Introduction to Real Analysis (6310)
Course Web Page and Syllabus
Fall 2012               
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.

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

(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


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