MATH 721 Homepage (Fall 2009)


Lecturer: Alexandru Ionescu, ionescu@math.wisc.edu

Handouts

Information
Problem Set 1. Due in class on Tuesday, Sep. 22. Graded exercises: 1, 2, 3, 5
Problem Set 2. Due in class on Tuesday, Oct. 6. Graded exercises: 2, 3, 4, 6
Problem Set 3. Due in class on Tuesday, Oct. 20. Graded exercises: 2, 3, 4, 5
Problem Set 4. Due in class on Thursday, Oct. 29. Graded exercises: 1, 2, 3, 6
Problem Set 5. Due in class on Thursday, Nov. 12

Syllabus

Sep. 03: Introduction, Riemann integration, measurability.
Sep. 08: The extended real number system, measurable functions.
Sep. 15: Sequences of measurable functions, positive measures.
Sep. 17: Integration of positive functions, the Lebesgue Monotone Convergence Theorem.
Sep. 22: Fatou's Lemma, integration of complex-valued functions.
Sep. 24: The Lebesgue Dominated Convergence Theorem, sets of measure 0.
Sep. 29: Linear functionals, Hausdorff locally compact topological spaces.
Oct. 01: Urysohn's Lemma, the Riesz Representation Theorem.
Oct. 06: The Lebesgue measure on R.
Oct. 08: Proof of the Riesz Representation Theorem.
Oct. 13: Regularity properties of Borel measures.
Oct. 15: Continuity properties of measurable functions.
Oct. 20: Convex functions, Jensen's inequality, Holder's inequality, Minkowski's inequality.
Oct. 22: Definitions of $L^p$ spaces, completeness of $L^p$ spaces.
Oct. 27: Dense subspaces of $L^p$ spaces, linear functionals.
Oct. 29: Weak convergence in $L^p$, duality of $L^p$ spaces, the Banach-Alaoglu theorem.
Nov. 03: Hilbert spaces: basic definitions and properties, orthogonal decompositions.
Nov. 05: Orthonormal sets in Hilbert spaces, the Bessel identity.