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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

MWF, 10:10-11:00, Vincent 1. 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 get feedback. No late homeworks will be accepted. 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!