Department of Mathematics, Massachusetts Institute of Technology
Tuesday and Thursdays, 1-2:30 in Room 4-102. The book is Adams and Guillemin 
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?
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
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
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
Sets of measure zero
(Special) Chebyshev's inequality
Weak law of large numbers
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.
A-G pp 53-58.
Countable additivity: Theorem 11.
Proof of Theorem 11 finished.
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.
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
The Banach space continued.
A Hilbert space,
Existence of a complete orthonormal basis in a separable Hilbert space.
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.
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
Metric on the space of Schwartz functions
Density of Scwhartz functions in
Extension of Fourier transform to
Density of Schwartz functions in
Harmonic oscillator, creation and annihilation operators.
Invariance under Fourier transform.
Completeness of eigenfunctions.