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Home page for AMCS 610: Analytic Techniques II/Functional
Analysis

Spring 2014

**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; for most of this semester we turn to the analysis of linear
equations in infinite dimensional. 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.

We use Peter D. Lax, Functional
Analysis as our
principal text book. Good additional references for functional analysis
are: Walter Rudin: Real
and Complex
Analysis and Functional
Analysis; E. Stein and R. Shakarchi, 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 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, 12:00-1:30 in
DRL 3C8.

- 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 Maxim Gilula. His office is 3C13 in DRL and
his office hours will be annouced.

### Syllabus

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)

- 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 16, 2014.

### Problem sets

- Problem set 1, due February 4, 2014.
- Problem set 2, due February 11, 2014.
- Problem set 3, due February 25, 2014.
- Problem set 4, due March 4, 2014.
- Problem set 5, due March 18, 2014.
- Problem set 6, due April 1, 2014.
- Problem set 7, due April 8, 2014.
- Problem set 8, due April 22, 2014.
- Problem set 9, due April 29, 2014 (you can actually turn it in before May 2).

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