The schedule for topics covered in lecture is as follows. Please do
the assigned reading prior to lecture. This will make the lectures
more effective for you.
August 30th Unit 1. Advanced calculus and differential equations.
Metric spaces, convergence, Cauchy sequences, completeness, the real
number system.
Loomis-Sternberg. 4.1, 4.3, 4.4, 4.7
Folland. 0.6
September 1st
Sequential compactness, the Bolzano-Weierstrass theorem and the Heine-Borel
theorem.
Loomis-Sternberg. 4.4, 4.7
Folland. 0.6
September 6th
Uniform convergency and continuity, equicontinuity, the Arzela-Ascoli
theorem.
Loomis-Sternberg. 4.5, 4.6
Folland. 4.6
September 8th
Banach spaces, linear functionals, the Hahn-Banach theorem.
Loomis-Sternberg. 4.7
Folland. 5.1, 5.2 Problem Set 1 due in lecture.
September 13th
Hilbert spaces, orthogonal projection, the Gram-Schmidt process.
Loomis-Sternberg. 5.1, 5.2
Folland. 5.5
September 15th
Differentiability.
Loomis-Sternberg. 3.5, 3.6, 3.7, 3.8, 3.9 Problem Set 2 due in lecture.
September 20th
Banach's Contraction Mapping Fixed Point Theorem. Newton's method. The
implicit and inverse function theorems.
Loomis-Sternberg. 4.9
September 22nd
Ordinary differential equations and initial value problems. Picard's
theorem on short-time existence and uniqueness.
Loomis-Sternberg. 6.1, 6.2 Problem Set 3 due in lecture.
September 27th MIDTERM 1 IN LECTURE
October 4th
Jordan canonical form, matrix exponentiation.
October 6th
Solutions of constant-coefficient linear ODEs.
Loomis-Sternberg. 6.3, 6.4, 6.5 Problem Set 4 due in lecture.
October 11th Unit 2. Measure theory.
Sigma-algebras.
Folland. 1.1, 1.2
October 13th
Measures.
Folland. 1.3 Problem Set 5 due in lecture.
October 18th
Outer measures.
Folland. 1.4
October 20th
Borel measures on the real line.
Folland. 1.5 Problem Set 6 due in lecture.
October 25th
Measurable functions.
Folland. 2.1
October 27th
Integration of nonnegative functions.
Folland. 2.2 Problem Set 7 due in lecture.
November 1st
Integration of complex functions.
Folland. 2.3
November 3rd MIDTERM 2 IN LECTURE
November 8th
Modes of convergence.
Folland. 2.4
November 10th
Product measures.
Folland. 2.5 Problem Set 8 due in lecture.
November 15th
Lebesgue measure on Euclidean spaces.
Folland. 2.6
November 17th
Signed measures.
Folland. 3.1 Problem Set 9 due in lecture.
November 22nd
The Lebesgue-Radon-Nikodym theorem.
Folland. 3.2
November 29th
Complex measures.
Folland. 3.3
December 1st
Differentiation of measures on Euclidean spaces.
Folland. 3.4 Problem Set 10 due in lecture.
Jason Starr
4-108 Math Tower
Department of Mathematics
Stony Brook University
Stony Brook, NY 11794-3651
Phone: 631-632-8270
Fax: 631-632-7631 Jason Starr