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| Date | Sections | What is covered. Comments | Homework assignments |
| 9/5 | s. 1.1 | Sets, relations | |
| 9/7 | s. 1.1 | Equivalence relations. Cosets, quotient space. Functions. | p. 23 # 2, 14, 15. Read s. 1.2 |
| 9/10 | s. 1.3 | Comparison of infinite sets. Countable and uncountable sets | 1. Prove that
the set of algebraic numbers is countable. 2. Is the set of all power series with rational coefficients countable? What about the set of polynomials with rational coefficients? |
| 9/12 | s. 1.3 | Comparison of cardinalities. Cantor-Berstein theorem | Prove that R^2 has cardinality continuum (i. e. the sam cardinality as the real line) |
| 9/14 | s. 1.5, 1.9 | Different ways of defining real numbers | Show that if X is an infinite set, and C is countable, then Card(X)=Card(X \union C) |
| 9/17 | s. 2.1 | Review of sequences | Click here to get the assignment |
| 9/19 | s. 6.3 | Metric spaces. Topology of metric spaces. | Click here to get the assignment |
| 9/21 | s. 6.1 | Interior, exterior, closure and boundary. Properties of open sets. | Click here to get the assignment |
| 9/24 | s. 6.1, 6.4 | Topological spaces. Limits and continuity in topological spaces. | Click here to get the assignment |
| 9/26 | s. 6.4 | Base of topology. Relative topology | Click here to get the assignment |
| 9/28 | s. 6.7 | Connected sets. | p. 150 # 1, 7, 20, 22 |
| 10/1 | s. 6.5 | Sequences and limits. Hausdorf spaces | Click here to get the assignment |
| 10/3 | s. 6.6 | Compactness. Sequential compactness | Click here to get the assignment |
| 10/5 | s. 6.6 | Compactness of a product. | Click here to get the assignment |
| 10/8 | Columbus day | ||
| 10/10 | S. 8.2 Differentiability | ||
| 10/12 | Midterm 1 | ||
| 10/15 | s. 8.1, 8.2 Review of linear algebra. Norms in R^n. Jacobi matrix. | Click here to get the assignment | |
| 10/17 | s. 8.4 Partial derivatives and differentiability | ||
| 10/19 | s. 8.1, 8.2. Matrix norms. Chain rule | Click here to get the assignment | |
| 10/22 | s. 8.3 Mean value theorem | ||
| 10/24 | s. 8.4 Higher order derivatives. Taylor's formula | Click here to get the assignment | |
| 10/26 | s. 8.5 Inverse function and implicit function theorems | p. 198 # 15, 16 | |
| 10/29 | s. 8.5 Proof of the inverse function and implicit function theorems | ||
| 10/31 | Manifolds. | Click here to get the assignment | |
| 11/2 | Manifolds. Tangent spaces | Click here to get the assignment | |
| 11/5 | Measures, algebras, sigma algebras | Click here to get the assignment
I'll be collecting #1, 4 from this assignment together with assignments from 10/31, 11/2 Wed. Nov. 7 |
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| 11/7 | Outer measures. | Click here to get the assignment | |
| 11/9 | s. 9.3 | Outer measures | |
| 11/12 | s. 9.4 | Proof of Caratheodory Theorem | Click here to get the assignment |
| 11/14 | Applications of Caratheodory theorem. | ||
| 11/16 | Uniqueness of extension. Complete measures. Borel sigma algebra. | Midterm 2 (take home) | |
| 11/19 | Uniqueness of translation invariant Borel measure. Hausdorff measures. | Compute Hausdorff measure H_p of the unit cube in R^n for all p. | |
| 11/21 | Thanksgiving | ||
| 11/23 | |||
| 11/26 | s. 9.5 | Hausdorff dimession. Metric outer measure. | Click here to get the
assignment. Will be collected Fri. 11/30 |
| 11/28 | s. 9.5, 9.6 | Metric outer measure. Non-measurable sets. | p. 219 # 6 (hint: use # 5), # 14 |
| 11/30 | s. 10.1 | Measurable functions. | Click here to get the assignment. |
| 12/2 | s 10.1, 10.2 | Definition of integral, distribution function. | Click here to get the assignment. |
| 12/5 | s. 10.2 | Monotone Convergence theorem, fatou lemma | Click here to get the assignment. |