Instructor Information
Dr. Jeffrey Ovall
I have been teaching mathematics for roughly 30 years. Functional Analysis is my favorite graduate-level course to teach.
I am committed to creating an accessible learning environment for all of my students. If you encounter any accessibility barriers in this course, please reach out to me via email. I will work with you and our campus partners to evaluate the issue and find solutions.
Office: Fariborz Maseeh Hall (FMH), 464R
Office Hours: MW 2-3pm, or by (rare) special appointment
Email: jovall@pdx.edu
Phone:(503) 725-3610
Sequence and Course Overview
The principal objects of study in this sequence, MTH
617-619, are linear mappings/operators between
infinite-dimensional normed vector spaces. Many of the most
significant results apply in Banach and Hilbert spaces:
e.g. Hahn-Banach Theorem(s), Baire Category Theorem, Open
Mapping Theorem, Closed Graph Theorem, Closed Range Theorem,
etc. A high-point of the sequence will be a careful study of
these "Great Theorems", their interrelations and some of their
applications; this will largely take place in MTH 618.
Spectral theory in Hilbert Space (compact selfadjoint
operators) and Banach Space (Fredholm theory) will also be
covered. In MTH 619, we will explore topics in the subject of your and my choosing.
MTH 617 will lay the groundwork for these significant results,
by exploring important properties in normed spaces
(particularly Banach spaces) and bounded/continuous and compact operators
between them. Important normed spaces, such as \(C(K,Y)\),
\(\ell^p\), \(L^p(\Omega)\), \(\mathcal{L}(X,Y)\) and
\(X'=\mathcal{L}(X,\mathbb{K})\) will be treated, and some
applications of interest will be highlighted.
MTH 617 and 618 will be "traditional" courses in the
sense that we will follow the assigned course textbook fairly
closely, and course grades will be based on assignments and
exams. For MTH 619, I will likely present some material not
discussed in our text, and there will be no assignments or
exams. Your course grade in MTH 619 will be based on a roughly
30 minute presentation on a Functional Analysis-related topic
that you choose (and I approve), the supporting notes for your
presentation, and engagement with the
presentations of others in the class.
Prerequisites: There are no formal pre-requisites, though two terms of Real Analysis at the advanced undergraduate level or beginning graduate level (e.g. MTH 411/511, 412/512) are recommended.
Course Meeting Times: MW 9-10:15am
Course Meeting Location: Stott Center (PSC) 128 (opens in new tab)
Course Reference Number (CRN): 13707 (Section 001)
Syllabus
Textbook and Topics
We will be using the textbook Linear and Nonlinear Functional Analysis with Applications (Second Edition), by Philippe G. Ciarlet (opens in new tab) . The publisher provides a good discount (~25%) to SIAM members. Portland State University is an academic partner institution of SIAM, so graduate students (Masters and PhD) are eligible for free membership. If you are not already a SIAM member, you may wish to take advantage of the free membership by signing up here (opens in new tab) here. If you find or suspect typos in the book, please let me know. I have a list of typos and corrections from the first edition. Many of these were corrected for the second edition, though not all. Topics: We will cover portions of Chapters 2-4. Some of the sections we plan to cover are:
- Normed Vector Spaces (Ch. 2)
- 2.2: Normed vector spaces, quotient spaces, F. Riesz Lemma, further properties
- 2.4: The space \(C(K,Y)\) and uniform convergence
- 2.5: \(\ell^p\) spaces
- 2.6: Lebesgue spaces \(L^p(\Omega)\)
- 2.10: F. Riesz Theorem (non-compactness of unit sphere in infinite dimensions), non-equivalent norms
- 2.12: Bounded/continuous linear operators, the spaces \(\mathcal{L}(X,Y)\), \(\mathcal{L}(X)=\mathcal{L}(X,X)\) and \(X'=\mathcal{L}(X,\mathbb{K})\)
- 2.13: Compact linear operators \(\mathcal{K}(X,Y)\), \(\mathcal{K}(X)=\mathcal{K}(X,X)\)
- 2.19: Convex Sets
- 2.20: Convex Functions
- Banach Spaces (Ch. 3)
- 3.1: Banach spaces (basic results)
- 3.2: First Banach space examples
- 3.4: Further significant examples
- 3.5: Dual Spaces
- 3.6: Series in Banach space
- 3.7: Banach fixed point theorem
- Hilbert Spaces (Ch. 4)
- 4.1: Inner-product spaces and Hilbert spaces (basic results)
Course Grade
Your course grade will be based on four assignments, one midterm exams and one comprehensive final exam. Your course grade will be assigned based on the percentage earned of 400 possible points:
- 4x25=100 for assignments
- 100 for the midterm exam
- 200 for the final exam
Scores on each assignment and exam may be adjusted (in your favor) in order to achieve a fair distribution of grades.
Extra Credit: Do not expect extra credit or make-up work.
Assignments are due on the Fridays indicated in Weekly Schedule. The problems for each assignment will be made available to you in both PDF and TEX files. You are free to modify the TEX files for your solutions, but hand-written solutions are acceptable if your hand-writing and organization are clear. I will select a (non-empty) subset of required problems on each assignment for grading. Solutions to all required and optional problems (and occasionally other problems from these sections) for each assignment will be made available a few days after the assignment due date.
Exam Dates: Because the dates of exams are given well in advance, make/adjust your travel plans accordingly. Only in exceptional cases (left to my discretion, but including observance of religious holidays) will an exam be given on an alternate date. Unless a missed exam is due to a properly documented sickness or family emergency, an alternate exam date will be given only if we have a written agreement to do so, made at least one week before the originally-scheduled exam date.
- Midterm: Monday, November 2, 9-10:15am
- Final: Monday, December 7, 9-10:50am
Weekly Schedule
The topics covered each day may be adjusted as the term progresses.
| Week | Monday | Wednesday | Assignment (due Friday before Midnight) |
|---|---|---|---|
| 1: Sep 28, 30 | Overview, Normed Spaces Part I (2.2) | Normed Spaces Part II (2.2) | |
| 2: Oct 5, 7 | The space \(C(K,Y)\) (2.4, 3.2) | \(\ell^p\) spaces (2.5, 3.4, 4.2) | |
| 3: Oct 12, 14 | \(L^p(\Omega)\) spaces (2.6, 3.4, 4.2) | \(L^p(\Omega)\) spaces (2.6, 3.4, 4.2) |
|
| 4: Oct 19, 21 | Norm equivalence (or not), compactness of unit sphere (or not) (2.10) | Continuous linear operators Part I (2.12) | |
| 5: Oct 26, 28 | Continuous linear operators Part II (2.12, 3.2) | Compact linear operators (2.13) |
|
| 6: Nov 2, 4 | Midterm (Chapter 2 material through 2.13) | Convex sets, functions (2.19, 2.20) | |
| 7: Nov 9, 11 | Convex functions, uniformly and strictly convex spaces (2.20) | Veteran's Day (no meeting) | |
| 8: Nov 16, 18 | Banach spaces (3.1) | Banach spaces, basic examples (3.1, 3.2) |
|
| 9: Nov 23, 25 | Dual spaces (3.5) | Series in Banach space (3.6) | |
| 10: Nov 30, Dec 2 | Banach Fixed-Point Theorem (3.7) | Hilbert Spaces Part I (4.1) |
|
| 11: Dec 7, 9 | Final Exam (9-10:50am) |
PSU Policies and Resources
Academic Integrity & Grading Policies
- PSU Academic Calendar (opens in new tab)
- PSU Grading System (opens in new tab)
- Student Code of Conduct (opens in new tab)
- Incomplete Grades Policy (opens in new tab)