Class: MWF 10:30-11:45, MB 124
Instructor: Dr. Zach Teitler
Office: MB 233A
Phone: 208-426-1086
E-mail: zteitler@boisestate.edu
Office hours: TBA and by appointment
Important dates:
Textbook:
Elementary Real Analysis, by Thomson, Bruckner, and Bruckner, 2nd ed, 2015.
See the web site of the book
to download and for errata.
Weekly schedule:
Dates | Topics (planned) | What we actually did |
---|---|---|
Week 1 | ||
Aug 24 | Properties of real numbers; suprema | ✔ |
Aug 26 | Suprema, sequences | ✔ |
Aug 28 | Sequences | ✔ |
Week 2 | ||
Aug 31 | Monotone convergence theorem | ✔ |
Sept 2 | Subsequences, Cauchy criterion | HW review and MCT |
Sept 4 | Upper and lower limits | Fekete's lemma, subsequences, Cauchy criterion |
Week 3 | ||
Sept 7 | Labor Day | |
Sept 9 | Cauchy criterion, upper and lower limits | ✔ |
Sept 11 | Infinite summation, unordered sums | Infinite summation, "unordered sequences" |
Week 4 | ||
Sept 14 | Unordered sums | ✔ |
Sept 16 | Cauchy criterion for series, absolute convergence, convergence tests | ✔ |
Sept 18 | Rearrangements | ✔ (Ohm's theorem, Dirichlet's theorem, other discussion) |
Week 5 | ||
Sept 21 | Rearrangements, Summability methods | Rearrangements (Riemann, Sierpinski, bounded permutations, Schlomilch) |
Sept 23 | Topology of real sets | Pringsheim's theorem on rearrangements. Cesaro and Abel summability methods |
Sept 25 | Compactness | Euler summation. Open sets |
Week 6 | ||
Sept 28 | Open sets | ✔ |
Sept 30 | Limit points, accumulation points, closed sets. Compactness | ✔ |
Oct 2 | Compactness | ✔ |
Week 7 | ||
Oct 5 | Limits and continuity of functions (review). Extremal properties | ✔ |
Oct 7 | Intermediate Value Property. Points of discontinuity | ✔ |
Oct 9 | Cantor set. Dense and nowhere dense sets. | ✔ |
Week 8 | ||
Oct 12 | Discontinuities of monotone functions. Countability of removable and jump discontinuities | ✔ |
Oct 14 | Oscillation, points of discontinuity | ✔ |
Oct 16 | G-deltas and F-sigmas; points of continuity are a G-delta | ✔ |
Week 9 | ||
Oct 19 | Baire category, basic properties | ✔ |
Oct 21 | Baire category theorem | ✔ |
Oct 23 | Q is not a G-delta | ✔ |
Week 10 | ||
Oct 26 | Uniform continuity, Riemann sums | ✔ |
Oct 28 | Continuous functions are Riemann integrable | ✔ |
Oct 30 | Basic properties of Riemann integration. Lebesgue criterion for Riemann integrability | Basic properties |
Week 11 | ||
Nov 2 | Proof of Lebesgue's criterion | Riemann criterion |
Nov 4 | Differentiation on R^n: partial and directional derivatives | Sets of measure zero |
Nov 6 | Integrals depending on a parameter | Riemann-Lebesgue lemma |
Week 12 | ||
Nov 9 | p-norms on R^n | |
Nov 11 | Derivatives, directional derivatives, partial derivatives | |
Nov 13 | Chain rule | |
Week 13 | ||
Nov 16 | Implicit function theorem | |
Nov 18 | Implicit function theorem | |
Nov 20 | Implicit and Inverse function theorems | |
Week 14 | ||
Nov 30 | Banach and sequence spaces | |
Dec 2 | Completeness of lp and c0 | |
Dec 4 | Containments of lp's; comparison of norms; non topological equivalence | |
Week 15 | ||
Dec 7 | Denseness of c00; parallelogram law (lp's are not Hilbert spaces, except p=2); other miscellaneous | |
Dec 9 | Separability; Hamel basis | |
Dec 11 | Schauder basis; continuity of linear functionals, dual spaces |
Written Homework:
Homeworks 1-5 (show/hide)
Assigned date | Assignment | Suggested problems | Due date |
---|---|---|---|
Wed, 11/11/15 | 11 |
Choose 3-4 problems from: 11.7.12, 11.7.13; 11.10.1, 11.10.2, 11.10.5, 11.10.6; 12.4.8, 12.4.10, 12.4.11, 12.4.13, 12.4.15, 12.4.18; 12.8.2, 12.8.3, 12.8.4 |
Wed, 11/18/15 |
Wed, 12/2/15 | 12 | Choose 3-4 problems as you wish. Recommended are chapters 11 and 12. | Wed, 12/9/15 |
You may use this homework template (383 KB) if you wish. It is completely optional.