Course Content: Banach spaces, Hahn–Banach, Baire category, open mapping, closed graph, and uniform boundedness theorems, Hilbert spaces, Lp spaces, duality, and interpolation, Fourier series, Fourier transform, convolution, distributions, Sobolev spaces, Radon measures, Riesz representation theorem.
Last update: November 8, 2026

Instructor: Steven Heilman, stevenmheilman(@-symbol)gmail.com
Office Hours: To be announced, KAP 406G
Lecture Meeting Time/Location: Mondays, Wednesdays and Fridays, 1:00–1:50 PM, GFS 108.
Recommended Textbook: Folland, Real Analysis.
Other Textbook: Knapp, Basic Real Analysis.

Midterm 1: Friday, February 19, 1:00–1:50 PM, in class.
Midterm 2: Monday, March 29, 1:00–1:50 PM, in class.
Final Exam: Wednesday, May 5, 2:00–4:00 PM, in class.
Other Resources: An introduction to mathematical arguments, Michael Hutchings; An Introduction to Proofs; How to Write Mathematical Arguments.
Email Policy:

My email address for this course is stevenmheilman(@-symbol)gmail.com
It is your responsibility to make sure you are receiving emails from stevenmheilman(@-symbol)gmail.com , and they are not being sent to your spam folder.
Do NOT email me with questions that can be answered from the syllabus.

Exam Resources: MATH 525B practice exams will be posted when available.

Exam Procedures: Students must bring their USCID cards to the midterms and to the final exam. Phones must be turned off. Cheating on an exam results in a score of zero on that exam. Exams can be regraded at most 15 days after the date of the exam. This policy extends to homeworks as well. All students are expected to be familiar with the USC Student Conduct Code. (See also here.)
Accessibility Services: If you are registered with the Office of Student Accessibility Services (OSAS), please discuss your accommodations with me at the beginning of the course and provide your accommodation letter as early as possible. Students requesting disability-related accommodations should register with OSAS. See OSAS for the registration process and current contact information.
OSAS: Grace Ford Salvatori (GFS) 120; Monday–Friday, 8:30 AM–5:00 PM.
213-740-0776; OSASFrontDesk@usc.edu.

Discrimination, sexual assault, and harassment are not tolerated by the university. For support and reporting options, see the Office of Civil Rights Compliance and USC Report & Response. These resources support the safety of the whole USC community.

Homework Policy:

Homeworks are due at 9 AM on the Fridays listed below.
Homeworks are submitted in Brightspace, under the "Assignments" tab. You are allowed unlimited submission "attempts" for an assignment, but only the last submission will be graded. To avoid internet issues, I recommend making your first submission of an assignment well in advance of the deadline.
Homeworks should be submitted as single PDF documents. One way to create a PDF document from paper homework assignments is the freely available Adobe Scan App
Late homework is not accepted.
If you still want to turn in late homework, then the number of minutes late, divided by ten, will be deducted from the score. (The time estimate is not guaranteed to be accurate.)
The lowest homework score will be dropped. This policy is meant to account for illnesses, emergencies, etc.
Do not submit homework via email.
A random subset of the homework problems will be graded each week. However, it is strongly recommended that you try to complete the entire homework assignment.
You may use whatever resources you want to do the homework, including computers, textbooks, friends, the TA, etc. However, I would discourage any over-reliance on the internet or AI tools, since their overuse could degrade your learning experience. By the end of the semester, you should be able to do the entire homework on your own, without any external help.
All homework assignments must be written by you, i.e. you cannot copy someone else's solution verbatim. However, I would very much encourage you to form study groups and do the homework together in small groups. Homework is the most important part of a graduate mathematics course, and I encourage you to take it very seriously.
Homework solutions will be posted on Friday after the homework is turned in.

Grading Policy:

The final course grade is weighted as the larger of the following two schemes.
- Scheme 1: homework (7%), class participation (5%), the first midterm (23%), the second midterm (27%), and the final (38%).
- Scheme 2: homework (7%), class participation (5%), the largest midterm grade (38%), and the final (50%).
The grade for the semester will be curved. However, I do not "curve down" since anyone who exceeds my expectations in the class by showing A-level performance on the exams and homeworks will receive an A for the class.
If you cannot attend one of the exams, you must notify me within the first two weeks of the start of the semester. Later requests for rescheduling will most likely be denied.
Class participation is not the same as attendance. I will never explicitly take attendance, but I will notice if someone is frequently absent. Things that increase your class participation grade include: asking good questions, paying attention in class, showing up on time or early to class, etc. Things that decrease your class participation grade include: excessive talking or disruptions during class, frequent absences, excessive texting/smartphone usage in class, frequent tardiness, etc.
You must take the final exam to pass the course.

Tentative Schedule: Topics and homework dates may change during the course. Lecture numbers correspond to the course outline. All dates below are in 2027.

Week	Monday	Tuesday	Wednesday	Thursday	Friday
1	Jan 11: Norms and normed spaces; completeness of L¹.	Jan 12	Jan 13: Product norms; quotient spaces; bounded linear maps; boundedness and continuity; completion.	Jan 14	Jan 15: Fundamental theorems of functional analysis; sublinear functionals; real Hahn–Banach theorem; Zorn’s lemma.
2	Jan 18: No class	Jan 19	Jan 20: Complex linear functionals; complex Hahn–Banach theorem; reflexivity; finite-dimensional normed spaces are reflexive.	Jan 21	Jan 22: Baire category theorem; Baire spaces. Homework 1 due.
3	Jan 25: Open mapping, inverse mapping, and closed graph theorems; Banach–Steinhaus theorem.	Jan 26	Jan 27: Uniform boundedness principle; locally convex vector spaces; Fréchet spaces.	Jan 28	Jan 29: Examples of locally convex spaces; Alaoglu’s theorem; Hilbert spaces; Cauchy–Schwarz inequality.
4	Feb 1: Parallelogram identity; Pythagorean theorem; orthogonal complements and the double orthogonal complement.	Feb 2	Feb 3: Riesz representation theorem for Hilbert spaces; Bessel’s inequality.	Feb 4	Feb 5: Parseval’s identity. Homework 2 due.
5	Feb 8: Application: Banach limits; L^p spaces; Young’s inequality.	Feb 9	Feb 10: Hölder’s inequality; Minkowski’s inequality; completeness of L^p.	Feb 11	Feb 12: Density of simple functions; relations between L^p and L^q spaces; interpolation; Nirenberg interpolation; duality of L^p.
6	Feb 15: No class	Feb 16	Feb 17: The canonical isometry L^q → (L^p)*; F. Riesz theorem.	Feb 18	Feb 19: Midterm 1
7	Feb 22: L^p duality: finite-measure and σ-finite cases.	Feb 23	Feb 24: L^p duality and reflexivity for 1 < p < ∞; M. M. Day theorem.	Feb 25	Feb 26: Chebyshev’s inequality; boundedness of integral operators; Young’s inequality as a corollary. Homework 3 due
8	Mar 1: Minkowski’s integral inequality; Young’s theorem.	Mar 2	Mar 3: Young’s inequality for weak L^p; interpolation theorems; Marcinkiewicz interpolation theorem; the distribution-function formula for the L^p norm.	Mar 4	Mar 5: Proof of Marcinkiewicz interpolation; Riesz–Thorin interpolation theorem; three-lines lemma.
9	Mar 8: Duality lemma; proof of the Riesz–Thorin interpolation theorem.	Mar 9	Mar 10: Fourier analysis; convolution; convolution and derivatives.	Mar 11	Mar 12: Approximate identities. Homework 4 due
10	Mar 15: No class	Mar 16: No class	Mar 17: No class	Mar 18: No class	Mar 19: No class
11	Mar 22: Heat equation; convergence of approximate identities; density of C_c^∞ in L^p for 1 ≤ p < ∞; Urysohn’s lemma.	Mar 23	Mar 24: Fourier series and Fourier transform; the L¹-to-L^∞ bound; the torus; Riemann–Lebesgue lemma.	Mar 25	Mar 26: Properties of the Fourier transform.
12	Mar 29: Midterm 2	Mar 30	Mar 31: Fourier transform of a Gaussian; integral duality identity; Fourier inversion theorem; Plancherel theorem.	Apr 1	Apr 2: Hausdorff–Young inequality. Homework 5 due
13	Apr 5: Fourier transform formulas; Schwartz class; the Schwartz space as a Fréchet space.	Apr 6	Apr 7: Continuity criteria for linear functionals on the Schwartz space; tempered distributions; operations on S′.	Apr 8	Apr 9: Reflection, translation, and multiplication of tempered distributions.
14	Apr 12: Fourier transform on tempered distributions; continuity on S′; Plancherel transform.	Apr 13	Apr 14: Convolution.	Apr 15	Apr 16: Sobolev spaces; Sobolev theorem; distributions. Homework 6 due
15	Apr 19: Calculus in D′(U); multiplication, reflection, translation, and support	Apr 20	Apr 21: Convolution and mollifiers.	Apr 22	Apr 23: Radon measures; outer and inner regularity; Riesz representation theorem.
16	Apr 26: Proof of the Riesz representation theorem.	Apr 27	Apr 28: Leeway	Apr 29	Apr 30: Leeway. Homework 7 due

Advice on succeeding in a math class:

Review the relevant course material before you come to lecture. Consider reviewing course material a week or two before the semester starts.
When reading mathematics, use a pencil and paper to sketch the calculations that are performed by the author.
Come to class with questions, so you can get more out of the lecture. Also, finish your homework at least two days before it is due, to alleviate deadline stress.
Write a rough draft and a separate final draft for your homework. This procedure will help you catch mistakes.
If you are having difficulty with the material or a particular homework problem, review Polya's Problem Solving Strategies, and come to office hours.

Homework

Homework .tex files

Supplementary Notes

525B Lecture Notes