Math 528A, Banach and Hilbert Spaces, Fall 2020

Information TypeData
Meeting TimeMWF, 1:00—1:50
Meeting Room On-line 
InstructorProfessor Marek Rychlik
OfficeMathematics 605
Homepage (Mirror)

Office Hours

PersonnelDay(s) of the WeekHourRoomComment
Marek RychlikMWF12:45pm—1:00pmOn-line Zoom Mini-office hour before class
Marek RychlikMWF2:00pm—2:15pmOn-line Zoom Mini-office hour after class
Marek RychlikM2:15pm—3:15pmOn-line Zoom Regular office hour

Required Texts

Functional Analysis , Edition: Any , Author: Lax, Peter D. , ISBN: 9780471556046, e-ISBN: 9781118626740, Publisher: Wiley, Date published: 04 Apr, 2002, Pages: 509, required.

Required Examinations And Other Grade Components

Homework, and take-home tests, shall be submitted as a typed paper, with the exception of these graphs and figures which cannot be easily drawn with software. Required software for preparing the homework paper is \(\LaTeX\). The work shall be submitted electronically, as a PDF document, through D2L, using the Dropbox feature of D2L.


  1. MATH 527B or MATH 523A (or equivalent coursework)


Homework is assigned throughout the semester. Two types of homework will be assigned:

  1. Homework which consists of selected exercises in the book.
  2. Custom homework created by the instructor.

Extra Credit Assignments

There will be some.


Overall Course Objectives and Expected Learning Outcome

  1. Learn in detail the fundamental concepts of functional analysis, in particular, of Banach spaces and Hilbert spaces.
  2. Learn how Banach and Hilbert spaces are used throughout many areas of mathematics, including mathematical analysis, ordinary and partial differential equations.
  3. Gain and improve problem-solving abilities utilizing notions learned in the course, including norms, seminorms, convex sets, Banach and Hilbert spaces, linear operators, spectral decomposition, weak and week-* convergence, and many others.

Course Outline

Week Dates Topics Sections Covered
1Aug 24—Aug 28 Linear Spaces. Linear Maps. Algebra of linear maps. Index of a linear map. Chapter 1, 2.1, 2.2
2Aug 31—Sep 4 Flag of nullspaces of powers of a linear operator. Nilpotent operators. Jordan canonical form. Index of a linear map. The Hahn-Banach Theorem. 2.1, 2.2, 3.1
3Sep 7Labor Day - no class.
3Sep 9—Sep 11 The Hahn-Banach Theorem. The extension theorem. Geometric Hahn-Banach theorem. Extensions of the Hahn-Banach theorem. Applications of the Hahn-Banach theorem. Extension of positive linear functions. Banach limits. Finitely additive invariant set functions. 3.1, 3.2, 3.3, 4.1, 4.2, 4.3
4Sep 14—Sep 18 Normed linear spaces. Norms. Noncompactness of the unit ball. Isometries. 5.1, 5.2, 5.3
5Sep 21—Sep 25 Hilbert space. Scalar product. Closest point in a closed convex subset. Linear functionals. Linear span. 6.1, 6.2, 6.3, 6.4
6Sep 28—Oct 2 Applications of Hilbert space results. Radon-Nikodym theorem. 7.1
7Oct 5—Oct 9 Measure theory review. Radon-Nikodym theorem. 7.1
8Oct 12—Oct 16 Dirichlet's problem. 7.2, 7.3
9Oct 21Midterm 1.
9Oct 24—Oct 28 Dirichlet's problem. 7.2, 7.3
10Oct 26—Oct 30 Duals of normed linear spaces. Bounded linear functionals. Extension of bounded linear functionals. Reflexive spaces. Support function of a set. 8.1, 8.2, 8.3, 8.4
11Nov 2—Nov 6 Applications of duality. Completeness of weighted powers. The Müntz approximation theorem. 9.1, 9.2
12Nov 11 Veteran's Day - no class.
12Nov 9—Nov 13 The weak and weak* topologies. Week convergence. Week sequential compactness. Week-*-convergence. 10.1, 10.2, 10.3
13Nov 16—Nov 20 The weak and weak* topologies. Weak convergence. Weak sequential compactness. Weak-*-convergence. 10.1, 10.2, 10.3
14Nov 23—Nov 27 The weak and weak* topologies. Week convergence. Week sequential compactness. Week-*-convergence. 10.1, 10.2, 10.3
15Nov 26—Nov 29 Thanksgiving recess.
16Nov 30—Dec 4 The weak and weak* topologies. Week convergence. Week sequential compactness. Week-*-convergence. 10.1, 10.2, 10.3
16Dec 2Midterm 2.
17Dec 7—Dec 9 Review. Problem solving.
17Dec 9 Last Day of classes
17Dec 10 Reading Day - no classes or finals
Finals Week (Dec 11-17)Dec 14 (Monday)Final Exam, 1:00 pm - 3:00 pm (take-home, to be made available on Dec. 7)

Course Policies

Attendance Policy

Students are expected to attend every scheduled class and to be familiar with the University Class Attendance policy as it appears in the General Catalog. It is the student's responsibility to keep informed of any announcements, syllabus adjustments or policy changes made during scheduled classes.

Expected Classroom Behavior

Students are expected to behave in accordance with the Student Code of Conduct and the Code of Academic Integrity. The guiding principle of academic integrity is that a student's submitted work must be the student's own. University policies can be found at

Threatening Behavior

See No prohibited behavior will be tolerated.

Administrative Drop

Students who miss the first two class meetings will be administratively dropped unless they have made other arrangements with the instructor.

Missed Exams

Students are expected to be present for all exams. If a verifiable emergency arises which prevents you from taking an in-class exam at the regularly scheduled time, the instructor must be notified as soon as possible, and in any case, prior to the next regularly scheduled class. Make-up exams and quizzes will be administered only at the discretion of the instructor and only under extreme circumstances. If a student is allowed to make up a missed exam, (s)he must take it at a mutually arranged time. No further opportunities will be extended. Failure to contact your instructor as stated above or inability to produce sufficient evidence of a real emergency will result in a grade of zero on the exam. Other remedies, such as adjusting credit for other exams, may be considered.

Accessibility and Accommodations

Disabled students must register with Disability Resources and be identified to the course instructor through the University's online process in order to use reasonable accommodations.

It is the University's goal that learning experiences be as accessible as possible. If you anticipate or experience physical or academic barriers based on disability, please let me know immediately so that we can discuss options. You are also welcome to contact Disability Resources 520-621-3268 to establish reasonable accommodations.

Please be aware that the accessible table and chairs in this room should remain available for students who find that standard classroom seating is not usable.

Policy on the grade of "I" (incomplete)

The grade of "I" will be awarded if all of the following conditions are met:

  • The student has completed all but a small portion of the required work.
  • The student has scored at least 50% on the work completed.
  • The student has a valid reason for not completing the course on time.
  • The student agrees to make up the material in a short period of time.
  • The student asks for the incomplete before grades are due, 48 hours after the final exam.


Changes to the Syllabus

The information contained in the course syllabus, other than the grade and absence policies, is subject to change with reasonable advance notice, as deemed appropriate by the instructor.