Math 240A/B/C -- Real Analysis (Fall/Winter/Spring 2016-17) Course
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Driver (email@example.com /
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: There
will be one in class
midterm on Monday October 31. The final is scheduled for Wednesday, December 7
from 8:00 -- 11:00 AM in APM 5402.
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. & 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
Tentative Topics coverd 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.
Introduced Holder's and Minikowski's
inequalities and showes l^p - spaces were normed spaces.
Introduced, elementary sets, algebras and described how to construct finitely
additive measure on algebras
We covered sigma - algebras including the Borel and product sigma-algebras
notion of countably additive measures
Gave 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 integrable
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)
More Multiplicative System Theorem and
the d-dimensional change of variables
theorem including polar coordinates.
Lp spaces and related results like Holder
and Minikowski inequalities and the different modes of convergence.
Developed the basics of Banach spaces
(=complete normed spaces) including operator spaces, dual spaces, and their
Hilbert space theory including
projection theorems, orthonormal bases, and adjoints of bounded operators.
Covered the notation of absolutely
continuous measures and and the Lebesgue-Radon-Nikodym
theorem in the absolutely continuous setting.
The dual of dual of Lp
Basics of orthonormal
bases on a Hilbert space and the applications to Fourier series including Plancherel theorem.
Hahn Banach Theorem and some of its consequences.
Notion of reflexive spaces and
integration theory of Banach valued funcitons.
Differential Calculus on Banach spaces and the
Inverse and implicit Function Theorems.
Basic ODE theory on Banach spaces
Convolution and smoothing operators
Proof of the d-dimensional change of variables formula
More Banach space results: Banach
Steinhaus Theorem, Open mapping theorem and the closed graph theorem.
More general point set
topology notions and basic notions of compactness.
Tychonoff's compactness theorem for
countable products of compact metric spaces.
Arzela-Ascolil compactness theorem.
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.
Math 240C (Spring 2017)
General Tychonoff's Compactness theorem.
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.
Absolute continuity, the fundamental
theorem of calculus, and integration by parts.
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.
Fourier Transform and its properties
with basic applications to PDE and Sobolev spaces.
Riesz Markov Representation theorem
Some basic Sobolev space theory.
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