240 Lecture Notes
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cMath 240A/B/C -- Real Analysis (Fall/Winter/Spring 2016-17) Course
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240C New Announcements
- Final Exam = Qual Exam.
Math 240 Qual is scheduled for Monday, May 22nd from 9:00 -- 12:00 AM in APM
You do NOT need a
for this exam.
The qualifying exam is cumulative over the whole years
Here is the Spring 2012 Qualifying Exam for Practice:
There will be no classes during the qual exam week of May 22 -- May 26, 2017.
No class Monday, May 29 as well as this is Memorial day,
Driver (firstname.lastname@example.org /
534-2648) in AP&M 5260.
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
9:50 AM in AP&M 5402.
will use the course lecture notes and also the book; "Real
Analysis, "Modern Techniques and Their Applications," 2nd
edition by Gerald B. Folland,
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 unless
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: 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.
will be one in class
midterm on Wednesday, May 9.
The final exam
will be the same as the qualifying exam
which is scheduled from 9:00 -- 12:00 AM on
Monday, May 22nd (in APM
6402). The qualifying exam is cumulative over the whole years
Office Hours: M. & W. & F. 10-11
AM in my office (AP&M
5260). [These are subject to change.]
Grading: The course grade will be computed
using the following formula:
Course Outline from
2016-2017 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 pseudo-metric 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
Introduced classes of subsets; 1) elementary sets, 2) algebras and 3)
additive measure on algebras from finitely additive function on an
Borel and product sigma-algebras
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
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
Product measure and the
theorems of Tonelli and Fubini.
Introduced d-dimensional Lebesgue measure
and showed it was translation invariant.
Math 240B (Winter 2017)
The d-dimensional change of variables
theorem including polar coordinates.
L^p spaces and related results like Holder
and Minikowski inequalities.
some different modes of convergence including L^p -convergence, convergence
in measure, and almost everywhere convergence.
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
Hahn Banach Theorem and some of its consequences.
Notion of reflexive spaces.
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.
theorem including the notion of absolute continuity (abstract
differentiation of measures)
The dual of L^p
spaces including (L^p)*=L^q.
Math 240C (Spring 2017)
More Banach space results: Banach
Steinhaus Theorem, Open mapping theorem and the closed graph theorem.
Maximal theorem, Lebesgue
sets, and differentiation of measures relative to Lebesgue measure.
Absolute continuity, the fundamental
theorem of calculus, and integration by parts.
Properties of convolution and its applications to
Fourier Transform and its properties
with basic applications to PDE and Sobolev spaces.
Some general point set
topology definitions and notations
Tychonoff's compactness theorem.
The notion of a compact
Weak convergence and sequential weak
compactness of the unit ball in Hilbert spaces.
Locally compact Hausdorff spaces
The general Stone Weierstrass theorem.
Tietze Extension Theorem,
Partitions of Unity,
Uryshon's metrization theorem.
Riesz Markov Representation theorem
Possible further topics
Some basic Sobolev space theory.
operators, the spectral theorem for self-adjoint compact operators.
Differential Calculus on Banach spaces and the
Inverse and implicit Function Theorems.
Basic non-linear ODE theory on Banach spaces
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