Goals: Develop a rigorous understanding of the real number line, convergence in metric spaces, the formal basis of calculus (review), and uniform convergence. Develop a beginning understanding of function spaces and the measure of subsets of the reals as preparation for a rigorous course in Lebesgue measure theory and integration as well as preparation for applied courses in approximation theory and physics courses that make use of function spaces. Students will further improve their theorem proving abilities. All proofs are to be written in grammatically correct sentences. Textbook: Real Mathematical Analysis, 2nd Edition, by Charles Pugh, Springer. You can get it from: Amazon, Springer, Powell's, etc. The textbook is meant to be read carefully. You cannot just skim through for the important parts like with many lower level textbooks. You should read all the exercises not just the assigned ones. This is holds for most any upper level textbook. Grading: Weekly homework, a midterm, and a final, will each count as one third of the final grade. Homework can be emailed to the instructor. The midterm and final exams will be given in the evening on campus. Proctoring arrangements will need to be made for students too far from campus to travel for the exams. Warning: We know solutions to textbook assignments can be found on line. Do not use them. Do not look at them. The point is to learn how to think for yourself. Your grade will be dropped one letter grade for each offense. Syllabus: Most of Chapter 1 you should know already. We will cover Section 2 carefully. We will cover all of Chapter 2 except the last section, and all of Chapters 3 and 4. We will cover parts of Chapter 6 as time permits. Prerequisites: Students should have had at least two semesters of calculus and a proof based course in the theory of calculus. At SIU Carbondale the respective course numbers are MATH 150, 250 and 352. The Illinois Articulation Initiative numbers for the first two are MTH 901 and 902. Hopefully you saved your textbooks and will review this material before starting this course. The first homework assignment is due the first day of class and is intended to insure you have done so. For additional help reviewing see: Get Ready for Math Grad School. Handouts and Supplemental Reading
Supported in part by a grant from the Illinois Board of Higher Education Copyright, Michael C. Sullivan, 2017 |