Home page for AMCS 609: Analysis II
Spring 2015
Instructor: Charles L. Epstein
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The Course
The main goal of analysis is the solution of equations.
In the first semester we covered 1-variable Complex Analysis
and the basics of Real Analysis. This semester we continue our
discussion of Real Analysis for 4 weeks and then turn to the
analysis of linear
equations in infinite dimensional vector spaces. Unlike the finite
dimensional
case, this requires the introduction of a topology, and so becomes a
topic in analysis. Amongst other things, we will develop the framework
needed to find solutions
of ordinary and partial differential equations.
For Real Analysis we use Real Analysis by E. Stein and R. Shakarchi as our principal textbook, and
Real Analysis by Gerald Folland, and Real Analysis by H. Royden as additional references; for
Functional Analysis we use Functional
Analysis by E. Stein and R. Shakarchi, and Functional Analysis, by Peter D. Lax as our
principal text books. Good additional references for functional
analysis
are: Walter Rudin: Real
and Complex
Analysis and Functional
Analysis; Ward Cheney, Analysis
for Applied Mathematics;
Michael Reed and Barry Simon, Methods
of Modern Mathematical Physics, Vol. 1; Tosio Kato, Perturbation Theory
of Linear
Operators. Good references for finite dimensional linear
analysis are: Peter Lax, Linear
Algebra.
A problem set
will be
assigned more or less every week on Tuesday, due the following week on Tuesday. I
very much prefer that students do the
problem sets
alone. We may
have a
take-home midterm and/or final exams.
- The class meeting is Tuesday and Thursday, 1:30-3:00 in
DRL 3C2.
- My office hour is Mondays 3:30-5PM. Contact me by e-mail
for an
appointment if you can not come during this time.
- My office in the Math Department is 4E7 DRL, tel. 8-8476.
- email: cle@math.upenn.edu.
Send e-mail if you have a question or need to contact me.
- The grader is Neel Patel. His office is 4E14 DRL.
email: neelpa@math.upenn.edu. His office hours are ???
Syllabus
- Real Analysis.
- Lebesgue Integration Cont.
- Fubini's Theorem
- Differentiation
- The Maximal Function
- Abstract Measure Theory
- Other Topics
- Functional Analysis. The numbers in parentheses are chapters numbers in Lax, Functional Analysis
- Review of finite dimensional linear algebra
- Normed linear spaces, definitions and examples (1, 2, 5)
- Convexity, the Hahn-Banach Theorem (3, 4, 13.1)
- Hilbert space and the Riesz Representation Theorem (6, 7)
- The Radon-Nikodym Theorem
- Duality (8, 9)
- Weak convergence (10, 11)
- Bounded linear maps (15, 16, 20)
- Compact operators (21, 22)
- Fredholm theory (21)
- Spectra of compact operators in a Hilbert space (28, 29)
- Solving elliptic equations using boundary layer theory
Announcement
- Our first classs will be Thursday, January 15, 2015.
Problem sets
- Problem set 1, due January 28, 2015.
- Problem set 2, due February 3, 2015.
- Problem set 3, due February 17, 2015.
- Problem set 4, due February 24, 2015.
- Problem set 5, due March 3, 2015.
- Problem set 6, due March 17, 2015.
- Problem set 7, due March 31, 2015.
- Problem set 8, due April 7, 2015.
- Problem set 9, due April 14, 2015.
- Problem set 10, due April 21, 2015.
- Problem set 11, due May 5, 2015.
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