** Instructor:** Bei Hu, Hurley 174A, b1hu@nd.edu

** 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:

- Linear Spaces, Inner Products, Normed linear spaces, Banach spaces and Hilbert Spaces.
- Fixed point theorems and the applications in partial differential equations;
- Orthogonality, Duality, Dirichlet principle, Lax-Milgram Theorem;
- Fourier series in Hilbert spaces;
- Eigenvalue problems, Fredholm Alternatives;
- Self-adjoint operators, Semi-groups;
- 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 (b1hu@nd.edu).

** 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:**

- Students in 50550 may try 60550 homework problems for extra points,
- Students in 50550 are required only to complete a subset of exam problems,
- Students in 60550 are required to do all homework problems and all exam problems.

** 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.