Instructor: Monica Visan, MS 6167. Email address: visan@math.ucla.edu
TA: Bjoern Bringmann, MS 6139. Email address: bringmann@math.ucla.edu
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: