Syllabus for Math 103

This syllabus is tentative and will be updated irregularly. The homework page will be updated on the regular basis.

 

Math 103 will concentrate on Measure Theory and Lebesgue integration with the goal of helping students to prepare for the Analysis certification exam. The exact syllabus will depend on the interests and backgrounds of the students enrolled. We shall start by reviewing Riemann integral and its properties. From there we will cover as much of Chapters 1-7 as time permits. Ideally the students in the class should have had undergraduate classes in abstract analysis.

 

Lectures

Sections in Text

Brief Description

Day 1,  September 22

 1.1-1.3

 Intuitive discussions about measure, sigma-algebras, examples, Borel algebras and its generating sets, tensor products of algebras, measures and their examples

Day 2,  September 27

 1.3-1.4

 Basic properties of measures< completion of a measure, outer measures and Caratheodory’s Theorem

Day 3, September 29

 1.4-1.5

premeasure and its extension, Borel and Lebesgue-Stieltjes measure associated to a function 

Day 4, October 4

 1.5

 Lebesgue measure, Cantor and generalized Cantor sets

Day 5, October 6

 2.1

 Measurable functions and their properties, supremum and infimum of measurable functions, decompositions of measurable functions and representations of them as limits of simple fucntions

Day 6, October 11

 2.2

 Integration of nonnegative functions, Monotone convergence theorem and it corollaries, Fatou’s Lemma

Day 7, October 12

x-hour instead of a class on 10/18

 2.3

 Integration of complex functions, L1-space, Dominated Convergence Theorem, applications to diffenretiation of integral with respect to parameter

Day 8, October 13

 2.3

Riemann integral and its relation to Lbesgue integral  Modes of Convergence, convergence in measure, almost everywhere convergence, L1-Egoroff’s Theorem 

October 18, No class

 

 

Day 9, October 20

 2.5

Rectangles, sections, Monotone class Theorem

Day 10, October 25

 2.5

 Fubini-Tonelli Theorem

Day 11, October 27

TENTATIVELY the take home Midterm exam is given out on this day and it will be due Thursday November 3

2.6 

 n-dimensional Lebesgue integral, Jordan content of a set

Day 12, November 1

 2.6

 Invariance of Lebesgue measure under rotation, behvaiour of Lebesgue integral under diffeomorphisms of Rn

Day 13, November 3

 3.1

 Signed measures, Hahn and Jordan decomposition Theorems

Day 14, November 8

 3.2

 Lebesgue-Radon-Nikodym Theorem

Day 15, November 10

 5.1

 Normed vector spaces, Banach spaces, quotient space and quotient norm, equivalences of norms, operator norm

Day 16, November 15

 5.1-5.2

continuous operators, Banach algebra, sublinear functionals and Hahn-Banach Theorem

x-hour November 16

Oral homework presentations

 5.2-5.3

Comples Hahn-Banch Theorem and its applications, Baire category Theorem 

Day 17, November 17

 5.3

 Open Mapping, Closed Graph Theorems, Uniform Boundness principle

Day 18, November 22

 5.5

 Hilbert Spaces, Schwarz inequality, Pythagorean Theorem, Hilbert spaces are dual to themselves

Day 19, November 29

Tentatively  the take home Final Exam will be distributed on December 1 and it will be due in the evening of December 7

 5.5

 Bessel inequality, existence of orthonormal bases. separable Hilbet spaces and their identification with l2(A).

 

 

Student with disabilities: Students with learning, physical, or psychiatric disabilities enrolled in this course that may need disability-related classroom accommodations are encouraged to make an office appointment to see me before the end of the second week of the term. All discussions will remain confidential, although the Student Disability Services office may be consulted to discuss appropriate implementation of any accommodation requested. At such a meeting please provide me with a copy of a disability registration form, which lists the accommodations recommended for the student by Student Disability Services within the Academic Skills Center http://www.dartmouth.edu/~acskills/disability/index.html The person you might want to contact at the Academic Skills center is the Student Disabilities (504) Coordinator Cathy Trueba 301 Collis Center - (603) 646-2014