Introduction to Real Analysis (6310) Course Web Page and Syllabus Fall 2012 Instructor: Dr. Janos Englander Course content: Presents the basic notions of analysis, e.g., limits, lim sup and lim inf, continuity, and the topology of the real line; develops the theory of Lebesgue measure and the Lebesgue integral on the line, emphasizing the various notions of convergence and the standard convergence theorems. Applications are made to the classical Lp spaces. Prereq.: MATH 4001 (Analysis II).  (You should be familiar with infinite series, sequences of functions, etc.) Instructor consent required for undergraduates. Text: "Real Analysis" by Royden and Fitzpatrick (4th edition)   Time: MWF 10:00-10:50 Room: ECCR 131 How to reach me: Office: MATH 324. Office hours: Monday and Friday 11-12 Phone: 303-492-4846 (with voice mail) electronic mail Enrollment questions: Please see the registrar's web page, or call their office. If you still have problems, then please see Tiffany (MATH office) if this is truly an enrollment question, and Barbara if this is more like a graduate studies question. Grading: Your grade will be based on your two midterms (each 25%) and on the final (50%). I will give you practice problems regularly. I will not grade them, but I will provide the solutions, and the questions in the exams will be taken form these practice problems (except perhaps a few warm up YES/NO questions). Grading cutoffs: (I. e. how I will convert your final numerical grade, expressed as a percentage, into a letter grade.) A: 90-100 A-: 80-89 B+: 77-79 B: 73-76 B-: 70-72 C+: 62-69 C: 55-61 C-: 50-54 Inadequate: 49 or lower. Exam Schedule: Midterms: Wednesday, October 3 (closed books/notes) Monday, November 5 (closed books/notes) (in class) SOLUTIONS Final exam: Friday, December 14 (last class, in classroom; closed book/notes) The final is exactly as long as a midterm. However its scope is much larger because it is cumulative. Practice Homework (see “Grading” above): (Numbers refer to Royden's book, 4th edition.) I strongly recommend you solve the problems on your own before opening the solutions. SET 1: p. 16, Problems 20, 22, 23, 24, 25, 26 (two sets are “equipotent” if they have the same cardinality) SOL.: page 1 page 2 SET 2: p. 20 Problem 34, p.24 Problem 46. p. 27-28 Problems 47, 48, 52, 53. SOL.: page 1 page 2 page 3 SET 3: p. 34 Problems 6-10. SOL.: click SET 4: p. 59 Problem 9.,p. 67 Problem 27, p. 71 Problem 8, p. 79 Problem 16, p. 85 Problem 25 SOL.: click SET 5: p. 90 Problem 36, p. 91 Problem 38, p. 96 Problem 46, p. 102 Problem 14, p. 106 Problem17 SOL.: click Eating/drinking policy: That’s fine. Texting policy: No texting in class. Proof of MCT: click A (BIASED) CLICKABLE LIST OF A FEW FAMOUS MATHEMATICIANS (Based on Balint Toth’s web page): Bachelier, Louis (1870 - 1946): Brownian motion Banach, Stefan (1892 - 1945): Banach-space Bernoulli, Jacob (Jacques) (1654 - 1705): Probability Birkhoff, George (1884 - 1944): Ergodic Theory Bochner, Salomon (1899 - 1982): Bochner’s Theorem on characteristic functions Borel, Émile (1871 - 1956): Borel-Cantelli lemma, notions in analysis Cantelli, Francesco Paolo (1875 - 1966): Borel-Cantelli lemma Cauchy, Augustin Louis (1789 - 1857): basics of analysis, Cauchy seq., etc. Chandrasekhar, Subrahmanyan (1910 - 1995): distributions in astronomy Chebyshev, Pafnuty Lvovich (1821 - 1894): Famous probability inequality, number theory Cramér, Carl Harald (1893 - 1985): large deviations Erdös, Pál (1913 - 1996): discrete mathematics Feller, William (1906 - 1970): Probability Finetti, Bruno de (1906 - 1985): de Finetti’s Theorem Gauss, Carl Friedrich (1777 - 1855): number theory, algebra, differential geometry Gnedenko, Boris Vladimirovich (1912 - 1995): Probability Hardy, Godfrey Harold (1877 - 1947): analysis, number theory Hölder, Otto (1859 - 1937): Hölder-inequality Ito, Kiyosi (1915 - 2008): stochastic analysis Jensen, Johan (1859 - 1925): Jensen-inequality Kac, Mark (1914 - 1984): probability, number theory Kakutani, Shizuo (1911 - . . . .): Individual Ergodic Theorem Khinchin, Aleksandr Yakovlevich (1894 - 1959): Iterated Logarithm Theorem in Probability Kolmogorov, Andrey Nikolaevich (1903 - 1987): Probability, functional analysis Koopman, Bernard (1900 - 1981): Dynamical systems Laplace, Pierre-Simon (1749 - 1827): Laplace-transform Lebesgue, Henri (1875 - 1941): The theory of Lebesgue measure and integration Le Cam, Lucien (1924 - 2000) : Probability Lévy, Paul Pierre (1886 - 1971): Probability Lindeberg, Jarl Waldemar (1876 - 1932): Probability Linnik, Yuri Vladimirovich (1915 - 1972): number theory, probability and statistics Littlewood, John Edensor (1885 - 1977): analysis, number theory Lyapunov, Aleksandr Mikhailovich (1857 - 1918): early form of Central Limit Theory Markov, Andrei Andreyevich (1856 - 1922): Markov-chains Mandelbrot, Benoit (1924 - 2010): fractals Maxwell, James Clerk (1831 - 1879): Maxwell’s laws in electro-magnetism, Maxwell-distribution Moivre, Abraham de (1667 - 1754): de Moivre--Laplace Theorem, Stirling’s-formula Neumann, János (1903 - 1957): set theory, operator theory, game theory, cybernetics (a. k. a. John von Neumann) Poisson, Siméon Denis (1781 - 1840): Poisson-process Pólya, György (1887 - 1985): analysis (classic problem book with Szego), random walks Rényi, Alfréd (1921 - 1970): Probability, random graphs, etc. Riemann, Bernhard (1826 - 1866): analysis, Riemann’s hypothesis about the zeta-function, differential geometry Riesz, Frigyes (1880 - 1956): foundations of functional analyis, Riesz-representation Scheffé, Henry (1907 - 1977): Scheffé’s Theorem Skorokhod, Anatoliy (1930– . . . .): Probability, Skorokhod-embedding Stirling, James (1692 - 1770): applications of “Stirling’s formula” Taylor, Brook (1685 - 1731): power expansions Wiener, Norbert (1894 - 1964): Brownian motion, cybernetics, Fourier analysis Yosida, Kosaku (1909 - . . . .): Functional analysis, ergodic theory SOME IMPORTANT CU POLICIES (1) If you qualify for accommodations because of a disability, please submit to me a letter from Disability Services in a timely manner (for exam accommodations provide your letter at least one week prior to the exam) so that your needs can be addressed. Disability Services determines accommodations based on documented disabilities. Contact Disability Services at 303-492-8671 or by e-mail at dsinfo@colorado.edu. If you have a temporary medical condition or injury, see Temporary Medical Conditions: Injuries, Surgeries, and Illnesses guidelines under Quick Links at Disability Services website and discuss your needs with me. (2) Campus policy regarding religious observances states that faculty must make reasonable accommodation for them and in so doing, be careful not to inhibit or penalize those students who are exercising their rights to religious observance. I am aware that a given religious holiday may be observed with very different levels of attentiveness by different members of the same religious group and thus may require careful consideration to the particulars of each individual case. For guidelines, see http://www.colorado.edu/policies/fac_relig.html If you have questions about the rules with regard to religious accommodations, please contact the Office of Discrimination and Harassment at 303-492-2797. A comprehensive calendar of the religious holidays most commonly observed by CU-Boulder students is at http://www.interfaithcalendar.org/ Campus policy regarding religious observances requires that faculty make every effort to deal reasonably and fairly with all students who, because of religious obligations, have conflicts with scheduled exams, assignments or required attendance. See full details at http://www.colorado.edu/policies/fac_relig.html (3) Students should be aware of the campus "Classroom Behavior" policy at http://www.colorado.edu/policies/classbehavior.html as well as faculty rights and responsibilities listed at http://www.colorado.edu/FacultyStaff/faculty-booklet.html#Part_1 These documents describe examples of unacceptable classroom behavior and provide information on how to handle such circumstances should they arise. Students and faculty each have responsibility for maintaining an appropriate learning environment. Those who fail to adhere to such behavioral standards may be subject to discipline. I will follow the guidelines stating that “professional courtesy and sensitivity are especially important with respect to individuals and topics dealing with differences of race, color, culture, religion, creed, politics, veteran's status, sexual orientation, gender, gender identity and gender expression, age, disability, and nationalities.” Class rosters are provided to me with the student's legal name. I will gladly honor your request to address you by an alternate name or gender pronoun. Please advise me of this preference early in the semester so that I may make appropriate changes to my records. See policies at http://www.colorado.edu/policies/classbehavior.html and at http://www.colorado.edu/studentaffairs/judicialaffairs/code.html#student_code (4) The University of Colorado Boulder (CU-Boulder) is committed to maintaining a positive learning, working, and living environment. The University of Colorado does not discriminate on the basis of race, color, national origin, sex, age, disability, creed, religion, sexual orientation, or veteran status in admission and access to, and treatment and employment in, its educational programs and activities. (Regent Law, Article 10, amended 11/8/2001). CU-Boulder will not tolerate acts of discrimination or harassment based upon Protected Classes or related retaliation against or by any employee or student. For purposes of this CU-Boulder policy, "Protected Classes" refers to race, color, national origin, sex, pregnancy, age, disability, creed, religion, sexual orientation, gender identity, gender expression, or veteran status. Individuals who believe they have been discriminated against should contact the Office of Discrimination and Harassment (ODH) at 303-492-2127 or the Office of Student Conduct (OSC) at 303-492-5550. Information about the ODH, the above referenced policies, and the campus resources available to assist individuals regarding discrimination or harassment can be obtained at http://www.colorado.edu/odh (5) All students of the University of Colorado at Boulder are responsible for knowing and adhering to the academic integrity policy of this institution. Violations of this policy may include: cheating, plagiarism, aid of academic dishonesty, fabrication, lying, bribery, and threatening behavior. All incidents of academic misconduct shall be reported to the Honor Code Council (honor@colorado.edu; 303-735-2273). Students who are found to be in violation of the academic integrity policy will be subject to both academic sanctions from the faculty member and non-academic sanctions (including but not limited to university probation, suspension, or expulsion). Other information on the Honor Code can be found at http://www.colorado.edu/policies/honor.html and at http://www.colorado.edu/academics/honorcode/

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