Math 510 - Introduction to Analysis I

This is the first graduate course on Analysis. Together with Math 511, it prepares graduate students for the Real Analysis Qualifying Exam.
Topics: Real number fields, sets and mappings. Basic point set topology, sequences, series, convergence issues. Continuous functions, differentiation, Riemann integral. General topology and applications: Weierstrass and Stone-Weierstrass approximation theorems. We will cover most of the material from Chapters 1-14 of Carothers' book.
 
Lectures Time & Location : Tue & Thur 12:30 - 1:45 p.m., SMLC 352
Office hours: Tue and Thur 11:00 - 11:50 a.m., and also by appointment, SMLC 212
Professor Info: Anna Skripka, skripka [at] math [dot] unm [dot] edu
 
Textbook: N. L. Carothers, Real Analysis, Cambridge University Press, 1st edition, 2000, ISBN-10: 0521497566.
Supplementary textbook: W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Science/Engineering/Math, 3d edition, 1976, ISBN-10: 007054235X.
Review of Advanced Calculus: W.R. Wade, An Introduction to Analysis, Prentice Hall; 4th edition, 2009, ISBN-10: 0132296381;
T. Tao, Analysis I, Hindustan Book Agency, 2nd edition, 2009, ISBN-10: 8185931941   related website.
Supplementary video lectures on selected topics (the content is not identical with the content of our course).
 
Prerequisites: proof based advanced calculus (Math 401 at UNM), proof based linear algebra (Math 321 at UNM), solid background in proof writing and mathematical rigor.
Grades: 40% homework, 60% midterm exam + final exam.
Policies: Your work will be graded on the clarity, completeness, correctness of your reasoning and presentation. Submit a neat, legible assignment, provide enough room for grader's comments in the margins, staple the pages of the homework together in the corner. On the homework assignments you are allowed to work in groups, but you must write up your own solutions in your own words. Submitting solutions obtained from third parties, including the internet, is strictly prohibited. No collaboration, consultation, books, notes, phones, computers, internet, etc. are allowed on the exams. You are expected to attend every classroom meeting and come prepared (study the previous lecture material, do exercises, read the textbook on the upcoming topic) as well as hand in a hard copy of your homework on time. No late homework will be accepted. In case of an emergency you are allowed to email a scanned copy of your homework as a SINGLE PDF file (multiple files or other formats will not be accepted) to the professor no later than the due date. A make-up exam will be given for a missed midterm only in case of a documented absence prescribed by the university (family emergency, serious medical problem, official UNM function). All objections to grades should be made within one week of the day the work in question is returned. You are responsible for picking up your graded work if you miss getting it in class.
 
TENTATIVE Schedule (will be updated at the end of each week)
 
Week
Important Dates
Chapters
Topics
HW Assignments
1
Aug 20 (Pretest), 22
Chapter 1
Review of Advanced Calculus.
HW 1
2
Aug 27, 29 (HW1)
Chapters 2, 3
Cardinality. Cantor set. Monotone functions. Metric spaces.
HW 2
3
Sep 3, 5 (HW 2)
Drop with no grade: Sep 6
Chapter 3
Normed spaces. ℓp spaces.
HW 3
4
Sep 10, 12 (HW 3)
Chapters 3, 4
Open and closed sets. Relative metric.
HW 4
5
Sep 17, 19 (HW 4)
Chapters 5, 6
Continuous functions. Connectedness.
HW 5
6
Sep 24, 26 (HW5)
Chapter 7
Completeness. Contraction mapping principle.
HW 6
7
Oct 1, 3
Chapter 8
Compactness. Uniform continuity.
HW 7
8
Oct 8 (HW6)
Chapter 10
Pointwise and uniform convergence.
9
Oct 15, 17 (Exam on Chapters 1-8)
Chapters 1-8
Review exercises.
HW on Chapters 1-8 for grade & practice
10
Oct 22, 24
Chapter 10
Interchanging limits. Series of functions. Space B(X). The Weierstrass M-test.
11
Oct 29, 31 (HW 7)
Chapter 11
Power series. Nowhere differentiable function. Weierstrass approximation theorem.
HW 8, Quiz 1
12
Nov 5, 7 (HW 8)
Withdrawal deadline: Nov 8
Chapter 11
Approximations in C(R). Equicontinuity. Compact subsets of C(X).
HW 9, Quiz 2
13
Nov 12, 14 (HW 9)
Chapter 13
Functions of bounded variation.
HW 10
14
Nov 19, 21 (HW 10)
Chapter 14
Riemann-Stieltjes integral with an increasing integrator.
HW 11, Quiz 3
15
Nov 26
Chapter 14
Properties of Riemann-Stieltjes integrable functions.
Quiz 4
16
Dec 3 (HW 11), 5
Chapter 14
Riemann-Stieltjes integral with a general integrator. Review.
Prepare for Final Exam
17
Final Exam: Dec 12 (Thursday), 10 a.m. - noon
Office hour: Dec 10, 12:30 - 2 pm
 
Academic Integrity: Each student is expected to maintain the highest standards of honesty and integrity in academic and professional matters. The University reserves the right to take disciplinary action, including dismissal, against any student who is found responsible for academic dishonesty. Any student who has been judged to have engaged in academic dishonesty in course work may receive a reduced or failing grade for the work in question and/or for the course. Academic dishonesty includes, but is not limited to, dishonesty on exams or assignments; claiming credit for work not done or done by others (plagiarism); and hindering the academic work of other students.
American Disabilities Act: In accordance with University Policy 2310 and the American Disabilities Act (ADA), reasonable academic accommodations may be made for any qualified student who notifies the instructor of the need for an accommodation. It is imperative that you take the initiative to bring such needs to the instructor's attention, as the instructor is not legally permitted to inquire. The student is responsible for demonstrating the need for an academic adjustment by providing Student Services with complete and appropriate current documentation that establishes the disability, and the need for and appropriateness of the requested adjustment(s). However, students with disabilities are still required to adhere to all University policies, including policies concerning conduct and performance. Contact Accessibility Services at 505-661-4692 for additional information.
Copyright Policy: All materials disseminated in class and on the course webpage are protected by copyright laws and are for personal use of students registered in the class. Redistribution or sale of any of these materials is strictly prohibited.
Disclaimer: The instructor reserves the right to change this syllabus. An up-to-date syllabus is posted on this webpage. It is your responsibility to know and understand the course policies.