Instructor: Hai Zhang
Office: 3442
Email: haizhang@ust.hk
Office Hours: Tue Thur 10:30AM—11:30AM
Lectures:
Tuesday and
Thursday 4:30PM-5:50PM, G003, CYT Bldg
Main references:
Applied Analysis,
by J K. Hunter and B Nachtergaele, World Scientific,
2001;
Functional Analysis,
by P D. Lax, John Wiely & Sons, Inc, 2002
Requisite: College Calculus, Linear algebra, Elementary
real analysis
Course Description and Objectives:
The aim of the course is to provide an introduction to the analysis techniques,
which are used frequently in applications. It is intended for beginning
graduate students who are interested in applied math and plan to take the
Advanced Calculus qualifying exam. The
following topics will be covered:
1. Contraction mapping theorem
1.1 Contraction mapping
theory
1.2 Applications in dynamic
system
1.3 Applications in integral
equations
2. Fourier series
2.1 Completeness of Fourier
basis
2.2 Fourier series of differentiable
functions
2.3 Applications in partial
differential equations
2.3 Applications to
isoperimetric inequality
2.4 Applications to ergodic theorem
3. Fourier transforms (including classic theory and modern theory on
distributions)
3.1 Fourier transform on
Schwartz functions
3.2 Fourier transform on L^1
functions
3.3 Fourier transform on L^2
functions
3.4 Fourier transform on
tempered distribution
3.5 Applications to translation
invariant operators
3.6 Applications to GreenŐs
functions
3.7 Poisson summation
formula
4. Basic of Hilbert Space theory
4.1 Scalar product
4.2 Orthogonal projection
4.3 Riesz
representation theorem
4.4 Applications to Randon-Nikodym theorem
5. Basic operator theory on Hilbert Spaces
5.1 Projections and
orthogonal projections
5.2 Self-adjoint
operators
5.3 Compact operators
5.4 Spectrum of bounded
operators
5.5 Spectral representation
of compact self-adjoint operators
5.6 Applications to Strum-Liouville problem
6. Hahn-Banach theorem and basic of Banach space theory
6.1 Hahn-Banach
theorem and separation theorem
6.2 Linear functionals and dual spaces
6.3 Applications of Duality (completeness
of weighted powers)
7. Convex analysis
7.1 Derivatives and
sub-gradient
7.2 Convex conjugate
7.3 Lagrangians
and saddle points
7.4 Karush-Kuhn-Tucker
type conditions
Remark: For the Advanced Calculus exam, only topics 1, 2, 3 will be covered.
Grading Scheme:
Homework 40%, Final Exam 60%
Exams:
Final exam: There will be no make-up exams, except for absolutely exceptional
situations.
No calculators, notes, or books will be permitted in in the exam.
Homework:
Homework will be assigned at the end of each chapter and two weeks are allowed
to work out the solution.
Make sure that your name, course section and university ID are included in the subtitle of
the homework before you hand in your homework to me. You are allowed and
encouraged to work together on homework assignments. However, you are expected to write the
homework solutions on your own and with your own words. Outright copying from
somebody else's assignment will be considered cheating.
DO
YOUR HOMEWORK. IT IS THE SECRET TO
SUCCESS.