Math 240B-C Home Page (Driver, 2012)


240 Lecture Notes
Test Data

Math 240B/C -- Real Analysis (Winter/Spring 2012) Course Information


You should also routinely login to your TED ( account for more course information. TED  is the web interface through which solutions and possibly other course materials will be communicated. Please check it frequently for announcements. The TED site will also contain a record of your grades for this course -- please make sure they are accurate.

New Announcements

  • 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 Brown's notes.

Instructor:         Bruce Driver (  /  534-2648) in AP&M 5260.

Grader:              Jesus Oliver ( / 534-9062) AP&M  6333

Meeting times: MWF 11:00 -- 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 edition.

Prerequisites: 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.

Homework:    Homework  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 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 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 5260).

Grading:         The course grade will be computed using the following formula:

Grade=.3(Home Work)+.3(Midterm)+.4(Final).

Tentative Course Outline

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 theorem.

  • Signed and complex measures including the Jordan and Hahn decompositions.

  • Lebesgue-Radon-Nikodym 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 operators.

  • Hilbert space theory including projection theorems, orthonormal bases, and adjoints of bounded operators.

  • Basics of Fourier series including Plancherel theorem.

  • 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 for measures.

  • 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 regularity.

  • 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 equations.

  • Differentiable manifolds.


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Last modified on Friday, 06 January 2012 12:45 PM.