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Real Analysis 8601-8602

Prerequisites for 8601: strong understanding of a year of undergrad real analysis, such as our 5615H-5616H or equivalent, with substantial experience writing proofs . Courses named Advanced Calculus are insufficient preparation. The necessary mathematical background includes careful treatment of limits (of course!), continuity, Riemann integration on Euclidean spaces, basic topology of Euclidean spaces, metric spaces, completeness, uniform continuity, pointwise limits, uniform limits, compactness, and similar.

General comfort with abstraction is a prerequisite.

Substantial experience writing proofs is a prerequisite. Ideally, students coming into this course have acquired a range of experience in proof writing, not only in a previous course in real analysis, but also in previous courses in abstract algebra, rigorous linear algebra, or point-set topology. For that matter, all the latter topics play roles in 8601-02.

Coherent writing, apart from proof-writing itself, is essential. Contrary to some myths, the symbols do not speak for themselves.

Diagnostics: A brief diagnostic questionnaire is available, for self-evaluation, by prospective students, of their readiness for 8601. The meanings of all the questions, and some ideas about the answers, should be very familiar to prospective students already. The conduct of the course will presume so. Also, within the first week or so of the actual course, we will have an in-class diagnostic midterm, a review of the prerequisites. Difficulty in that exam would be a sign that one is not prepared for the course. By that point, it would likely be very difficult to catch up, and it would surely be wise to switch to the 5xxx-level analysis course.

Note: even with a somewhat thin background, systematic study over the summer can greatly improve one's preparedness for this course. However, one should not under-estimate the level of effort required to emulate one or more serious year-long courses in abstract mathematics in a few months over the summer.

Prerequisites for 8602: 8601 or equivalent.

Fall 2022 - Spring 2023

Text will be notes posted here, supplemented by whatever books or other notes you like.

grades will be determined by in-class midterms , scheduled as in the table below. There is no final exam. You are not competing against other students in the course, and I will not post grade distributions. Rather, the grade regimes are roughly 90-100 = A, 75-90 = B, 65-75 = C, etc., with finer gradations of pluses and minuses. So it is possible that everyone gets a "A", or oppositely. That is, there are concrete goals, determined by what essentially all mathematicians need to know, and would be happy to know.

There will be regular homework assignments preparatory to exams, as scheduled below, on which I'll give you feedback about mathematical content and writing style. The homeworks will not contribute to the course grade, and in principle are optional, but it would probably be unwise not to do them and thus fail to get feedback. No late homeworks will be accepted, for logistical reasons. Homework should be typeset, presumably via (La)TeX, and emailed to me as a PDF.

Office hours: Mon 11:15-12:00, Wed 11:15-12:00. Send email anytime!

Exam and homework-example schedule, spring 2023:

Sunday Monday Tuesday Wednesday Thursday Friday Saturday
Jan 18 Jan 20
Jan 23 Jan 25 Jan 26
Jan 30 Feb 01 Feb 03
Feb 06 hmwk 05 Feb 08 Feb 10 exam 05
Feb 13 Feb 15 Feb 17
Feb 20 Feb 22 Feb 24
Feb 27 hmwk 06 Mar 01 Mar 03 exam 06
Mar 06 Mar 08 Mar 10
Mar 13 Mar 15 Mar 17
Mar 20 Mar 22 Mar 24
Mar 27 Mar 29 Mar
Apr 03 hmwk 07 Apr 05 Apr 07 exam 07
Apr 10 Apr 12 Apr 14
Apr 17 Apr 19 Apr 21
Apr 24 hmwk 08 Apr 26 Apr 28 exam 08
May 01 last class

Unless explicitly noted otherwise, everything here, work by Paul Garrett, is licensed under a Creative Commons Attribution 3.0 Unported License. ... [ garrett@umn.edu ]

The University of Minnesota explicitly requires that I state that "The views and opinions expressed in this page are strictly those of the page author. The contents of this page have not been reviewed or approved by the University of Minnesota."