240 Lecture Notes
Math 240B/C -- Real Analysis (Winter/Spring 2012) Course
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- There is no class on Monday June 4, 2012 but you may be interested in
Tom Laetsch's thesis defense on Monday at 10:00AM.
- There will be class on Wednesday (Wave equation basics) and on Friday
(Quantum Harmonic Oscillator) of next week.
- Here is a list of the key topics for Math 240:
Math 240 Topics_Ver2.pdf
- For another interesting application of Tychonoff's theorem, see section
5. in Ken
Driver (firstname.lastname@example.org /
534-2648) in AP&M 5260.
Grader: Jesus Oliver (
email@example.com / 534-9062)
Meeting times: MWF
11:50 AM in AP&M 5402.
Textbook: I will mainly follow the lecture notes which will be available from this
web-site. We will also use Gerald B. Folland, "Real
Analysis, "Modern Techniques and Their Applications," 2nd
Students are assumed to have taken Math 240A
or its equivalent (see description below). Students are also assumed, at the
very least, to have taken a two-quarter sequence in undergraduate real analysis covering in a
rigorous manner the theory of limits, continuity and the like in Euclidean
spaces and general metric spaces. The theorems of Heine-Borel (compactness in
Euclidean spaces), Bolzano-Weierstrass (existence of convergent subsequences),
the theory of uniform convergence, Riemann integration theory should have been
covered. One quarter of undergraduate complex analysis is also recommended.
is assigned, collected, and partially graded
regularly. Each homework is due at the beginning of each Friday class.
Please print your names and
student ID numbers on your homework. Please staple together your homework
late homework will be accepted unless a written verification of a valid
excuse (such as hospitalization, family emergency, religious observance,
court appearance, etc.) is provided.
Test times: There
will be one in class
midterm on Wednesday February 15 in class. The final is scheduled for
Monday, March 19 from 11:30 - 2:30 PM.
Note: neither rescheduled nor make-up exams will be allowed unless a written
verification of a valid excuse (such as hospitalization, family emergency,
religious observance, court appearance, etc.) is provided.
Office Hours: M. & W. at 12:00 -
1:00PM (tentative, subject to change) in my office (AP&M
Grading: The course grade will be computed
using the following formula:
Math 240A (Highlights of what Bo Li Covered in Fall 2011)
Reviewed: limit operations, countability, set theoretic manipulations, and
some basics metric space ideas
Introduced sigma - algebras, measures
Caratheodory's construction of measures with
examples of measure on the real line including Lebesgue measure.
Measurable functions and
their properties under limits and other calculus operations.
Construction of the integral from a measure
Explored the relationship
of the Lebesgue integral with the Riemann integrable
General properties of the integral (Fatou's lemma, monotone convergence,
Lebesgue dominated convergence, Tchebychev inequality, Jensen's inequality)
Product measure and the
theorems of Tonelli and Fubini.
Introduced n --
dimensional Lebesgue measure and described the associate change of variables
Signed and complex
measures including the Jordan and Hahn decompositions.
theorem including the notion of absolute continuity (abstract
differentiation of measures)
Maximal theorem, Lebesgue
sets, and differentiation of measures relative to Lebesgue measure.
Math 240B (Winter 2012)
You reviewed the main limit and
differentiation past the integral theorems from last quarter.
Multiplicative System Theorem and its
related density results.
Absolute continuity, the fundamental
theorem of calculus, and integration by parts.
Basics of Metric spaces and Normed
spaces (including dual spaces) including the notions
of completeness, closure, and compactness.
Arzela-Ascolil compactness theorem.
Lp spaces, Holder inequality, integral
Hilbert space theory including
projection theorems, orthonormal bases, and adjoints of bounded operators.
Basics of Fourier series including
Introduction to general point set
topology notions and basic notions of compactness.
Tychonoff's compactness theorem for
countable products of compact metric spaces.
Compact Operators, Hilbert Schmidt
operators, the spectral theorem for self-adjoint compact operators.
Weak convergence and sequential weak
compactness of the unit ball in Hilbert spaces.
Hahn Banach Theorem and some of its consequences.
Notion of reflexive spaces and
integration theory of Banach valued funcitons.
(Material not covered on 240B final.) The dual of dual of Lp
spaces and Jensen's inequalities.
Math 240C (Spring 2012)
More Banach space results: Banach
Steinhaus Theorem, Open mapping theorem and the closed graph theorem.
Density and approximation theorems
including the use of convolution and the Stone Weierstrass theorem.
Locally compact Hausdorff spaces,
Uryshon's Lemma, Tietze Extension Theorem, Partitions of Unity, Alexandrov's
compactification, Uryshon's metrization theorem.
Tychonoff's Compactness theorem.
Fourier Transform and its properties
with basic applications to PDE and Sobolev spaces.
Riesz Markov Representation theorem
Some basic Sobolev space theory.
Some Calculus on Banach spaces and the
Inverse Function Theorem.
Possible further topics
A little complex analysis.
Distribution theory and elliptic
The Spectral Theorem for bounded self-adjoint
operators on a Hilbert space.
Unbounded operators and the Spectral
Theorem for self-adjoint operators.
Properties of ordinary differential