Instructor: Hai Zhang
Office: 3442
Email: haizhang@ust.hk
Office Hours: Tue Wed 10:30AM—11:30AM
Lectures:
Tuesday and
Thursday, 9AM-10: 20AM, Room 1026, LSK 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. 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
2.4 Applications
to isoperimetric inequality
2.5 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
Tempered distribution
3.5
Fourier transform on tempered
distribution
3.6
Applications to invariant operators
3.7
Applications to GreenŐs functions
3.8
Applications to Poisson summation formula
3.9
Applications to central limit theorem
4. Basic of Hilbert Space theory
4.1
Scale product
4.2
Orthogonal projection
4.3
Riesz
representation theorem
4.4
Orthonormal bases
4.5
Applications to least square problems
4.6
Applications to Randon-Nikodym
theorem
5. Basic operator theory on Hilbert Spaces
5.1
Projections and orthogonal projections
5.2
Adjoint
operators
5.3
Self-adjoint
operators
5.4
Spectrum of bounded operators
5.5
Compact operators
5.6
Diagonalization
of compact self-adjoint
5.7
Applications to Strum-Liouville problem
6. Hahn-Banach theorem and basic of Banach space theory
6.1
Hahn-Banach
theorem
6.2
Normed linear space
6.3
Linear functionals, weak convergence and dual spaces
6.4
Applications to RungeŐs theorem
6.5
Linear operators and open
mapping theorem
6.6
Applications
7. Convex analysis
8. Differential calculus
(elementary nonlinear functional analysis).
For each topic, we shall introduce a variety of examples from
different fields to motivate and demonstrate the ideas. We shall keep the
presentation as simple as possible, but without sacrificing the mathematical
rigor. The course is also intended for beginning graduate students who are interested
in applied math and plan to take the Advanced Calculus qualifying exam.
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 any exams.
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.
Tentative schedule
Lecture 01, Sep 01: Course overview and
Section 1.1
Lecture 02, Sep 06: Section 1.2, 1.3
Lecture 03, Sep 08: Section 2.1
Lecture 04, Sep 13: Section 2.1
Lecture 05, Sep 15: Section 2.2
Lecture 06, Sep 20: Section 2.3, 2.4
Lecture 07, Sep 22: Section 2.5
Lecture 08, Sep 27: Section 3.1
Lecture 09, Sep 29: Section 3.2, 3.3
Lecture 10, Oct 04: Section 3.4
Lecture 11, Oct 06: Section 3.5
Lecture 12, Oct 11: Section 3.6, 3.7
Lecture 13, Oct 13: Section 3.8, 3.9
Lecture 14, Oct 18: Section 4.1, 4.2
Lecture 15, Oct 20: Section 4.3, 4.3
Lecture 16, Oct 25: Section 4.4, 4.5
Lecture 17, Oct 27: Section 5.1, 5.2
Lecture 18, Nov 01: Section 5.3, 5.4
Lecture 19, Nov 03: Section 5.5, 5.6
Lecture 20, Nov 08: Section 5.7
Lecture 21, Nov 10: Section 6.1
Lecture 22, Nov 15: Section 6.2
Lecture 23, Nov 17: Section 6.3
Lecture 24, Nov 22: Section 6.4
Lecture 25, Nov 24: Section 6.5
Lecture 26, Nov 29: Section 6.6
DO
YOUR HOMEWORK. IT IS THE SECRET TO
SUCCESS.
Academic
Integrity:
The University places a strong emphasis on academic integrity. Please
check the website
http://tl.ust.hk/integrity/student-1.html for
regulations. In any case, no cheating will be tolerated. Honesty is the best
policy. Talk to me if you have trouble or are worried about grades. Exceptions
may be made, but you have to talk to me early!
Working together on homework:
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.
How to write solutions:
When you solve problems for homework and exams, you must provide complete
solutions with all steps described in detail.
You must not miss steps in solutions just because they look obvious to you.