** Course content:** This is a first graduate
course on Analysis. It is also the course that prepares graduate students
for the __Real Analysis Qualifying__.
Students should have at this point enough computational background (calculus of one variable and multivariable calculus -at least
2 and 3 variables), and
I expect most of you to have been exposed at least once to rigorous
epsilon-delta proofs, if not you might consider taking Math 401/501 instead which is being offered at exactly the same time as Math 510.
I expect familiarity with the real an complex numbers.

We will start the
course with a quick overview of the real and complex numbers numbers. We will then review some basic point set topology, metric spaces, compact and connected sets. Next topic are sequences and series, mostly of complex numbers, however the basic concepts of convergence will be described in the more general setting of
metric spaces. We will consider functions defined and with values on arbitrary metric spaces, and we will define limits, continuity, and connections to compactness and connectedness. We will then specialize to real valued functions defined on an interval or the real line. We will study differentiation properties and Riemman-Stieltjes integrals of such functions.
At this point we will be ready to study sequences and series of real-valued functions, and their interplay with integrations and differentiation, here the concept of * uniform convergence* is crucial. Time permitting we will study power series, Fourier series, and the classical Stone-Weierstrass approximation theorem.

**Prerequisites:** Advanced calculus and linear algebra, or permission from the instructor.

**Required Textbook:** * Principles of Mathematical Analysis*
by Walter Rudin, McGraw-Hill Science/Engineering/Math, 3d edition, 1976, ISBN-10: 007054235X.

We will cover chapters 1-7, and time permiting parts of chapter 8.

**Recommended Books:**
There are many books on analysis, some are classical, some present fresher views of the subject.
Reading from more than one source will enhance your learning, and will help you build the big picture. I recommend * Real Analysis * by N.L. Carothers, Cambridge University Press, 1st edition, 2000, ISBN-10: 0521497566, that was used last year and some of you may own (Chapters 1-14).

For a review I highly recommend * Analysis I and II * by Terence Tao, Hindustan Book Agency, 3rd edition, 2016, ISBN-10: 9789380250649 and ISBN-10: 9789380250656 (earlier editions work too), the textbooks I use for Advanced Calculus I and II (Math 401/501 and 402/502) and that some of you are already familiar with because you took the class(es).

** Exams:** There will be two midterms and a final exam.

**
Homework:** Homework problems will be posted in UNM Learn weekly or bi-weekly, and will be
graded and returned to you promptly also via UNM Learn. Please try to meet the deadlines.
Problems from past real Analysis Qualifying exams will be weaved into the homework, hopefully by the end of the
course you will have built a folder with solutions to most of those problems for future reference. This exercise is useful if you do the problems yourself, I encourage you to seek collaboration with fellow classmates where you discuss the problmes and then each one of you writes separately your own rendition of a solution. Your homework will be graded based on the clarity, completeness and correctness. Many problem solutions are available online, I will discourage you to use such recources, the point here is that you struggle and you find your own solutions, this is how you will learn. We will have opportunities to discuss homework problems and give you guidance and hints so you can make progress on your own. By now you should know that to learn mathematics you need to practice, not practicing your mathematical skills is like a violin player not practicing scales or a soccer player not practicing dribbling drills.

** Grades:** The final grade will be determined by
homework
two midterms, and a final exam. Grading policies to be discussed in class.

** Modified Fall 2020 Academic Calendar/Schedule for Main Campus and Branch Campuses:**
UNM modified the Fall calendar to reduce travel and associated public health risks. The traditional "Fall Break" has been replaced with two Break Days: Wednesday, October 7 and Tuesday, November 3. The last day for in-person teaching on campuses is Wednesday, November 25. A remote instruction week is scheduled for Nov. 30 to Dec. 4 and a remote final exams week for Dec. 7-12. Please see registrar. The detailed Fall 2020 course schedule is being constantly updated. The Fall course schedule provides information about course delivery modality with indications about times/locations for face-to-face and/or remote classes. This information is being used to map and plan a socially distant Fall campus experience. This video for students explains how to understand indications on the Fall schedule: vimeo.

** Covid-19 ** We find ourselves in an unprecedented times. You have the ability to prevent the spread of COVID-19 and to preserve the health of fellow students, your instructor, staff and the community by following UNM health protocols. The UNM Provost Administrative Directive on Mandatory Student Face Covering and Symptom Reporting of July 9, 2020 requires that all students on UNM-Main and UNM branch campuses wear face masks in the face-to-face classroom and on campus unless they have a specific mask accommodation (confidentially documented with the Accessibility Resource Center). UNM Provost Administrative Directive is consistent with Governor Lujan Grisham's Public Health Emergency Order, as amended, and the Public Health Order of the New Mexico Health Secretary. It also requires daily participation in symptom screening through covidscreen, which will be sent via UNM e-mail.

**Accomodation Statement:**
Accessibility Services (Mesa Vista Hall 20121, 277-3506) provides academic support to students who have dissabilities. If you think you need alternative accesible formats for undertaking and completing coursework, you should contact this service right away to assure your needs are met in a timely manner. If you need local assistance in contacting Accessibility Services, see the Bachelor and Graduate Programs office.

**Academic Integrity:** The University of New Mexico believes that academic honesty is a foundation principle for personal and academic development. All university policies regarding academic honesty apply to this course. Academic dishonesty includes, but is not limited to, cheating or copying, plagiarism (claiming credit for the words or works of another from any type of source such as print, Internet or electronic database, or failing to cite the source), fabricating information or citations, facilitating acts of academic dishonesty by others, having unauthorized possession of examinations, submitting work of another person or work previously used without informing the instructor, or tampering with the academic work of other students. The University's full statement on academic honesty and the consequences for failure to comply is available in the college catalog and in the Pathfinder.

**Gender Discrimination:** In an effort to meet obligations under Title IX, UNM faculty, Teaching Assistants, and Graduate Assistants are considered "responsible employee" by the Department of Education (see pg 15 - http://www2.ed.gov/about/offices/list/ocr/docs/qa-201404-title-ix.pdf ). This designation requires that any report of gender discrimination which includes sexual harassment, sexual misconduct and sexual violence made to a faculty member, TA, or GA must be reported to the Title IX Coordinator at the Office of Equal Opportunity (oeo.unm.edu). For more information on the campus policy regarding sexual misconduct, see: https://policy.unm.edu/university-policies/2000/2740.html

**Citizenship and/or Immigration Status:** All students are welcome in this class regardless of citizenship, residency, or immigration status. Your professor will respect your privacy if you choose to disclose your status. As for all students in the class, family emergency-related absences are normally excused with reasonable notice to the professor, as noted in the attendance guidelines above. UNM as an institution has made a core commitment to the success of all our students, including members of our undocumented community. The Administration's welcome is found on the website https://undocumented.unm.edu

Return to: Department of Mathematics and Statistics, University of New Mexico

Last updated: July 30, 2020