Introduction to Analysis

Overview

Lectures: Tu, Th 9:30 - 11:00am, Room 70 Evans

Instructor: Jan Reimann

Office: 705 Evans Hall
Office hours: Tu 11 - 1, We 1-2, and by appointment
Discussion session: We 4:30-6pm

Email:
Personal Website: http://math.berkeley.edu/~reimann/

Course Description

In this course we will study the foundations of real analysis. This means we will get acquainted with the real number system, how it can be defined axiomatically. We will then use the basic properties of the real numbers to study fundamental notions of analysis such as sequences and series. A thorough understanding of these is a prerequisite for any higher level mathematics.

We will also see how this notions can be treated in a more general spaces, not necessarily based on real numbers, namely metric spaces. This will give us a first glimpse at how mathematics uses generalization.

Next, we study functions over the real numbers. The basic concept here is continuity. In particular, we will learn how to describe the intuitive concept of continuity ("a function without jumps") formally using the notion of sequences, and prove results about continuity rigorously. As continuity is a purely topological concept, we will see that we can treat it much more generally in metric spaces.

After that, we will start approximating functions by other functions. This brings us to sequences and series of functions. A fundamental way to approximate is using power series, which can be thought of as "infinite" (or better, sequences of) polynomials.

In the final part of the course we will revisit concepts already known from Calculus, differentiation and integration. This time, we will investigate them from another angle, much less interested in doing actual calculations than in proving basic properties.

Throughout the class, strong emphasis will be put on writing rigorous mathematical proofs.

Literature

The textbook for the course will be K.A. Ross, Elementary Analysis: The Theory of Calculus. We will follow it more or less closely. Chapters 1-4 will be the core topics of the course, and so we will study them most thoroughly. The last part of the course will cover parts of chapters 5 and 6.

I will occasionally draw material from two other texts, Rudin's Principles of Mathematical Anaylsis and Real Mathematical Analysis by Pugh, especially supplements on metric spaces and examples.

Exams

There will be two midterms: the first on Tuesday, September 26 (in class), the second on Tuesday, November 7 (also in class). Note that the first midterm is before the five-week drop-out deadline.

Information on the first midterm (09/26): This will be a closed book exam(sorry, no cheat sheets!). Bring your blue books. There is a sample midterm available.

Information on the second midterm (11/07): Again, this will be a closed book exam(sorry, no cheat sheets!). The material covered will be Ross, §§ 13-15, 17-21, 23-26. Bring your blue books. There is a sample second midterm available.

The final exam is held on Thursday December 14, 8-11 am, 166 Barrows. As before, a closed book exam (no cheat sheets). The final exam will cover all material covered in class (see log below), with an emphasis on the second part of the semester. Bring your blue books. There is a sample final available.

The final grade will be determined as follows: Only yhe better one of the two midterm scores will count towards the final grade. The final grade will be the better one calculated by the following two methods: (1) 20% homework, 40% midterm score, 40% final exam; (2) 1/3 homework, 1/3 midterm, 1/3 final.

Homework

Homework will be assigned on Tuesday and will be due on the following Tuesday in class. Homework will be graded and the lowest score will be dropped. Late homework will not be accepted.

A note on academic honesty: Collaboration among students to solve homework assignments is welcome. This is a good way to learn mathematics. So is the consultation of other sources such as other textbooks. However, every student should hand in an own set of solutions, and if you use other people's work or ideas you should indicate the source in your solutions.
(In any case, complete and correct homework receives full credit.)

Date File Due Solutions
09/05 Homework 1 (pdf) 09/12
09/12 Homework 2 (pdf) 09/19
09/19 Homework 3 (pdf) 09/26
10/03 Homework 4 (pdf) 10/10
10/10 Homework 5 (pdf) 10/17
10/17 Homework 6 (pdf) 10/24
10/24 Homework 7 (pdf) 10/31
10/31 Homework 8 (pdf) 11/07
11/14 Homework 9 (pdf) 11/21
11/29 Homework 10 (pdf) 12/05

There is a GSI for this course, Aubrey Clayton. His office hours are: Mo 10-12, 2-5 and Tu 9:30-11, 1:30-5 in 891 Evans.

Brief Summary of Lectures

Date Material Covered Suggested Textbook Reading
08/29 Introduction; the rational numbers;
Pythagoras' proof of the irrationality of square roots;
approximations by rational numbers;
the field axioms
Chap. 1, § 1-3
08/31 Consequences of the field axioms;
discussion of the notion of an axiom;
orders and ordered fields;
properties of ordered fields
Chap. 1, § 3
09/05 lower and upper bounds, infima and suprema;
completeness;
the real numbers as a complete ordered field;
consequences of completeness: the Archimedean property, density of the rationals;
absolute value and distance function
Chap. 1, § 3,4, and 6
09/07 absolute value and distance function;
sequences of real numbers; convergence;
examples of convergent and divergent sequences
Chap. 1, § 3, 6; Chap. 2, § 7, 8
09/12 natural numbers and induction;
stability properties of limits;
examples; monotone and bounded sequences
Chap. 1, § 1; Chap. 2, § 8, 9, 10
09/14 lim sup and lim inf; Cauchy sequences;
subsequences
Chap. 2, § 10, 11
09/19 Bolzano-Weierstrass Theorem;
cardinality; finite, countable and uncountable sets
Chap. 2, § 11; Rudin Chap. 2, pp. 24 - 30
09/21 comparing cardinalities;
Cantor-Schroeder-Bernstein Theorem;
Cantor's diagonal method and uncountability of the set of reals
Rudin Chap. 2, pp. 24 - 30
09/26 1st Midterm
09/28 Review midterm;
Infinite series; convergence; Cauchy criterion for series; geometric and harmonic series
Chap. 2, § 14
10/03 Tests for convergence of series: comparison, root, ratio test;
alternating series; integral tests;
metric spaces
Chap. 2, § 13, 14, 15
10/05 Examples of metric spaces;
convergence and completeness in metric spaces;
topology; open balls, open sets, closed sets, examples
Chap. 2, § 13
10/10 Characterizations of open and closed sets; interior, boundary, and limit points;
the Cantor set;
compactness; examples; statement of the Heine-Borel Theorem
Chap. 2, § 13
10/12 Proof of the Heine-Borel Theorem
Chap. 2, § 13
10/17 Continuous functions, stability properties;
Continuous functions on compact sets;
Intermediate value theorem
Chap. 3, § 17, 18
10/19 Applications of the Intermediate Value Theorem;
uniform continuity; uniform continuity and compactness;
continuity in arbitrary metric spaces
Chap. 3, § 18-21
10/24 Power series, radius of convergence;
sequences of functions; pointwise and uniform convergence;
uniform convergence and continuity
Chap. 4, § 23-25
10/26 uniform convergence of power series; Abel's Theorem'
Weierstrass approximation theorem;
Chap. 4, § 26-27
10/31 exponential function; complex numbers;
definition and properties of sin and cos;
11/02 properties of trigonomeric functions;
review of material for midterm;
11/07 second midterm
11/09 review of second midterm;
differentiability;
Chap. 5, § 28
11/14 chain rule for differentiation;
mean value theorem
Chap. 5, §§ 28,29
11/16 mean value theorem; examples
Chap. 5, § 29
11/21 Taylor series
Chap. 5, § 31
11/28 Definition of the Riemann integral
Chap. 6, § 32
11/30 Properties of the Riemann integral
Chap. 6, § 33
12/05 Fundamental theorem of calculus
Chap. 6, § 33
12/07 Integration and differentiation of power series
Chap. 4, § 26