MATH 721 Homepage (Fall 2003)


Lecturer: Alexandru Ionescu, ionescu@math.wisc.edu
Grader: Jernej Tonejc, tonejc@math.wisc.edu

Handouts
Information, also in pdf-file.

Problem Sets
Problem Set 1 (also in pdf-file). Due in class on Thursday, September 18.
Problem Set 2 (also in pdf-file). Due in class on Tuesday, October 7.
Problem Set 3 (also in pdf-file). Due in class on Tuesday, October 21.
Problem Set 4 (also in pdf-file). Due in class on Thursday, November 6.
Problem Set 5 (also in pdf-file). Due in class on Thursday, November 20.
Problem Set 6 (also in pdf-file). Due in class on Tuesday, December 2, or Thursday, December 4.
Problem Set 7 (also in pdf-file). Due on Tuesday, December 16, by 5 P.M.

Solutions to Problem Sets
Solutions to Problem Set 1 (also in pdf-file).
Solutions to Problem Set 2 (also in pdf-file).
Solutions to Problem Set 3 (also in pdf-file).
Solutions to Problem Set 4 (also in pdf-file).

Syllabus
Sep. 2: Definition of sigma-algebras, the sigma algebra generated by a family of sets, Borel sets, positive measures.
Sep. 4: Measurable functions.
Sep. 9: Limits of measurable functions, sums and products of measurable functions, simple functions.
Sep. 11: Integrals of functions, the monotone convergence theorem.
Sep. 16: Fatou's lemma, integration of complex functions, the Lebesgue dominated convergence theorem.
Sep. 18: Metric spaces, open and closed sets, compact sets, continuous functions (review).
Sep. 23: Urysohn's lemma, the partition of unity lemma.
Sep. 25: The Riesz representation theorem.
Sep. 30: The Riesz representation theorem (part 2).
Oct. 2: The Riesz representation theorem (part 3).
Oct. 7: Inner and outer regularity of Borel measures, Euclidean spaces.
Oct. 9: Construction of the Lebesgue measure on Euclidean spaces, uniqueness of measures lemma.
Oct. 14: Uniqueness of measures lemma (part 2), L^p norms, L^p spaces.
Oct. 16: Holder inequality, Minkowski inequality, L^p as a normed space.
Oct. 21: Completeness of the L^p spaces, product measures.
Oct. 23: Fubini's theorem.
Oct. 28: Fubini's theorem (part 2), the Minkowski inequality for integrals.
Oct. 30: Duality of L^p spaces.
Nov. 4: Duality of L^p spaces (part 2).
Nov. 6: The Lebesgue-Radon-Nikodym theorem.
Nov. 11: The Lebesgue-Radon-Nikodym theorem (part 2), weak convergence.
Nov. 13: Separability of L^p( R^n ), the Banach-Alaoglu theorem.
Nov. 18: The Banach-Alaoglu theorem (part 2), density in L^p of C^\infty_0.
Nov. 20: Convolutions, boundedness of operators defined by L^1 kernels.
Nov. 25: Schwartz functions on R^n, distributions.
Nov. 27: No class.
Dec. 2: Examples of distributions, continuous maps on the space of Schwartz functions.
Dec. 4: Operations with Schwartz functions.
Dec. 9: Operations with distributions, supports of distributions, the Fourier inversion formula.
Dec. 11: The Fourier inversion formula (part 2), the Plancherel theorem.