MATH 420 - REAL ANALYSIS
TR 4:00 p.m. - 5:50 p.m.
Instructor: Dr. Milan Luki´c
Office: MC 521
Office Hours: MF 10:00-10:50, TW 2:00-3:00 or by appointment
Phone: (608) 796-3659 (Office); 787-5464 (Home)
Course Description. (from the catalog)
Study of selected topics from real variable theory such as: real numbers; topology of the real line; metric spaces; Euclidean spaces; continuity; differentiation; the Riemann-Stieltjes integral; series.
Prerequisite: grades of C or higher in MATH 260 and MATH 320.
Text Elementary Real and Complex Analysis by Georgi E. Shilov, Dover Publications,
Inc, New York 1973.
The content. Analysis is about the concepts of function, derivative, and integral (quoted from the preface for the Shilov’s book). As a starting point in this study we will look into the structure of real line and explore some of the axioms of the real number system.
The primary focus in the initial part of the course will be on the Completeness Axiom and various equivalent forms of it. We will study in detail the concepts of limit, derivative, integral , and the series. Although you have already study all those concepts in your Calculus courses, I am sure that you will realize that there is a lot of room to improve our basic understanding of those concepts. Real Analysis is one of the key courses in the foundations of mathematics, and most of the time in this course will be spent into looking back into the foundations of Calculus. However, we would also like to use our improved understanding of
Calculus to move toward the greater level of abstractions, and to learn something about some of the more “modern” developments of Analysis. In particular, we will study the concept of Metric Spaces a little bit.
1. Program Assessment - Course Objectives Here is a concise list of course goals following the format of Math Department
Assessment Plan. In Section
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Course-level Outcomes. Students shall study foundations of the real number system, limits, continuity, derivative and integral.
Content: Axioms of real line, limits, derivative, Riemann integral. Infinite sequences and series, uniform convergence.
Mathematical Reasoning: Students will try to achieve an in-depth understanding of the ideas presented. This includes making sure statements and claims made are checked for accuracy, and an appropriate justification is given.
Problem Solving: A number of calculus problems will be revisited, and many new solved. This course will challenge and develop students’ problems solving skills to the limit.
Communication: The main mode of communication in this course will be written (homework, exams). However, in-class participation (oral communication) is essential, as well.
Technology: Use of a Computer Algebra System (Derive, Maxima) and
2. Course Philosophy and Procedure
In order to succeed in this course, you should really immerse yourself totally in doing mathematics. The key strategy in solving problems is “not to give up”. This course is truly a problem solving course.
Most of the course time will be spent on limits, derivatives, integrals, ... . You will see again some of the stuff that you are familiar with, but sooner or later you will stumble over some basic stuff that you actually do not know very well.
You should certainly build up on your strengths (the stuff you more or less know), but make sure that you do not neglect working on the weaknesses, as well. Use homework and exams as directions, but you should really guide yourself in the work of filling in the gaps in your knowledge in order to be able to meet the goals set up by the homework and exam problems.
Mathematics is not a spectator sport. It is learned by doing. Viterbo University is striving to be a Learner-centered institution. That entails an expectation of maturity and taking responsibility for their learning on the part of students. I see my job as one helping you succeed in this learning process.
In spite of my best efforts, I may not always manage to say things the way which best leads to your full comprehension. You can also help me by providing as much of a feedback as you can. I will try to do a formal evaluation survey around the middle of the semester. Other than that, I find the questions in class, and especially when someone comes to my office for assistance, very helpful.
As a further assistance to you:
• About a week prior to any exam, you will receive a practice exam which will be, in terms of format and type of problems, very much like the actual exam.
• I am asking you to keep a The Learner’s Journal. This is to be a separate notebook that should contain a record of your study/practice on daily basis.
I would also like you to keep a time log - date, hour from-to - for each study session. I would prefer that you use a pen for writing in that journal. If you are going to use pencil, then please do not use erasers, and in any case, do not tear pages out. For a learning to take place, you have to try to do
MATH 420 - REAL ANALYSIS SPRING 2005 3 something. In trying, you are likely to make mistakes. The real learning will start taking place once you start understanding and correcting your mistakes.
You turn that journal in together with your exam, and then you will be graded for the portion of that journal that covers the period preceding that current exam. Up to 30% of the exam score is possible to earn this way.
The elements that will play the key role in grading the journal are
– Organization - readability: In order to evaluate, I have to able to read it first. I should not have a difficult time navigating through those notes.
– Mathematical correctness.
– The quality of the work and the amount of time spent on studying.
• Take-home problems: These assignments should test/help a better integration of material. Some will include more difficult problems. One of those assignments will be a group HW. In general, I encourage you to find some time to study together, but unless stated otherwise, the HW is to be written up on your own.
I will try to space those assignments so that you could have some time to catch up. This should leave significant room for exploring the book on your own, and I encourage you to find your own balance between solving some problems in full, and just sketching solutions to some. You should try to read, meaning to the point where you really understand the question, most of the book problems.
The work in class, your book, HW, and practice exams should give you a pretty clear idea what is that you are expected to learn. It is your job to, perhaps through trial and error, find learning strategies that work best for you. Remember, learning is something you do, rather than something I do to you.
Help. I am used to you asking questions in class, coming to my office, working in groups, asking questions by e-mail. Hope all of that will continue.
There is a growing mount of Internet resources, too. Just go to Yahoo, or Google and search for “real analysis”. In particular, you may want to look at
http://www.shu.edu/projects/real It is an interactive Real Analysis textbook. Grading. Approximate distribution: Homework 25%, Midterm 20%, class participation 5%, journal 15%, and the final exam 35% of the grade.
Some of the HW will involve some use of technology (CAS and LATEX). The details will be given later.
One of the writing assignments will be graded in two parts - the second part will require you to come to my office and explain your reasoning, answer some questions.
In all your work, written and oral, it is essential to provide explanations, justify your reasoning.
My grading scale is
A=80%, B=60% , C=40%, D > 30% .
The following exceptions to that scale are possible:
• An A on the final exam (more than 180/200 points) will mean an A for the final grade as well.
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• If one is passing the course by the time of the final exam, but earns less than 30% points (a score less than 60/200), that will result in an F for the final grade.
3. Some details and examples
Specific Course Goals. To study rigorous foundations of calculus; extend basic knowledge of functions, limits, derivatives and integrals. Students are expected to learn to state definitions and theorems precisely, and be able to prove theorems stated. In particular, the proofs involving the concept of limit are going to be of central importance.
The process of working toward those goals will involve looking back into everything you have learned in mathematics so far, and to subject those concepts to the following key questions:
• What do I really know about ...?
• What does ...mean?
• Is what I just said about ...true?
• How do I know if (why) it is true?
Let me try to clarify this a little bit by looking into an example.
Example: For some Math 155 students, conquering a problem such as
p8 + p32 −
p18 = 3p2
represents a major undertaking; something they consider worth of including into their portfolio for the semester.
Now, when you look at what is involved into justifying the above result, a number of problems present themselves. For example, we have to use the rule
pAB = pApB , A,B 0.
Can you prove that rule?
Another key rule here is
AC + BC = (A + B)C .
How about proving this one It is much more difficult question than one before.
In fact, it is so basic that there is nothing more basic to use as a help in proving it. So, we have to accept that rule as an axiom. Note: I would like to encourage you to read [4, Chapter 4] here!
Moreover, the rules are used by applying them to “existing” mathematical objects (in this case, real numbers). But, what do we mean by p2. Can you prove that such a number exists? What do we even mean by asking a question like that?
The goal of this course is not only to learn how to answer the questions as those (that is how to do certain rigorous proofs), but we would also like to develop the corresponding mathematical awareness, so that we do not overlook those “simple” questions when solving problems.
Ultimately, this kind of training leads to deepening our understanding of what mathematics is about. We would like to use that improved understanding to bring about other two basic goals of this course, which are:
• Improve our overall mathematical skills, in particular, the skills in handling Algebra, Trigonometry, and Calculus problems.
MATH 420 - REAL ANALYSIS SPRING 2005 5
• Move beyond the concepts we are familiar with. We will study the concept of metric spaces.
Fall 1999 - Final. To illustrate the goal of “improving Calculus skills”, let’s consider the final exam I gave to the Real Analysis class in Fall 1999.
Problem 1. • State the definitions of a lower/upper bound of a set of real numbers.
• State the definition of infimum.
• State the theorem about existence of an infimum of a set bounded below.
• Extra credit: Should I say “the infimum”?
• Prove that the set
S = 3,
4, . . . has an infimum, and find that infimum.
• State the Completeness Axiom.
• Prove that the Completeness Axiom is equivalent to the Infimum Theorem
Problem 2. Define
n2 + 1
n! xn .
(a) Use the ratio test to show that f is defined for all real x.
(b) Prove that f is continuous at x = 0.
Problem 3. State the definition of the derivative f0(a) of a function f at the point a of its domain.
Use that definition to find the derivative of f(x) = 3x at an arbitrary point of
Problem 4. Find,
sin (ln (1 + x))
ln (1 + sin (x)) .
State and prove all the rules used in the process. In the case of L’Hospital’s rule, just state it. The proof would be an extra credit.
Problem 5. Probably most of the functions we have encountered so far would be continuously differentiable. That is, if f is differentiable at x = a, then f0 is a continuous function at
x = a.
Is this statement a theorem?
Hint: Show that the function
f(x) = (x2 sin 1
x , if x 6= 0
0, if x = 0
is a counterexample.
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A student that has passed a Calculus sequence should have no difficulty in understanding almost all of the questions on that final. Being able to answer them completely is a different matter.
Americans with Disability Act. If you are a person with a disability and require any auxiliary aids, services or other accommodations for this class, please see me and Wayne Wojciechowski in Murphy Center Room 320 (796-3085) within ten days to discuss your accommodation needs.
 Leonhard Euler, Introductio in Analysin Infinitorum Book I, 1748. Springre-Verlag (out of print). See UWL library.
 Leonhard Euler, Institutiones Calculi Differentialis. English translation: Foundations of Differential Calculus, Springer 2000.
 David Bressoud, Radical Approach to Real Analysis, Mathematical Association of America,
 Barry Mazur, Imagining Numbers, in particular the p−15, Picador, 2003.
Classes begin: January 17, 2005.
Midterm break: March 5-13.
Easter Break: March 24 − 28.
Last day of class: Friday, May 6.
No class: -
• Friday, April 15; due to my absence - attending a conference.
Final Exam: Tuesday May 10, 5:45-7:45
This syllabus is tentative and may be adjusted during the semester.
Have a good semester !