ACMS 50550/60550: Applied Functional Analysis, Spring 2023

Instructor: Bei Hu, Hurley 174A,

Class: TTh 11:00 am - 12:15 pm, Pasquerilla Center 116

Textbook: Applied Functional Analysis by Eberhard Zeidler, Third printing 1999, Springer-Verlag, ISBN 0-387-94442-7.
There are two volumes by this author, we need the series 108, the series 109 is NOT our textbook.
The book is also available here.

Pre-requsite: (ACMS 20750 or MATH 20750) and (ACMS 20620 or MATH 20610). i.e., Undergraduate multivariable calculus, Linear algebra, Ordinary Differential Equations and Partial Differential Equations.

Syllabus: We will cover selected materials:

  1. Linear Spaces, Inner Products, Normed linear spaces, Banach spaces and Hilbert Spaces.
  2. Fixed point theorems and the applications in partial differential equations;
  3. Orthogonality, Duality, Dirichlet principle, Lax-Milgram Theorem;
  4. Fourier series in Hilbert spaces;
  5. Eigenvalue problems, Fredholm Alternatives;
  6. Self-adjoint operators, Semi-groups;
  7. Boundary value problems - Laplace equations, heat equations, wave equations, and other equations;

Office Hours: There is an open office hour policy. You are welcome to stop by my office at any time. However, I may have other duties and may not always be available. You can always make an appointment with me by email (

Homework: Homework problems will be assigned here and will be collected once a week on Tuesdays. The main purpose of the homework is to help you learn the material. Experience shows that students who take their homework seriously do very well in the course because they have a better understanding of the material. You are encouraged to submit your homework, no matter how late it will be, but we do not accept homework after the last day of classes.

Exams: One midterm exam (in class), and one final exam (take home).

Grades: Your final grade is based on 35% homeworks, 25% midterm, and 40% final.

ACMS 50550 and ACMS 60550:

More on this course: in our undergraduate Partial Differential Equations (PDEs) course, either ACMS 20750 or MATH 40750, we solved our linear problems using separation of variables; for the time dependent problems, “the separation of variables” method gives the well-posedness and the asymptotic behavior at the same time; but the method works only on regular shaped domains. The same “separation of variables” can be applied for general linear problems, if the associate operators are either compact operators, or Fredholm operators. Fortunately, a lot of linear operators associated with the PDEs are compact operators. This is the beauty of the spectrum theory – a unified framework for a lot of the problems in PDEs.

Graduate students: these materials are fundamental in studying PDEs.

Undergraduate students: if you intend to go to graduate school and study PDEs, this is a nice addition.