lecture 1 (01/18/24) The real number system
lecture 2 (0120/24) Density of the rationals, existence of square roots.
lecture 3 (01/22/24) Metric spaces, Cauchy-Schwarz, triangle inequality of the Euclidean metric, open balls in metric spaces
lecture 4 (01/24/24) Properties of open sets. Bounded sets. Closed and bounded sets in R. Convergence of sequences. Convergent sequences are bonded.
lecture 5 (01/26/24) Convergent sequences are bounded. A subset C is closed iff it contains all the limits of sequences in C.
Properties of sequences in R. Interior, closure, boundary.
lecture 6 (01/29/24) Properties of the interior, closure, boundary. Monotone sequences in R. Iim sup, lim inf.
lecture 7 (01/31/24) Iim sup, lim inf. Completeness: Cauchy sequences and their properties, notion of a complete metric space.
lecture 8 (02/02/24) Every bounded sequence is R has a convergent subsequence. R is complete. Rn is complete with respect to d1, norms on Rn and the associated metrics. Equivalent norms and metrics. Equivalence of norms and convergence and completeness.
lecture 9 (02/05/24) Rn is complete with respect to d2 and d∞ . Notion of a topology. Equivalent metrics define the same topology. Convergence of a sequence in a topological space. Open covers and compactness.
lecture 10 (02/07/24) Definition of compact, closed set ∩ compact set = compact, in metric spaces compact sets are closed and bounded, there are metric spaces where closed and bounded sets are not compact. In general topological spaces compact sets need not be closed. Nested sequences of closed compact sets. Sequential compactness.
lecture 11 (02/09/24) In metric spaces, a set is compact iff it's sequentially compact iff it's complete and totally bounded. In Rn a set is compact iff it's closed and bounded.
lecture 12 (02/12/24) Continuity of functions between metric spaces and in general. Composite of two continuous maps is continuous. Images of compact sets under continuous maps is compact. A continuous real-valued function on a compact set achieves max and min. Cluster points, limits.
lecture 13 (02/14/24) Function is continuous iff it maps convergent sequences to convergent sequences. Sums, products and ratios of continuous functions. Uniform continuity. A continuous function on a compact set is uniformly continuous.
lecture 14 (02/16/24) Uniform convergence of functions. Uniform limit of continuous functions is continuous.
(02/19/24) Review for the first midterm. Please report any issues on Campuswire.
(02/21/24) First midterm exam.
lecture 15 (02/23/24) Connectedness and path connectedness. In particular [0,1] is connected.
lecture 16 (02/26/24) Path connected spaces are connected, but not conversely. Intermediate value theorem.
lecture 17 (02/28/24) differentiation, differentiable implies continuous, chain rule, sum, product and ratio. Derivative is 0 at extremal points.
lecture 18 (03/01/24) Rolle's, Mean Value theorems, derivatives and monotonicity, continuous bijections between compact metric spaces have continuous inverses.
lecture 19 (03/04/24) Inverse function theorem, Taylor's theorem
lecture 20 (03/06/24) Proof of Taylor's theorem. Analytic and flat functions.
lecture 21 (03/08/24) Upper and lower sums, Darboux integral.
lecture 22 (03/18/24) Cauchy criterion for integrability, Riemann integral, properties of integrals.
lecture 23 (03/20/24) Properties of integrals.
lecture 24 (03/22/24) The composite g○f h(x) = g(f(x)) is integrable if g is continuous and f is integrable. Fundamental theorem of calculus (versions 1 and 2), integration by parts.
lecture 25 (03/25/24) Change of variables formula for integrals. Natural logarithm and the exponential functions.
lecture 26 (03/27/24) Integrals and derivatives of limits of sequences of functions. Differentiation under the integral sign.
lecture 27 (03/29/24) Differentiation under the integral sign. Series.
lecture 28 (04/01/24) Rearrangement of terms of absolutely convergent series, convergence tests: comparison, root, ratio.
lecture 29 (04/03/24) Ratio and Dirichlet tests. Power series, radius of convergence.
(04/05/24) Review for the second midterm. Please report any issues on Campuswire.
(04/08/24) Second midterm exam.
lecture 30 (04/10/24) Weierstrass M-test. Integrating a series of functions term by term.
lecture 31 (04/12/24) Integration and differentiation of power series. Power series of definition of sin(x) and cos(x).
The next few lectures will follow Jim Belk's measure theory course
lecture 32 (04/15/24). Sums over sets. Lebesgue outer measure. Definition of measurable sets.
lecture 33 (04/17/24). (Countable) unions of measurable sets are measurable. Intervals are measurable.
lecture 34 (04/19/24). Intervals are measurable and their Lebesgue measure is the length. Sets of (outer) measure 0 are measurable. Countable sets are measurable. The notion of a measurable function.
lecture 35 (04/22/24). Integral of a simple function. Definition of the Lebesgue integral of a nonnegative measurable function.(In lecture 35 I am "borrowing" arguments from Adams and Guillemin's Measure theory and probability, pp 60-64
lecture 36 (04/24/24). Measurable functions with more care. Measurable functions as monotone limits of simple functions.
lecture 37 (04/26/24). Definition of an integral of a measurable function. The sum of two measurable functions is measurable, Lebesgue monotone convergence theorem.
lecture 38 (04/29/24). Definition of L1(E); it's a vector space and integration is linear. Monotonicity of the Lebesgue integral.
(05/01/24) Review for the final exam. Please report any issues on Campuswire.
Last modified: Monday, April 29 14:41:53 CST 2024