# 18.103, Spring 2004

Richard Melrose

Department of Mathematics, Massachusetts Institute of Technology

rbm@math.mit.edu

Tuesday and Thursdays, 1-2:30 in Room 4-102. The book is Adams and Guillemin [1]

Lecture 1:
February 3.

Read A-G, Chapter 1, pages 1-4, 24-26.

Problems, A-G, Chapter 1, p.11, Problems 1, 2; p;39 Problems 1 (first part), 2 Here are the problems

1.1.1. Prove that the set of Bernoulli sequences is uncountable by the Cantor diagonal argument.

1.1.2. a) Let Show that can be written in the form with or Show that this expansion is unique when we restrict to nonterminating series.

b) Show that for any (positive) integer can be written in the form where Show that the expansion is unique when we restrict to nonterminating series.

1.3.1. Let be an uncountable set. Let be the collection of all finite subsets of Given let be the number of elements in Show that is a ring and that is a measure on

1.3.2. Let be an infinite set and let be the collection of sets consisting of the finte sets and the complements, in of finte sets. Let be the function if is finite, if the complement of is finite. Is a measure?

Lecture 2:
February 5.

Read A-G, pp 27-31, pp 4-9

Problems: p.39, nos. 3, 4; p.12 nos. 4,5. Here are the problems

1.3.3. a) Let be an infinte set and the collection of all countable subsets of Is a ring?

b) Let be a measure on Show that there exists a function such that

 (1)

c) Show that the funcion in part b) has the following two properties. (1) the set is countable and (2)

d) Show that if has the properties in part c) the formula (1) defies a measure on

1.3.4. Let be the real line and Given let if, for some positive contains the interval and otherwise Show that is an additive set function but is not countably additive.

1.1.4. Show that

 or

for any sequence When is it

1.1.5. Define the Rademacher functions on the whole real line by requiring them to be periodice of period one - so setting With this definition show that and by induction,

Countable additivity and subadditivity

Outer measure

Sets of measure zero

Measurable sets

Rademacher functions

Lecture 3:
February 10.

(Special) Chebyshev's inequality

Weak law of large numbers

Lecture 4:
February 12.

Homework due Friday 20th: Chapter 1, p. 41, nos 11, 13; p. 42 nos 19, 20.

A-G, pp 33-39.

The -ring of measurable sets.

The law of large numbers.

Lecture 5:
February 19 (Monday schedule on February 17).

A-G pp 53-58.

Measurable functions

Lecture 6:
February 24. The integral

Countable additivity: Theorem 11.

Lecture 7:
February 26. Linearity of the integral.

Proof of Theorem 11 finished.

Monotone convergence.

Additivity for positive functions.

The space of integrable functions (on with respect to the measure on the -ring

Linearity of the integrable on integrable functions.

Lecture 8:
March 2.

A-G pages 77-80.

Fatou's lemma (for a sequence of non-negative measurable functions

Lebesgue's Dominated convergence theorem. Pointwise convergence of a sequence of measurable functions with absolute value bounded by a fixed integrabe function implies convergence of the integrals to the integral of the limit.

The Banach space

Lecture 9:
March 4.

The Banach space continued.

A Hilbert space,

Lecture
. 10:] March 9.

Hilbert spaces

Bessel's inequality

Existence of a complete orthonormal basis in a separable Hilbert space.

Lecture 11:
March 11.

Outline of proof that an function on is determined by its Fourier coefficients.

Equality of Riemann and Lebesgue integrals of a continuous function on

Convergence of the Fourier series of a function a differentiable function.

Lecture 12:
March 16.

Parallelogram law in a Hilbert space; converse explained but not done.

A non-emtpy closed convex set in Hilbert space has a unique point closest to the origin.

Linear functions on a normed space are continuous if and only if they are bounded.

Riesz Representation Theorem:- Any continuous linear functional on a Hilbert space is of the form for some

Lecture 13:
March 18. First in-class test. Covers all material in Lectures 1-10.

Lecture 14:
March 30.

Lecture 15:
April 1. (I will be away and Professor Helgason will lecture)

Schwartz functions

Fourier transform

Inverse transform

Lecture 16:
April 6.

Metric on the space of Schwartz functions

Density of Scwhartz functions in

Extension of Fourier transform to

Lecture 17:
April 8.

Density of Schwartz functions in

Harmonic oscillator, creation and annihilation operators.

Invariance under Fourier transform.

Completeness of eigenfunctions.

Weak solutions.

Lecture 18:
April 13.

Lecture 19:
April 15.

Lecture 20:
April 22 (April 20 is a holiday).

Lecture 21:
April 27. Second in-class test.

Lecture 22:
April 29.

Lecture 23:
May 4.

Lecture 24:
May 6.

Lecture 25:
May 11.

Lecture 26:
May 13.

Richard B. Melrose 2004-05-24