Instructor: Monica Visan, MS 6167. Email address: visan@math.ucla.edu
TA: Stan Palasek, MS 3965. Email address: palasek@math.ucla.edu
Office Hours: Mon: 9:30-10:30am, Wed: 2:30-3:30pm, or by appointment.
Problem session: Tu: 11:00-11:50am in MS 6627.
Textbook: Principles of Mathematical Analysis, 3rd Edition by W. Rudin and Metric Spaces, Cambridge University Press, by E. T. Copson.
Midterm: Wednesday, November 7th, in class.
The midterm counts for 20%
towards the final grade.
Final: Monday, December 10th, 8:00am-11:00am. 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 | Basic Logic, Peano Axioms, mathematical induction. |
| 2 | Mathematical induction, equivalence relations, construction of the rational numbers. |
| 3 | Ordered fields. |
| 4 | Rational numbers form an ordered field. |
| 5 | Upper and lower bounds, the least upper bound and the greatest lower bound, the least upper bound property. |
| 6 | Construction of the real numbers, the Archimedean property and consequences. |
| 7 | "Uniqueness" of an ordered field with the least upper bound property. |
| 8 | Sequences: bounded, monotonic, convergent. Examples. |
| 9 | Limit Theorems. Sequences diverging to ±∞. |
| 10 | The Cauchy criterion. |
| 11 | The limsup and liminf of a sequence. |
| 12 | The limsup and liminf of a sequence. |
| 13 | Series: convergent, absolutely convergent. The Cauchy criterion, the comparison and root tests. |
| 14 | The ratio test, the Abel, Leibnitz, and dyadic criterions. |
| 15 | Rearrangements. |
| 16 | Functions: domain, range, image, preimage, injective, surjective, bijective, inverse. |
| 17 | Cardinality of a set: finite, infinite, countable, at most countable, uncountable. Examples. |
| 18 | A set is infinite if and only if it is equipotent to one of its proper subsets. The Schroder-Bernstein theorem.. |
| 19 | A countable union of countable sets is countable. Finite and countable cartesian products of countable sets. Examples. |
| 20 | Metric spaces, examples, the Holder and Minkowski inequalities. |
| 21 | Interior, adherent, isolates, and accumulation points. Open and closed sets and their properties. |
| 22 | The exterios, boundary, and frontier of a set. Dense and nowhere dense sets. Subspaces of a metric space. |
| 23 | Complete metric spaces: definition and characterization. |
| 24 | Complete metric spaces: examples. |
| 25 | Baire's category theorem. |
| 26 | Separated sets and properties. |
| 27 | Connected sets: equivalent definitions. |
| 28 | Connected subsets of the real line. |
| 29 | Decomposition of a set into its connected components. |
Homework problems: