Math 240A-C Home Page (Driver, F2016-S17)


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


You should also routinely login to your TED =Triton ED ( 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.

240C New Announcements

  • Final Exam = Qual Exam. Math 240 Qual is scheduled for Monday, May 22nd from 9:00 -- 12:00 AM in APM 6402. You do NOT need a Blue Book for this exam.
  • The qualifying exam is cumulative over the whole years material!!
  • Here is the Spring 2012 Qualifying Exam for Practice:  240qual_s2012_test.pdf
  • 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,

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:  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.

  • There 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 material!!


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=Qual Exam).

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 space.

  • Introduced classes of subsets; 1) elementary sets, 2) algebras and 3) sigma-algebras

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

  • Borel and product sigma-algebras

  • 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 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.

  • 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.

  • Lebesgue-Radon-Nikodym 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)

  • 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 smoothing operators.

  • 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.

      Arzela-Ascolil compactness theorem

      The notion of a compact operator.

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

  • Locally compact Hausdorff spaces

    • Uryshon's Lemma

    • The general Stone Weierstrass theorem.

    • Tietze Extension Theorem,

    • Partitions of Unity,

    • Alexandrov's compactification,

    • Uryshon's metrization theorem.

  • Riesz Markov Representation theorem for measures.

Possible further topics

  • Some basic Sobolev space theory. 

  • Hilbert Schmidt 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 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.