The principal source of information about this course is now available (to registered students) on WebCT. In particular, the assignments are found there.
The course will cover roughly the first 7 chapters of the textbook. However, we will not look very closely at Chapter 0 or the first 6 sections of Chapter 1. These focus on an axiomatic treatment of set theory, and an axiomatic introduction to real numbers respectively. I would like to get to the interesting analysis as quickly as possible, so I will leave the foundations of set theory to other courses, and assume the fundamental properties of the real number line well known. I will, however, review several key point that were probably covered in Mathematics 281, but are particularly important for this course. The book has an interesting layout. The body of the chapter introduces the concepts, gives examples, and states and explains theorems. The proofs of the theorems are presented in an appendix at the end of each chapter. This is a good way to learn the material. It encourages the reader to be quite clear about what the theorem claims and how it can be used before getting buried in the proof. In the readings indicated below, I have not included the page references for the proofs. It will be understood, however, that when an assigned reading includes a theorem, then the proof should be looked up in the appendix and should be studied as well.
From the Introductory chapter in the textbook: Sets and functions between sets, infinite sets and cardinality, countable sets; Schröder - Bernstein Theorem (not in the text - a handout will be provided); proof that the rationals are countable and that the interval [0,1] is not.
Reading for week 1:
From Chapter 1 in the textbook: Review the main properties of the number systems (natural numbers, integers, rationals and reals) completeness of the real line, density of the rationals in the set of reals, sup and inf, Bolzano-Weierstrass Theorem for the set of reals, cluster points of a sequence.
metric spaces, normed linear spaces (=normed vector spaces) and inner product spaces, non-standard examples of each (including the Hausdorff metric, see readings below), Cauchy-Schwarz inequality.
Reading for week 2:
From Chapter 2 in the textbook: The first part of this material will be dealt with in greater generality than is done in the textbook. See the boxed comment on the bottom of page 106 in the text in this regard. In the lectures we will define the concept of a topological space and give examples of topological spaces, including subspace topology and pullback topology induced by a map into a topological space; We will show that metric spaces are examples of topological spaces. We will discuss open sets, interior of a set, closed sets, accumulation points, closure of a set, boundary of a set.
Reading for week 3:
From Chapter 3 in the textbook: compact sets, sequential compactness, Bolzano-Weierstrass Theorem, totally bounded sets, Heine-Borel Theorem, sketch of the proof of completeness of the space of compact subsets of a complete metric space with the Hausdorff metric between sets (this last item not found in the textbook).
Reading for week 4:
From Chapter 3 in the textbook: nested set property, four definitions of continuity (from Ch. 4 - as given in Theorem 4.1.4); (back to chapter 3:) path-connected sets, connected sets, limits. I will make it clear in class why I find it convenient to include Section 4.1 here rather than later, though of course the order on which the book covers this is perfectly fine.
From Chapter 4 in the textbook: composition of continuous functions, continuous images of compact and connected sets, Maximum-Minimum Theorem, Intermediate Value Theorem.
Reading for week 5:
From Chapter 4 in the textbook: uniform continuity and the Uniform Continuity Theorem.
From Chapter 5 in the textbook: uniform convergence, continuity of a uniform limit of a sequence or series of functions, Weierstrass M-test, the Cauchy Criterion and uniform Cauchy sequences.
Reading for week 6:
From Chapter 5 in the textbook: integration and differentiation of series, the space of bounded continuous maps as a metric space, and as a normed vector space if the image space is a normed vector space, with completeness if the image space is complete; equicontinuity and the Arzela-Ascoli Theorem.
Reading for week 7:
From Chapter 5 in the textbook: The Contraction Mapping Principle with applications to the existence theorem for ordinary differential equations, to integral equations, and to iterated function systems and the Collage Theorem (not in the textbook - see the readings below). To experiment with iterated function systems, a web search using the key words "iterated function systems" will turn up numerous websites.
Reading for week 8:
From Chapter 5 in the textbook: the Stone-Weierstrass Theorem.
From Chapter 6 in the textbook: Review of differentiable mappings between Euclidean spaces, the derivative as a linear map, techniques for the calculation of the derivative matrix, especially for mappings on spaces of matrices, say X -> Xm where X is an nxn matrix.
Reading for week 9:
From Chapter 6 in the textbook: review of the chain rule and the product rule; the Mean Value Theorem; higher derivatives as multilinear maps; Taylor's Theorem; differentiability of the exponential function on square matrices.
Reading for week 10:
From Chapter 7 in the textbook: Inverse Function Theorem and Implicit Function Theorem, with applications.
Reading for weeks 11 and 12:
| 5 homework assignments | 70% |
| Final exam | 30% |