Lecture plan
Date | Covered |
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4/2 | Welcome + introduction. Real numbers and their axioms (three types of properties, about: their algebra, their ordering, and the supremum property). Uniqueness of supremum (=least upper bound). Sequences and their convergence. Uniqueness of limits. Monotone bounded sequences are convergent. Lim sup and lim inf. |
4/4 | More on lim sup and lim inf of bounded sequences (the sequence converges if and only if they agree). Subsequences and the Bolzano-Weierstrass theorem. Countable and uncountable sets. |
4/9 | More on countable and uncountable sets. Criteria for countability. Brief mention of the "Continuum hypothesis". Series, their convergence and sum. Series with non-negative terms. |
4/11 | The geometric series. Absolute convergence. Rearrangements of series with non-negative terms. Tests for convergence and non-convergence. The comparison test. The root test. Power series and their radius of convergence. The Riemann zeta function. |
4/16 | Cauchy sequences in R. Continuous real functions. Metric spaces: definitions and several examples. |
4/18 | More on metric spaces: convergence of sequences. Examples. The metric spaces l^{1}, l^{2}, l^{∞} and convergence in there. Continuity of functions between metric spaces. |
4/23 | Lecture given by Prof. Leon Simon. Open sets, closed sets, examples. Continuity. |
4/25 | Inverse images (set-theoretic properties). More on continuity: A function is continuous if and only if the inverse image of any open set is open. A function is continuous if and only if the inverse image of any closed set is closed. A function is continuous if and only if it preserves limits. Examples: two metrics on R^{n} which has the same open sets (and hence the same continuous functions). In the "Standard" metric on Z, all subsets are open, so any function out of that metric space is continuous. |
4/30 | The relative metric on a subset of a metric space. Open and closed sets in the relative metric. The Heine-Borel theorem on open covers of [a,b]. |
5/2 | Midterm exam. |
5/7 | Compactness. Consequences of compactness (continuous real-valued functions are bounded and attain their maximum.) Relationship between closedness and compactness. |
5/9 | The image of a compact space is compact. A continuous bijection from a compact space has continuous inverse. A metric space is compact if and only if all sequences have a convergent subsequence. A subset of R^{n} is compact iff it's closed and bounded. The closed unit ball in l^{∞} is not compact. |
5/14 | Completeness. Examples of complete and non-complete spaces. Compact spaces are complete. l^{2} is complete. |
5/16 | Introduction to integration. Partitions of a rectangle in R^{n} into smaller rectangles. Step functions, and how to integrate them. The Riemann integral of a function defined on a rectangle. Some notes by prof. Leon Simon. |
5/21 | More on step functions. The Riemann integral of a bounded function defined on a rectangle. Continuous functions are Riemann integrable. |
5/23 | Lebesgue measure zero. The Lebesgue integral of a function in L_{+}(R). |
5/28 | Proof that the Lebesgue integral of a function in L_{+}(R) is well-defined (Lemma 3.4 in Leon Simon's notes). |
5/30 | The Lebesgue integral of a function in L^{1} and some properties. |
6/4 | Review. |