Catalog Course Description: Measures on locally compact spaces. Riesz representation theorem. Elements of functional analysis. Normed linear spaces. Hahn-Banach, Banach-Steinhaus, open mapping, closed graph theorems. Weak topologies. Banach-Alaoglu theorem. Function spaces. Stone-Weierstrass and Ascoli theorems. Basic Hilbert space theory. Application to Fourier series. Additional topics: Haar measure on locally compact groups. Prereq: MATH 423.
Text: "Real Analysis" by Folland, second edition (1999, 978-0471317166).
Notes: (1) Here is
a link to errata (list of misprints, mostly). (2) This is the same text as that used in MATH 423 last semester. (3) If you want to use an earlier edition, you may try AT YOUR OWN risk.
COURSE STRUCTURE & GRADES
The course: This is the second course of the MATH 423-424 sequence.
MATH 423 covers general theory of measure and integration.
The second course, MATH 424, is really an introduction to functional analysis,
with MATH 423 providing the needed background.
Functional analysis is a branch of mathematical analysis that emerged in 20th century, which is
fundamental in numerous applications of mathematics. It is sometimes described as the
meeting point of algebra, analysis and geometry.
The sequence is required
of most mathematics graduate students. It can also be useful to graduate students
in other mathematical sciences (such as statistics
and operation research) or in physics and engineering.
Finally, the sequence can be taken by advanced undergraduates.
Exams & Grades: Your Final Grade in the course will be based on
Attendance/Assignments/Various Forms of Class Participation (40%, may include in-class presentations),
Midterm Exam (20%; tentatively the week of March 15th) and Final Exam (40%; default date 5/18/2021).
Students with special needs should contact
Division of Student Affairs.
Integrity: It is OK (and indeed encouraged) to discuss homework assignments with fellow students. However, any submitted work must be your own.
Merely copying someone else's work is unethical,
a waste of time, and may be penalized.
(This includes copying solutions found on the internet.)
See
CWRU academic integrity policy.
SYLLABUS & HOMEWORK
We will cover approximately Chapters 5-7 from the textbook and some material
from Chapter 4 as needed, particularly sections 4.6 and 4.7.
(Chapter 4 provides background in general topology; the material may be reviewed
in class if and when needed, but primarily it is covered in MATH 461.)
We may venture into Chapter 8 and/or other topics, time permitting.
