# Lectures

Tuesday and Thursdays, 11-12:30 in Room 4-153.

Note that the text, [1], is Rudin's ``Principles of Mathematical Analysis''. The problems are from the 3rd edition.

There are two sections of 18.100B, plus a section of 18.100A. This web page is only for my section. The lecturer for the other section of 18.100B is Professor S. Helgason.

The timetable here should not be completely relied upon! As set out below it will mean that we move at a quite brisk pace. Probably I will have to modify it a bit as I go along. The dates of the tests and on which homework is due will not change but it is possible that the material they cover will change a little.

Lecture 1:
February 3. Where do we start?

Reading:- Rudin Pages 1 - 11.

Problems:- Rudin Chapter 1, Problems 1,3,5.

NOTE: There is a homework due tomorrow. You can in fact get full marks by handing in a page with your name on it. However, try the first three questions so that I can help you with setting out proofs. Our grader has not yet been selected.

First `Proof' - that there is no rational with square 2.

Naive set theory, union and intersection, Cartesian product.

Fields, rational numbers.

Ordering.

Lecture 2:
February 5. The real numbers.

Problems:- Rudin Chapter 1, Problems 8,9,10.

Least upper bound property.

Archimedean property of real numbers.

Euclidean spaces.

Schwarz inequality.

Triangle inequality.

Complex numbers.

Lecture 3:
February 10. Countability.

Problems:- Rudin Chapter 2, Problems 2,3,4.

Maps, surjectivity, injectivity, bijectivity.

Equivalence of sets

Finite, countable, uncountable, at-most-countable and infinite sets.

Countability of the integers (duh).

A countable union of countable sets is countable.

Cartesian product of two countable sets is countable.

Countability of the rationals.

The uncountability of the set of sequences with values in

Amusement for the over-prepared. Prove Sylvester's theorem. Suppose and are non-negative integers. Show that every integer larger than can be expressed as a linear combination with and non-negative integres.

Lecture 4:
February 12. Metric spaces, open sets.

Problems:- Rudin Chapter 2, Problems 9a, 9b, 9c, 11,

Euclidean metric, discrete metric and supremum metric.

Open balls in a metric space.

Open subsets of a metric space.

Unions and finite intersections of open sets are open.

Open balls are open (duh).

Limit points and closed sets.

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

Problems:- Rudin Chapter 2, Problems 10, 22, 23.

Complements of closed sets are open and vice versa.

Closure of a set.

Relatively open subsets.

Compact sets are closed.

Lecture 6:
February 24. Compact sets.

Problems:- Rudin Chapter 2, Problems 12, 16, 25.

Countable intersection property

Infinite subsets of compact sets have limit points

Lecture 7:
February 26. Compact subsets of Euclidean space

Problems:- Rudin Chapter 2, Problems 24, 26, 29.

Compactness of the unit cube.

Heine-Borel theorem

Weierstrass' theorem

Connectedness of sets.

Lecture 8:
March 2. Completeness.

Problems:- Rudin Chapter 2, Problems 19, 20, 21.

Sequential compactness.

Convergence of sequences.

Cauchy sequences.

Completeness.

Completeness of Euclidean spaces.

Lecture 9:
March 4. Sequences and series.

Problems:- Rudin Chapter 3, Problems 2, 7, 12, 16.

Completeness of compact spaces.

Sequential compactness.

Did not do series, root, ratio tests, absolute convergence.

Lecture 10:
March 9. Continuity.

Problems:- Rudin Chapter 4, Problems 1, 4, 15.

Limits of functions at a point.

Continuity of functions at a point.

Continuity of composites.

Continuity of maps.

Lecture 11:
March 11. Continuity and sets.

Problems:- Rudin Chapter 4, Problems 1, 4, 15.

Continuity and open sets.

Continuity and closes sets.

Continutiy and components.

Lecture 12:
March 16. Continuity and compactness.

Problems:-

Continuity and compactness.

A continuous function on a compact set has a maximum

Continuity and connectedness.

A continuous function on an interval takes intermediate values.

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

Lecture 14:
March 30. Differentiability.

Problems:-

Differentiability and the derivative.

Differentiability implies continuity.

Sums and products.

Chain rule

Maxima and minima.

Lecture 15:
April 1. (Professor Helgason will lectures, since I will be away.) Mean value theorem.

Mean value theorems.

Increasing and decreasing functions

l'Hopital's rule.

Higher derivatives

Taylor's theorem.

Lecture 16:
April 6. Riemann-Stieltjes integral defined.

Problems:-

Upper and lower sums.

Upper and lower integrals.

Integrability.

Refinement

Integability criterion.

Lecture 17:
April 8. Integrability of a continuous function.

Continuous functions are Riemann-Stieltjes integrable.

Monotonic functions are R-S integrable w.r.t. continuous length functions.

Finite discontinuities.

Continuous function of R-S integrable function is R-S integrable.

Lecture 18:
April 13. Riemann-Stieltjes integral.

Properties of the integral

Lecture 19:
April 15. Fundamental theorem of calculus.

Integration by parts

FTC version 1

FTC version 2

Lecture 20:
April 22 (April 20 is a holiday). Sequences of functions.

Problems:-

Pointwise convergence of sequences of functions

Uniform convergence

Cauchy criterion

Uniform convergence and continuity

Lecture 21:
April 27. Second in-class test
Lecture 22:

April 29. Uniform convergence.

Problems:-

The metric space of bounded continuous functions on a metric space.

Uniform convergence and integration.

Uniform convergence and differentiation.

Lecture 23:
May 4. Equicontinuity

Equicontinuity and compactness.

Stone-Weierstrass theorem.

Lecture 24:
May 6. Power series

Convergent Taylor series.

Analytic functions.

Lecture 25:
May 11. Fundamental theorem of algebra.

Exponential, logarithm and trigonometric functions.

Fundamental theorem of algebra.

Lecture 26:
May 13.

Final review, with some indications of what more we could have done with a little time. I will give you some idea of the structure of the final examination. Also I will try to give you an idea of the relationship of the material in this course to other mathematics courses.

Final Exam
Is on Thursday May 20, 1:30PM-4:30PM in Walker.

Richard B. Melrose 2004-05-18