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Math 240B  Real Analysis (Fall/Winter/Spring 201617) Course
Information
(http://www.math.ucsd.edu/~bdriver/240AC_201617/index.htm)
You should also routinely login to your
TED =Triton ED (https://tritoned.ucsd.edu)
account for more course information. TED
is the web interface through which solutions and possibly other course materials
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240B New Announcements
 Finals week office hours are Monday and Tuesday, 910:30AM.
 Final Exam is
Wednesday, Wednesday, March 22 from 8:00  11:00 AM in APM 5402.
 please bring a
Blue Book to write your exam in.
 See homework page for a correction to last homework problem of the
quarter.
 Problem 2.95 (=26.15 in the lecture notes) is no longer due this
week!
 Midterm this quarter will be on Friday, February 10th
 please bring a
Blue Book to write your exam in.
 Holidays this quarter are Monday January 16th and Monday February 20th.
Instructor: Bruce
Driver (bdriver@ucsd.edu /
5342648) in AP&M 5260.
Grader: Jacqueline
Warren (
email /
53449072) AP&M
6446 *This course does not have a TA per se but only a
grader. Please only contact the TA about grading issues.
Course Meeting times: MWF
9:00 
9:50 AM in AP&M 5402.
Textbook: We
will use the course lecture notes and also the book; "Real
Analysis, "Modern Techniques and Their Applications," 2^{nd}
edition by Gerald B. Folland,
Prerequisites:
Students are also assumed, at the
very least, to have taken a twoquarter 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 HeineBorel (compactness in
Euclidean spaces), BolzanoWeierstrass (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.
Homework:
Homework
is assigned, collected, and partially graded
regularly. Each homework is due at the beginning of each Wednesday class unless
otherwise instructed.

Please print your names and
student ID numbers on your homework. Please staple together your homework
pages.

No
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 Friday, February 10. The final is scheduled for Wednesday,
March 23
from 8:00  11:00 AM (likely in APM 5402).
Note: neither rescheduled nor makeup 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. & F. 1011
AM in my office (AP&M
5260). [These are subject to change.]
Grading: The course grade will be computed
using the following formula:
Grade=.3(Home Work)+.3(Midterm)+.4(Final).
Course Outline from
20162017 Academic Year
Tentative Topics covered or to be covered
(order may change!)
Math 240A (Fall 2016)

Reviewed: limit operations, countability, set theoretic manipulations.

Covered basic notions of metric and pseudometric spaces

Introduced normed and Banach spaces

Introduced sums over arbitrary sets and prove the prototypical theorem: DCT,
MCT, Fatou's Lemma, Tonneli and Fubini Theorems.

Holder's and Minikowski's
inequalities for l^p  spaces showing in particular that l^p is a normed
space.

Introduced classes of subsets; 1) elementary sets, 2) algebras and 3)
sigmaalgebras

Construction of
finitely
additive measure on algebras from finitely additive function on an
elementary class.

Borel and product sigmaalgebras

Introduced
countably additive measures and covered a
special case of Caratheodory's construction of measures.

Used the measure construction theory to describe all
Radon measure on the real line and in particular constructed Lebesgue
measure.

Measurable functions and
their properties under limits and other calculus operations.

Construction of the integral from a measure

Mentioned the relationship
of the Lebesgue integral with the Riemann integral.

General properties of the integral (Fatou's lemma, monotone convergence,
Lebesgue dominated convergence, Tchebychev inequality, Jensen's inequality)

Multiplicative System Theorem and its
many applications

Product measure and the
theorems of Tonelli and Fubini.

Introduced ddimensional Lebesgue measure
and showed it was translation invariant.
Math 240B (Winter 2017)

The ddimensional change of variables
theorem including polar coordinates.

L^p spaces and related results like Holder
and Minikowski inequalities.

Covered
some different modes of convergence including L^p convergence, convergence
in measure, and almost everywhere convergence.

Jensen's inequalities.

More Multiplicative System Theorem and
density results in L^p spaces.

Developed the basics of Banach spaces
(=complete normed spaces) including operator spaces, dual spaces, and their
noms.

Hahn Banach Theorem and some of its consequences.

Notion of reflexive spaces.

Bochner type
integration theory of Banach valued functions including
the fundamental theorem of calculus.

Basic linear ODE theory on Banach spaces

Hilbert space theory including
projection theorems, Riesz Theorem, and adjoints of bounded operators.

Basics of orthonormal
bases on a Hilbert space and the applications to Fourier series including Plancherel theorem.

Signed and complex
measures including the Jordan and Hahn decompositions.

LebesgueRadonNikodym
theorem including the notion of absolute continuity (abstract
differentiation of measures)

The dual of L^p
spaces including (L^p)*=L^q.

Convolution inequalities
Math 240C (Spring 2017)

Convolution as a smoothing operator.

Maximal theorem, Lebesgue
sets, and differentiation of measures relative to Lebesgue measure.

Absolute continuity, the fundamental
theorem of calculus, and integration by parts.

Fourier Transform and its properties
with basic applications to PDE and Sobolev spaces.

More Banach space results: Banach
Steinhaus Theorem, Open mapping theorem and the closed graph theorem.

Some general point set
topology definitions and notations.

Tychonoff's compactness theorem.

ArzelaAscolil compactness theorem.

Compact Operators, Hilbert Schmidt
operators, the spectral theorem for selfadjoint compact operators.

Weak convergence and sequential weak
compactness of the unit ball in Hilbert spaces.

The general Stone Weierstrass theorem.

Locally compact Hausdorff spaces,
Uryshon's Lemma, Tietze Extension Theorem, Partitions of Unity, Alexandrov's
compactification, Uryshon's metrization theorem.

Riesz Markov Representation theorem
for measures.

Some basic Sobolev space theory.
Possible further topics

Differential Calculus on Banach spaces and the
Inverse and implicit Function Theorems.

Basic nonlinear ODE theory on Banach spaces

A little complex analysis.

Distribution theory and elliptic
regularity.

The Spectral Theorem for bounded selfadjoint
operators on a Hilbert space.

Unbounded operators and the Spectral
Theorem for selfadjoint operators.

Properties of ordinary differential
equations.

Differentiable manifolds.
