INTRODUCTION TO HIGHER ANALYSIS: REAL VARIABLES(Math 347) for Fall 1998-99

# INTRODUCTION TO HIGHER ANALYSIS: REAL VARIABLES (Math 347)

## 143 Altgeld Hall, 11, MWF

### Instructor: Mihail Kolountzakis

E-mail: `kolount@math.uiuc.edu`

Text: K.A. Ross, Elementary analysis: the theory of calculus, Springer Verlag.

Material to be covered (approximately): The entire book.

A brief review of the book (from the Mathematical Reviews) This book is intended for the student who has a good, but naive, understanding of elementary calculus and now wishes to gain "a thorough understanding of a few basic concepts in analysis such as continuity, convergence of sequences and series of numbers, and convergence of sequences and series of functions". The basic properties of the real number system are discussed in the first chapter; no proofs are given but the construction from Peano's axioms to the set of real numbers is excellently set out and well motivated. Assuming, then, these basic properties, with the existence of least upper bounds as the completeness axiom, all the basic concepts referred to above are derived and discussed with exemplary thoroughness and precision. A few of the more difficult early proofs are preceded by a "discussion", which makes the short formal proofs easily understood -- a student could actually read this book and comprehend the material{fact} As well as covering the basic concepts, the author also covers decimal representation, Weierstrass' approximation theorem via Bernshtein polynomials, and the Riemann-Stieltjes integral, and there are optional sections generalizing results to metric spaces -- here the Heine-Borel theorem, compact sets, and connected sets are introduced. There are many nontrivial examples and exercises which illuminate and extend the material. The author has tried to write in an informal but precise style, stressing motivation and methods of proof, and, in this reviewer's opinion, has succeeded admirably. --Reviewed by William Eames

Grading policy: There will be two 2-hour-long tests plus the final. Each of the two tests counts for 20% of the grade and the remainder 60% is the final. The homework will not directly contribute to your grade.

Office hours (at 234 Illini Hall): anytime I'm in the office (or anywhere else you find me) or by appointment.

W, Aug 26: Organizational stuff, introduction to mathematical induction as a method of proving propositions with a natural number as a parameter.

F, Aug 28: More examples of the use of mathematical induction. The rational numbers. The square root of 2 is not rational. The rational zeros Theorem.

HWK 1: 1.2, 1.3, 1.6, 1.9, 1.11, 2.5

M, Aug 32: The real numbers. maxima/minima, upper/lower bounds and suprema/infima of sets of real numbers.

HWK 2: 4.1-4.4 (do these together), 4.6, 4.7, 4.8, 4.11, 4.12, 4.14, 4.16

There will be a short (optionally anonymous) quiz on Friday (15-20 min).

W, Sep 2: We mostly talked about hwk problem 4.1. We proved the Archimedean Property and introduced the symbols +infinity and -infinity.

Grader for this course is Dimitris Kalikakis (Illini Hall 223). Turn your hwk to him (optional) no later than a week after it has been assigned (there will be one envelope outside his door for incoming and one for graded homework). I remind here that no grade will be earned directly from hwk and that you're doing it in order to learn, and the purpose of the grading is to show to you if you're doing well or not. So do not expect too accurate a marking of the paper (or maybe there will be no numerical grade assigned at all).

HWK 3: 5.1, 5.2, 5.6 (turn these in with HWK 2).

CORRECTION: (which concerns graduate students only)
I must have been very very confused when I proposed that those of you who want extra credit (to 1 unit) could get it by writing a computer program. What I mean is that what I had in mind was my other (!) class, Math 243, which (since it speaks about geometry in 3-space) is perfectly suitable for such a project. However Math 347 is not at all suitable for this, and I have to retract my offer of having some of you writing a program. The deal should be, either to do some extra reading (which we'll choose together) and then come and explain it to me, or to take an "enhanced" final exam (and the two tests as well) with some extra problems (which will be more difficult than the others).
Each person who wants extra credit should e-mail me with his choice of the two above methods, preferably within a week from today 9/3/98).

F, Sep 4: We spoke about countable and uncountable sets and gave Cantor's diagonal argument to prove that the set of real numbers is not countable.

W, Sep 9: Limits of sequences. We gave the definition an applied it to several concrete cases in which we proved that a certain number is a limit using the definition directly. We also saw several sequences which have no limit. We proved the uniqueness of the limit of a sequence.

HWK 4: 7.1(b), 7.3, 7.4, 7.5 (turn these in by Monday).

F, Sep 11: We more or less covered the material in Section 8, but not the problems therein, over some of which we shall go on Monday. Please, read and try to comprehend this Section by then. As a self-test, try to reproduce the proof of example 5 without looking at the book. Also, a week from today we shall have a short quizz, mainly to test how far along you are in learning how to write a mathematical proof from which nothing essential is missing.

M, Sep 14: Section 9. Some theorems that help us to calculate limits. We started doing the "basic examples" in paragraph 9.7.

HWK 5: 8.1, 8.7, 8.9, 8.10

W, Sep 16: We finished Section 9 (we discussed mostly the concept of convergence to infinity and we did some of the problems in Section 8). We shall have a short quiz on Friday.

F, Sep 18: We discussed in detail the convergence of n^{1/n}. Proved that every monotone sequence has a finite or infinite limit. Had a short quiz. Most common error: when you said that the minimum of a finite set exists, you should have added that it is NOT zero, the reason being that 0 is not in the finite set. Except for roughly 5 people (to whom it will be obvious that it is them I mean) you did quite well on this one.

HWK 6: (Section 9) 4, 5, 8, 9, 11, 12, 14, 18.

M, Sep 21: We defined the liminf and limsup of an arbitrary sequence of real numbers and proved that a sequence is convergent (both infinities included as possible limits) if and only if its liminf is equal to its limsup. Further we defined Cauchy sequences and saw that a sequence is Cauchy if and only if it is convergent to a finite limit.

HWK 7: (Section 10) 1, 6, 7, 8, 9, 10.

W, Sep 23: We proved that every sequence has a monotonic subsequence which converges (Bolzano-Weierstrass), defined what accumulation points are (in your book: subsequential limits) and eventually proved that a seq. converges iff it has only one accumulation point. We also saw that the limits of accumulation points are accumulation points (of the original sequence) themselves.

HWK 8: (Section 11) 1, 2, 3(1st), 4(4th), 8.

F, Sep 25: We defined open and closed sets of real numbers and proved several properties of them. Last thing we did was to prove that a finite intersection of open sets is open and that a set is open if and and only if its complement is closed. For Monday, please write down a proof of this last statement and use it to prove that a finite union of closed sets is closed.

The first exam is tentatively scheduled for Th, Oct 8, sometime after 5pm. It will be two hours long and the material will be eveything that has been covered until then. More details will follow.

M, Sep 28: We went over section 12 and did some problems as well.

HWK 9: (Section 12) 3, 4, 5, 8, 9, 10, 12.

The first exam will be on Thursday, Oct. 8, at 7-9pm in 124 Burrill Hall.

W, Sep 30: We saw the definition of metric spaces and several examples. We proved that the Euclidean distance on R^d is a metric, which involves the use of the Cauchy-Schwartz inequality (we proved that). We defined completeness and proved that R^d is complete and the Bolzano-Weierstrass theorem for R^d.

HWK 10: (Section 13) 1, 7, 9.

F, Oct 2: We defined open and closed sets in a metric space. Also the set of limit points of a set, the interior and the boundary of a set. We proved that a decreasing sequence of bounded and closed sets in R^n has non-empty intersection.

HWK 11: (Section 13) 10, 11, 12, 13, 14.

M, Oct 5: We finished section 13 with a discussion of compactness in general metric spaces and in the Euclidean space. The test on Thursday will cover up to section 13.

W, Oct 7: We talked about the convergence of series, absolute convergence, the ratio test and the root test and their relative strength. Finally we say some series which converge but not abolutely.

HWK 12: (Section 14) 1, 2, 3, 5, 7, 8, 12, 14.

F, Oct 9: We talked about the integral test for series as well as about alternating series.

HWK 13: (Section 15) 1, 2, 3, 4, 6, 7.

Grades for the first test are here.

M, Oct 12: Skipped section 16 (decimal expansion). Continuous functions from one metric space to another. In particular real valued functions of a real argument.

HWK 14: (Section 17) Problems 6--14.

W, Oct 14: We discussed some properties of continuous functions, such as that they are always bounded on compact sets and that they assume their sup and inf therein, and the intermediate value theorem and several consequences of it.

HWK 15: (Section 18) 4, 5, 7, 8, 9, 12.

F, Oct 16: We introduced the concept of uniform continuity, proved that every continuous function on a compact set is uniformly continous and used this to define the (Riemann) integral of a continuous function defined on a closed interval.

M, Oct 19: We proved that a function continuous in an open interval is uniformly continuous if and only if it can be extended continuously at the endpoints. We also show a new equivalent defintion of continuity for functions from a metric space to another (preimages of all open sets must be open).

HWK 16: (Section 19) 1, 4, 5, 6, 7, 11, (Section 20) 11, 14, 18.

W, Oct 21: We finished section 21. We saw that for a continuous function from a metric space to another the preimages of open (resp. closed) sets are open (resp. closed), and that the images of compact sets are compact. We also saw several counterexamples to several similar statements.
We shall have a short quiz on Friday.

HWK 17: (Section 21) 1, 2, 4, 5, 8, 9, 10, 11.

F, Oct 23: We discussed some homework problems and had a short quiz.

M, Oct 26: Pointwise convergence of a sequence of functions to another and how several properties of the functions in the sequence fail to propagate to the limit function. Power series and their radius of convergence. Sveral examples. Deifinition of uniform convergence of a sequence of functions to another. The metric space C[0,1] with the "sup" distance function that gives uniform convergence.

HWK 18: (Section 23) 1, 2, 6, 7, 8, 9.

W, Oct 28: We proved that a uniform limit of continuous functions is continuous. We also saw several examples and counterexamples.

HWK 19: (Section 24) 1-5, 10, 11, 14, 17.

F, Oct 30: Uniform covergence on a finite interval implies convergence of the integrals to the integral of the limit function. We saw that this is not true for unbounded intervals. Defined what it means for a sequence of functions to be uniformly Cauchy and saw that this is equivalent to the sequence possesing a uniform limit. Proved the Weierstrass M-test which helps us prove that some series of functions converge uniformly.

HWK 20: (Section 25) 2, 3, 5, 6, 12, 13, 14.

M, Nov 2: We proved that power series can be differentiated or integrated termwise, in their interval of convergence.

For Graduate students only: For those of you who want to receive the extra credit for the course your task will be the follwing. I am going to put on reserve in the Math Library the book "Introduction to topology and modern analysis", by G. F. Simmons. You should read from there Chapter Two (Metric Spaces). You should read all the material and do all problems (or, at least, all but very few). You will then be examined (either orally or in writing, and not before December the 1st) on your understanding of the material (which includes the ability to solve problems).
Those of you who are going ahead with this project (i.e. those of you who want the extra credit), should let me know by the end of the week (11/6/98) by e-mail that you are going to do so.

The 2nd test will take place on Nov 19, 7-9pm, at 112 Chem Annex. The material to be examined is everything that I shall have tought by the previous Monday.

W, Nov 4: We proved Abel's theorem and discussed Taylor series expansion of a function. Also stated the Weierstrass approximation theorem (did not prove it) and saw several ways that assumptions it makes cannot be relaxed.

HWK 21: (Section 26) 2-7.

F, Nov 6: The derivative of a function at a point. Rules of differentiation and their proof.

NO CLASS for the week of the 9th of November. We resume on Monday the 16th.

M, Nov 16: We covered the first 4 problems from the sample exam that I gave you 10 days ago.

EXTRA CLASS on T, Nov 17, at 7pm in 145 Altgeld.

W, Nov 18: The mean value theorem and its applications.

HWK 22: (Section 29) 1, 3, 5, 9, 13, 14, 16, 18.

The 2nd test will take place on Nov 19, 7-9pm, at 112 Chem Annex. The material to be examined is everything that I shall have tought by the previous Monday.

F, Nov 20: We went over the problems that were on the test of yesterday.

NO CLASS on Monday, Nov 30.

2nd test grades are here. The maximum grade is 45 (problem 2b was not taken into account).

W, Dec 2: We started section 32 (the definition of the Riemann integral of a function). We defined the lower and upper sums corresponding to a certain partition of the interval [a,b], and the lower and upper integrals of a bounded function on [a,b]. If these are the same the function is called Riemann integrable.

Extra class on Tuesday and Wednesday, Dec. 8 and 9, in Altgeld Hall 145 at 7pm. Please come with questions.

F, Dec 4: We completed section 32 (the Darboux and Riemann integral) and showed that the two definitions of integrability are equivalent.

HWK 23: (Section 32) 2, 3, 7, 8.

The Final Exam will take place on Wednesday, December 16, 8-11am, in the same room where the class is being tought.

M, Dec 7: We proved that monotonic and continuous functions are integrable, that sums of integrable functions are integrable, the trinagle inequality for integrals, and more.

HWK 24: (Section 33) 3, 4, 5, 7, 8, 9, 10, 11, 13, 14.

Extra class on Tuesday and Wednesday, Dec. 8 and 9, in Altgeld Hall 145 at 7pm. Please come with questions.

W, Dec 9: The Fundamental Theorem of Calculus.

HWK 25: (Section 34) 2, 3, 5, 6, 10, 12.

Your final grade will be computed as follows. It will be the maximum of your final exam and 0.6*final + 0.2*(first test) + 0.2*(second test).