Math 131BH: Analysis Honors. Lec 1.

Lectures: MWF 11:00-11:50am in MS 5127.

Instructor: Monica Visan, MS 6167. Email address:

TA: Bjoern Bringmann, MS 6139. Email address:

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 final exam in order to pass the class. The final exam will count for 60% towards the final grade.

Exam rules:

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%.

Lecture Topics
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.
13 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.
17 Taylor's theorem.
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.

Homework problems: