Download Modern Analysis Course at Columbia University and more Lecture notes Algebra in PDF only on Docsity!
Columbia University in the City of New York | New York, N.Y. 10027
DEPARTMENT OF MATHEMATICS 508 Mathematics Building 2990 Broadway
Fall Semester 2005
Professor Ioannis Karatzas
W4061: MODERN ANALYSIS
Description
The algebra of sets; ordered sets, the real number system, Euclidean space. Finite, countable, and uncountable sets. Elements of general topology: metric spaces, open and closed sets, complete- ness and compactness, perfect sets. Sequences and series of real numbers, especially power series; the number e. Continuous maps.
Functions of real variable: continuity and differentiability, the chain and L’Hopital rules. The Riemann integral: characterizations, mean-value theorems, the fundamental theorem of calculus. Uniform convergence; its relevance in continuity, integration and differentiation. Sequences and series of functions; double series.
Approximations: the Stone-Weierstrass theorem, Bernstein polynomials. Euler/Mac Laurin, De Moivre, Wallis and Stirling. Taylor approximations, Newton’s method. The DeMoivre/Laplace and Poisson approximations to the bimomial distribution; examples.
Monotone functions, functions of finite variation. Infinitely-differentiable functions. Continuous functions which are nowhere differentiable. Convex sets, their separation properties. Convex functions, their differentiability and their relevance.
_________________
Prerequisites: Calculus IV (Math V1202) and Linear Algebra (Math V2010).
Required Text: W. RUDIN: Principles of Mathematical Analysis. Third Edition, 1976. McGraw-Hill Publishing Co. New York.
Detailed Lecture Notes , generously made available by Professor P.X. Gallager, will be distributed regularly.
Homework will be assigned and discussed regularly, in recitation sections by TA’s.
There will be a Mid-Term and a Final Examination.
COURSE SYLLABUS (Tentative)
. Lecture #1: Wednesday, 7 September. Algebra of subsets. . Lecture #2: Monday, 12 September. Algebra of maps.
Assignment #1: To be turned in Monday, 19 September.
. Lecture #3: Wednesday, 14 September. Partitions. Equivalence relations. Cardinal numbers. . Lecture #4: Monday, 19 September Countable and uncountable sets. . Lecture #5: Wednesday, 21 September Properties of the rational number system. Notions of total ordering, field, totally ordered field, upper bound, supremum, the least-upper-bound property. Cuts , as subsets of the rationals. The Dedekind construction of the reals.
Reading: Chapter 1 of Rudin, including the Appendix.
. Lecture #6: Monday, 26 September Metric and Topological Spaces. Open and closed sets; interior and closure of a set; properties. . Lecture #7: Wednesday, 28 September Notions of limit points of sets; perfect sets. Equivalent characterization of closed sets, in terms of their limit points. Continuous functions – global and local notions, relationship.
Reading: Chapter 2 of Rudin, pp.24-36. Assignment #2: To be turned in Wednesday, 5 October.
. Lecture #16: Monday, 30 October The definition and properties of the Riemann integral. . Lecture #17: Wednesday, 2 November Definition and properties of the derivative. Intermediate and mean-value theorems. The fundamental theorem of calculus.
Reading: Chapter 5 of Rudin, pp. 103-111. Chapter 6 of Rudin, pp. 123-134.
Assignment # 6: To be handed in on Wedn. 9 November. Problems # 1, 2, 4, 6, 7, 10, 11 of Chapter 5 in Rudin. Problems # 2, 4 of Chapter 6 in Rudin.
. Lecture #18: Monday, 7 November University Holiday . Lecture #19: Wednesday, 9 November Mean Value Theorems for Integrals. Uniform convergence and integration; uniform convergence and differentiation. The Holder and triangle inequalities for the integral.
Assignment # 7: Not to be handed in. Read pp. 147-154 in Rudin. Exercises 1-7 on p. 19.6 of Prof. Gallager’s notes. Problems # 22, 24, 26, 27 of Chapter 5 in Rudin. Problem # 15 of Chapter 6 in Rudin.
. Lecture #20: Monday, 14 November Differentiation under the integral sign. Double integrals; improper integrals. The Gamma function.
Assignment # 8: Due Monday, 21 November. Exercises 1-3 on p. 22.6 of Prof. Gallager’s notes. Problems # 7, 8, 9, 16 (pp. 138-141) of Chapter 6 in Rudin. Problem # 4, 7, 12 (pp. 165-167) of Chapter 7 in Rudin.
. Lecture #21: Wednesday, 16 November Properties of the exponential and Gamma functions. Transcendence of e.
Assignment # 9: Not to be handed in. Problems # 1, 4, 5 (a,b), 6, 9 (pp. 196-197) of Chapter 8 in Rudin. Exercises 1, 2, 3 on p. 24.5 of Prof. Gallager’s notes.
. Lecture #22: Monday, 21 November Euler-MacLaurin summation formula. Formula of Wallis. The Stirling approximation. . Lecture #23: Wednesday, 23 November Taylor approximation. Newton’s method.
Assignment # 10: Not to be handed in. Problems # 15, 16, 17, 18, 19, 25 (pp. 115-118) of Chapter 5 in Rudin.
. Lecture #24: Monday, 28 November The Binomial theorem, the Binomial distribution. Computation of moments. The weak law of large numbers. . Lecture #25: Wednesday, 30 November Newton’s Binomial Series. Bernstein’s proof of the Weierstrass approximation theorem. The Gauss-Laplace function. Statement and significance of the DeMoivre-Laplace limit theorem.
Assignment # 11: Not to be handed in.
. Lecture #26: Monday, 5 December Proof of the DeMoivre-Laplace limit theorem. The Poisson approximation to the binomial probabilities. . Lecture #27: Wednesday, 7 December Irrationality – and computation – of \pi. Computation of e. . Lecture #28: Monday, 12 December The Laplace asymptotic formula. . Lecture #29: Wednesday, 14 December Problem-solving session.
