# Math 131AH: Analysis (Honors). Lec 1.

Lectures: MWF 11:00-11:50am in MS 5138.

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:

• 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.
• The lowest homework score will be omitted.

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:

• Homework 1 is due in class on Friday, Oct. 5.
• Homework 2 is due in class on Friday, Oct. 12.
• Homework 3 is due in class on Friday, Oct. 19.
• Homework 4 is due in class on Friday, Oct. 26.
• Homework 5 is due in class on Friday, Nov. 2.
• Homework 6 is due in class on Friday, Nov. 9.
• Homework 7 is due in class on Friday, Nov. 16.
• Homework 8 is due in class on Monday, Nov. 26.
• Homework 9 is due in class on Friday, Nov. 30.
• Homework 10: not to be turned in. You are strongly encouraged to solve it in order to better prepare for the final.