Instructor: Monica Visan, MS 6167. Email address: firstname.lastname@example.org
TA: Bjoern Bringmann, MS 6139. Email address: email@example.com
Office Hours: Mon: 9:30-10:30am, Wed: 1:00-2:00pm, or by appointment.
Problem session: Tu: 11:00-11:50am in MS 5137.
Textbooks: Principles of Mathematical Analysis, 3rd Edition by W. Rudin and Metric Spaces, Cambridge University Press, by E. T. Copson.
Midterm: Wednesday, February 13th, in class.
The midterm counts for 20%
towards the final grade.
Final: Monday, March 18th, 11:30am-2:30pm. You have to take the
order to pass the class. The final exam will count for 60% towards the
Homework: There will be weekly homework. It is due on Fridays in class. The homework will count for 20% towards the final grade. Further information is given below.
Grading: Homework 20%; Midterm 20%; Final 60%.
|1||Compactness and sequential compactness.|
|2||The Heine-Borel theorem.|
|3||Compactness and the finite intersection property. Cartesian products of compact spaces.|
|4||Continuous functions: equivalent definitions.|
|5||Continuity and compactness. Uniform continuity.|
|6||Continuity and connectedness. Path connectedness.|
|7||Convergent sequences of functions. Dini's theorem.|
|8||Spaces of functions. The Arzela-Ascoli theorem.|
|9||The Arzela-Ascoli theorem. Necessity of the hypotheses.|
|10||The oscillation of a real function and continuity.|
|11||The points of continuity of a pointwise limit of continuous functions on a space with the Baire property form a dense set.|
|12||The Weierstrass approximation theorem. Stone-Weierstrass.|
|14||The derivative of a function. Sums and products of differentiable functions. The chain rule.|
|15||Mean value theorems. The intermediate value property for derivatives.|
|16||The derivative of the inverse. L'Hospital's rule.|
|18||Uniform convergence and differentiation.|
|19||The Darboux integral.|
|20||Criteria for Darboux integrability.|
|21||The Riemann integral. Equivalence of the two integrability notions.|
|22||Monotonic and continuous functions on compact intervals are Riemann integrable. Properties of the Riemann integral.|
|23||Integration by parts, the change of variable formula, and the fundamental theorem of calculus.|
|24||Uniform convergence and integration.|
|25||The Lebesgue criterion for Riemann integrability.|
|26||Improper Riemann integrals: the Cauchy and Abel criteria.|
|27||The contraction mapping theorem and aplications.|
|28||Continuous 1-periodic functions: convolution, approximation to the identity.|
|29||Dirichlet and Fejer kernels. Cesaro convergence of the Fourier series.|