**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:**

- Bring student ID to the midterm and the final.
- There will be no make-up exams.
- No calculators, notes, or books will be permitted in any exam.

**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.

- No late homework will be accepted.
- Write your name at the top of the first page.
- Staple your pages!
- The lowest homework score will be omitted.

**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:**

- Homework 1 is due in class on Friday, Jan 11.
- Homework 2 is due in class on Friday, Jan 18.
- Homework 3 is due in class on Friday, Jan 25.
- Homework 4 is due in class on Friday, Feb 1.
- Homework 5 is due in class on Friday, Feb 8.
- Homework 6 is due in class on Friday, Feb 15.
- Homework 7 is due in class on Friday, Feb 22.
- Homework 8 is due in class on Friday, Mar 1.
- Homework 9 is due in class on Friday, Mar 8.
- Homework 10 is due in class on Friday, Mar 15.