Math 4332, Section 12670 (Math 6313, Section 14673) --- BLECHER

Spring Semester 2015

The Course: Introduction to Real Analysis

THIS PAGE CONTAINS ERRORS AND MAY BE MISSING SOME IMPORTANT FACTS, SO WILL CHANGE

Text: No text is required since we will be using notes provided by instructor, however some recommended books are: Tom Apostol, Mathematical Analysis, 2nd Ed., Addison Wesley. W. Rudin, Principles of Mathematical Analysis, McGraw Hill. Kenneth A. Ross, Elementary Analysis: The Theory of Calculus, Springer-Verlag. T. Tao, Analysis I, II.

Time and Place: TuTh 10-11:30am in MH 127.

Instructor: Dr. David Blecher (Email: dblecher@math.uh.edu, PGH 622, Phone 713-743-3451) Be sure to write Real Analysis or 4332 in the email title, else I probably will not open it

Communications: Important information re the course or tests will be given in class, or by email sent to your official email address in the uh database (if you are not receiving these emails tell me immediately). If you email me, be sure to write Real Analysis or 4332 in the email title, else I probably will not open it.

Office Hours: in PGH 622 TuTh 11:40am-12:45 (this may change so watch this spot) (or call 743-3451 for appointment).

Prerequisites: Math 4331.

TA's/Graders: : Ricky Ng (Office hours Tu3-4 in PGH 626, CASA hours MW1-3pm). His email is rickula@math.uh.edu

Final exam: The final exam is scheduled for the last day (or days) of class, April 30 (and possibly also April 28). The date may possibly change to Thurs May 14, 11-2pm, in class, but right now this is looking unlikely since I was asked by the College to avoid this date.

Last day to drop a class or withdraw without receiving a grade (W deadline) : April 6.

Last day to drop a class with some refund and without hours counting: towards the Enrollment Cap for Texas Residents. : Wednesday February 4.

First day of class : Tuesday January 20.

Holidays : Spring break holiday. March 16-21.

Last day to add a class: Tuesday January 27.

Last day of class : Monday May 4.

Course Description:: In this course we continue to develop the theory underlying calculus, and some other important aspects of mathematical analysis. Much emphasis will be placed on rigorous proofs and the techniques of mathematical analysis. You will continue to learn how to read and write proofs, and doing this will constiture a large part of your grade (like in the prequel Math 4331). The tests and exam will be based on the notes given in class, and on the homework. The homework assignment for each chapter is fairly lengthy, and you should attempt all problems. However, a smaller number of problems will be deemed `central', and you are required to turn in some of these for grading. There will be three tests during the semester and a comprehensive final exam at the end of the semester. There are no makeups, thus leave your home several hours in advance of a test so that there is time to eg. catch a taxi in case of unexpected car trouble. If you are sick etc I will need a doctors note for my records. The three one-hour exams exams will be administered in class. There will also usually be weekly quizzes administered in class, of approximately 10 minutes in duration.

Course Outline: :

Chapter I: Infinite series of real numbers. Various tests for convergence. Double series.

Chapter II: Sequences and series of functions. Uniform convergence. Weierstrass M-test. Connection with integration and differentiation. Power series. Taylor series. Elementary functions. Weierstrass approximation theorem.

Chapter III: Fourier Series. Convergence in mean. Pointwise convergence of Fourier series. Fejers theorem.

Chapter IV: Multivariable differential calculus. Differentiability. The inverse and implicit function theorems.

You are encouraged to work with others, form study groups, and so on. However you should not simply copy homework; your homework should be written in your own words, as your understanding of the solution. Thus no two students work should look the same, since student A's understanding of the logic or flow of the argument will be different to student B's. Homework graders will be looking for academic dishonesty (cheating on your homework, e.g. misrepresenting the work of others as your own (plagiarism)), and if found the entire assignment will receive a (nondroppable) zero. Please read the sections of the student handbook discussing academic dishonesty and the disciplinary actions it entails. There will be a mock exam for each test. Please bring comments or complaints to my attention as soon as possible. Don't wait until the end of the semester to bring up a matter which we could deal with and solve early on.

ADVICE FOR STUDYING FOR TESTS:

1) Put in the time. 2) Read thru the classnotes several times making sure you understand everything, and how everything fits together. Memorize the definitions, statements of main facts/theorems, etc. 3) Make a list of things you don't understand and ask me or your TA. 4) Go carefully thru old tests and quizzes, and the keys, making sure you learn from your mistakes. 5) Go thru the homeworks, and do plenty more. 6) Do the mock exams, but keep one to do under test conditions as a reality check. 7) Keep checking my website for new postings if any.

Assessment

   300 points   determined by semester tests 1, 2, 3 (100 points each);
   100 points   determined by in-class quizzes and assigned homework
   200 points   determined by the final exam.
   600 points total
The instructor may drop the lowest of each type of grade (e.g.\ lowest test, lowest few homeworks, etc) for students who have not been showing gross irresponsibility in the class. By `gross irresponsibility' I mean for example who have quit turning in homeworks or taking quizzes). The instructor may change the `formula' for the grades above at his discretion if doing so will benefit the class as a whole.

Attendance and other course policies

Failure to attend class regularly without excuse is grounds for dismissal from the course. Coming to class you will hear a lot of math `culture', how we think about beginning certain problems, how to think through computations, how to express your answers, and so much more. If you do not come to class you are missing out on a lot of very important conscious and subconscious learning and culture. In addition, not coming to class is usually a slippery slope that the student soon falls off of. Classroom behavior: Arrive on time, ready to pay attention. Turn off your cellphone. Do not play with your phones, surf the web, talk to your neighbor, or be disruptive. If you must leave early be sure to sit at the rear of the class (if there is an exit at the rear, else sit by the exit), so as to not appear rude and disturb the class atmosphere when you leave. Except for the start and end of class, please restrict your questions in class to things relevant to what the instructor is discussing, to avoid confusing all the other students. Specific policies concerning attendance and any other policy issues that may arise will be communicated to you by your instructor. Whenever possible, and in accordance with 504/ADA guidelines, we will attempt to provide reasonable academic accommodations to students who request and require them.

Incompletes: only given to students with at least a C average who are unable to take the Final for unforseeable, unpreventable, documented circumstances.

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