Math 171 - Fundamental Concepts of Analysis - Spring 2013

Lecture plan
Date Covered
4/2Welcome + 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.
4/4More 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.
4/9More on countable and uncountable sets. Criteria for countability. Brief mention of the "Continuum hypothesis". Series, their convergence and sum. Series with non-negative terms.
4/11The 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.
4/16Cauchy sequences in R. Continuous real functions. Metric spaces: definitions and several examples.
4/18More on metric spaces: convergence of sequences. Examples. The metric spaces l1, l2, l and convergence in there. Continuity of functions between metric spaces.
4/23Lecture given by Prof. Leon Simon. Open sets, closed sets, examples. Continuity.
4/25Inverse 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 Rn 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.
4/30The 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].
5/2Midterm exam.
5/7Compactness. Consequences of compactness (continuous real-valued functions are bounded and attain their maximum.) Relationship between closedness and compactness.
5/9The 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 Rn is compact iff it's closed and bounded. The closed unit ball in l is not compact.
5/14Completeness. Examples of complete and non-complete spaces. Compact spaces are complete. l2 is complete.
5/16Introduction to integration. Partitions of a rectangle in Rn 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.
5/21More 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).
5/28Proof that the Lebesgue integral of a function in L+(R) is well-defined (Lemma 3.4 in Leon Simon's notes).
5/30The Lebesgue integral of a function in L1 and some properties.
6/4Review.