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Math 240A/B/C -- Real Analysis (Fall/Winter/Spring 2016-17) Course Information


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New Announcements

  • Final Exam is Wednesday, December 7 from 8:00 -- 11:00 AM in APM 5402. -- please bring a Blue Book to write your exam in.
  • Tentative Finals Week office hours, 10-12 on Monday and Tuesday of finals week.

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

Grader:              Jacqueline Warren ( email  / 534-49072) 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," 2nd edition by Gerald B. Folland,

Prerequisites: 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 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 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:

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

Course Outline from 2016

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

  • Covered the 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 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 integrable

  • 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 d-dimensional Lebesgue measure and showed it was translation invariant.

Math 240B (Winter 2017)

  • More Multiplicative System Theorem and density results.

  • Covered the 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 noms.

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

  • Introduced complex measures

  • 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 spaces.

  • 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.

  •  Jensen's inequalities.

  • 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.

  • 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 for measures.

  • Some basic Sobolev space theory.


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 Thursday, 23 March 2017 02:14 PM.