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.
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.
Reading:- Rudin Pages 11-17.
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.
Reading:- Rudin Pages 24-30.
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.
Reading:- Rudin Pages 31-35.
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.
Reading:- Rudin pages 34-36.
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.
Reading:- Rudin Pages 36-38.
Problems:- Rudin Chapter 2, Problems 12, 16, 25.
Countable intersection property
Infinite subsets of compact sets have limit points
Reading:- Rudin Pages 38-40.
Problems:- Rudin Chapter 2, Problems 24, 26, 29.
Compactness of the unit cube.
Heine-Borel theorem
Weierstrass' theorem
Connectedness of sets.
Reading:- Rudin Pages 42-43, 47-55.
Problems:- Rudin Chapter 2, Problems 19, 20, 21.
Sequential compactness.
Convergence of sequences.
Cauchy sequences.
Completeness.
Completeness of Euclidean spaces.
Reading:- Rudin Pages 55-69, 71-75.
Problems:- Rudin Chapter 3, Problems 2, 7, 12, 16.
Completeness of compact spaces.
Sequential compactness.
Did not do series, root, ratio tests, absolute convergence.
Problems:- Rudin Chapter 4, Problems 1, 4, 15.
Reading:- Rudin pages 83-86.
Limits of functions at a point.
Continuity of functions at a point.
Continuity of composites.
Continuity of maps.
Reading:- Rudin pages 85-93.
Problems:- Rudin Chapter 4, Problems 1, 4, 15.
Continuity and open sets.
Continuity and closes sets.
Continutiy and components.
Reading:- Rudin pages 89-93.
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.
Reading:- Rudin pages 103-107.
Problems:-
Differentiability and the derivative.
Differentiability implies continuity.
Sums and products.
Chain rule
Maxima and minima.
Reading:- Rudin pages 107-110.
Mean value theorems.
Increasing and decreasing functions
l'Hopital's rule.
Higher derivatives
Taylor's theorem.
Reading:- Rudin pages 120-124.
Problems:-
Upper and lower sums.
Upper and lower integrals.
Integrability.
Refinement
Integability criterion.
Reading:- Rudin pages 124-127.
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.
Reading:- Rudin pages 128-133.
Properties of the integral
Reading:- Rudin pages 133-136.
Integration by parts
FTC version 1
FTC version 2
Reading:- Rudin pages 143-151
Problems:-
Pointwise convergence of sequences of functions
Uniform convergence
Cauchy criterion
Uniform convergence and continuity
April 29. Uniform convergence.
Reading:- Rudin pages 150-154.
Problems:-
The metric space of bounded continuous functions on a metric space.
Uniform convergence and integration.
Uniform convergence and differentiation.
Reading:- Rudin pages 154-161.
Equicontinuity and compactness.
Stone-Weierstrass theorem.
Reading:- Rudin pages 172-180
Convergent Taylor series.
Analytic functions.
Reading:- Rudin pages 180-185.
Exponential, logarithm and trigonometric functions.
Fundamental theorem of algebra.
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.
Richard B. Melrose 2004-05-18