Readings are noted on the day when they should be completed. Daily homework problems are noted on the day when they are assigned; they will be discussed on the following day. Weekly (graded) homework assignments are noted on the day when they are assigned, and the due date is provided.
| Week | Date | Reading/Topic | Daily Homework | Weekly Homework |
|---|---|---|---|---|
| 8/25 - 8/29 | 8/25 | No reading (first day of class) | I.9, I.10 | HW #1: II.1, II.7, II.10, II.11 (due Wed 9/3) | 8/27 | Chapter 2: Axiomatic description of the real numbers | II.9 | 8/29 | Chapter 2: Construction of the real numbers | II.14 |
| 9/1 - 9/5 | 9/1 | No class (Labor Day) | 9/3 | Chapter 3.1: Metric spaces | HW #2: III.1(b)(c), III.4, III.5, III.15 (due Wed 9/10) | 9/5 | Chapter 3.2: Open and closed sets in metric spaces | III.2 (just the sentence about open balls), Show: a set U in a metric space X is open iff U is a union of a collection of open balls |
| 9/8 - 9/12 | 9/8 | Chapter 3.3: Sequences, convergence, limits | 9/10 | Chapter 3.4: Cauchy sequences, completeness, completion of a metric space |
Show that the space of continuous functions
on [0,1] with the metric d(f,g) = integral of |f(x)-g(x)| from x=0 to x=1 is not complete. |
HW #3: III.12, III.18, III.24, III.32 (due Wed 9/17) | 9/12 | Chapter 3.5: Compactness | Show that every closed subspace of a complete metric space is complete |
| 9/15 - 9/19 | 9/15 | Chapter 3.5: Compactness (continued) | Show that the set of sequences of real numbers
which are bounded by one, equipped with the supremum metric, is bounded but not totally bounded. |
9/17 | Chapter 3.5: Compactness (continued) | Give an example of two disjoint closed sets A,B in R^2 so that dist(A,B)=0 | HW #4: III.31, III.33, two extra problems (due Fri 9/26) | 9/19 | Chapter 3.6: Connectedness |
| 9/22 - 9/26 | 9/22 | Chapter 4.1,4.2: Continuity | Show that the function f defined by f(x) = 1 for x < 0 and
f(x) = x for x >= 0 is not continuous at x = 0 (Give three different proofs using each of the three definitions of continuity discussed in class) |
9/24 | Chapter 4.1,4.2,4.3: Continuity (continued) | no homework this week | 9/26 | Homeomorphisms, bi-Lipschitz maps and isometries | IV.1 |
| 9/29 - 10/3 | 9/29 | Chapter 4.4: Continuity and compactness, Extreme Value Theorem, uniform continuity |
10/1 | MIDTERM EXAM I | 10/3 | Chapter 4.5: Continuity and connectedness, Intermediate Value Theorem, Peano curves |
HW #5: IV.3, IV.8, IV.16, IV.22(a)(b), IV.27 (due Fri
10/10) NOTE: IV.16 is a particularly important result! |
| 10/6 - 10/10 | 10/6 | Chapter 4.6: Sequences of functions, pointwise convergence vs. uniform convergence | 10/8 | Chapter 4.6: Function spaces, uniform metric, Kuratowski Isometric Embedding Theorem |
Show that balls (open or closed) in the metric space (C_b(X),D) of bounded real-valued functions on a metric space X are connected sets. | 10/10 | Contraction Mapping Principle in complete metric spaces, Picard Existence and Uniqueness Theorem for first order ODE (Guest lecturer: Jiri Lebl) |
| 10/13 - 10/17 | 10/13 | Chapter 5.1, 5.2: Definition of the derivative, basic properties, Product, Quotient, Chain Rules | V.2 | HW #6: V.1(a)(b), V.5, two extra problems due Mon 10/20 | 10/15 | Chapter 5.3: Mean Value Theorem, higher order derivatives, polynomial approximation, geometric interpretations of derivatives | 10/17 | Smoothness classes C^k, bump functions, partitions of unity |
| 10/20 - 10/24 | 10/20 | The Riemann-Darboux integral | HW #7: VI.11, VI.13, VI.16, VI.22, one extra problem due Wed 10/29 | 10/22 | Examples of integration, continuous functions are integrable, nets and Moore-Smith limits | Is the function f defined on [0,1] by f(x)=1/q if x=p/q is a nonzero rational in lowest terms and f(x)=0 if x is irrational or zero Riemann integrable? | 10/24 | Properties of Riemann integrable functions, monotone functions are integrable, characterization of integrability via step functions | Prove that the Riemann integral is additive as a function of the interval [a,b] |
| 10/27 - 10/31 | 10/27 | Indefinite integrals, Fundamental Theorem of Calculus, integration by parts, change of variables | 10/29 | Defining functions via indefinite integrals, the Gamma function | HW #8 three extra problems due Wed 11/5 |
10/31 | Discontinuity sets of integrable functions, sets of (Lebesgue) measure zero | Show that subsets of sets of measure zero are sets of measure
zero Show that countable unions of sets of measure zero are sets of measure zero Show that f is continuous at x if and only if the oscillation of f at x is equal to zero |
| 11/3 - 11/7 | 11/3 | (Optional lecture) An introduction to the Lebesgue integral | 11/5 | MIDTERM EXAM II | 11/7 | Chapter 7.1: Integration and differentiation of sequences of functions |
| 11/10 - 11/14 | 11/10 | Chapter 7.2: Series, absolute vs. conditional convergence | HW #9: VII.5, VII.11, VII.35, two extra problems due Mon 11/17 | 11/12 | Chapter 7.2: Rearrangements, a series definition for e, irrationality of e | 11/14 | Chapter 7.2, 7.3: Series of functions, Weierstrass M-test, power series |
| 11/17 - 11/21 | 11/17 | Chapter 7.3: Power series (continued), real analytic functions | 11/19 | MIDTERM EXAM III | 11/21 | NO CLASS: HAPPY THANKSGIVING! |
| 12/1 - 12/5 | 12/1 | O/o notation, uniform approximation, Weierstrass Approximation Theorem, Stone-Weierstrass Theorem | HW #10: on handout due Wed 12/10 | 12/3 | Weierstrass Approximation Theorem (proof), accuracy of uniform approximations | 12/5 | Existence of best approximants |
| 12/8 - 12/10 | 12/8 | Compactness for subsets of C(K), equicontinuity, Arzela-Ascoli theorem | 12/10 | Weierstrass' examples of continuous nowhere differentiable functions, random walks, Brownian motion, Peano curves |