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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.
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!
Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
Sept 07 | Sept 09 | |||||
Sept 12 diagnostic exam | Sept 14 | Sept 16 | ||||
Sept 19 | Sept 21 | Sept 23 | ||||
Sept 26 hmwk 1 | Sept 28 | Sept 30 exam 1 | ||||
Oct 03 | Oct 05 | Oct 07 | ||||
Oct 10 | Oct 12 | Oct 13 | ||||
Oct 17 exam 2 | Oct 19 | Oct 21 exam 2 | ||||
Oct 24 | Oct 26 | Oct 28 | ||||
Oct 31 | Nov 02 | Nov 04 | ||||
Nov 07 hmwk 3 | Nov 09 | Nov 11 exam 3 | ||||
Nov 14 | Nov 16 | Nov 18 | ||||
Nov 21 | Nov 23 | Thanksgiving | Nov 25 | |||
Nov 28 | Nov 30 hmwk 04 | Dec 02 | ||||
Dec 05 hmwk 4 | Dec 07 | Dec 09 Exam 4 | ||||
Dec 12 | Dec 14 last class |
Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
Jan 22 | Jan 24 | |||||
Jan 27 | Jan 29 | Jan 31 | ||||
Feb 03 | Feb 05 | Feb 07 hmwk 05 due | ||||
Feb 10 | Feb 12 | Feb 14 exam 05 | ||||
Feb 17 | Feb 19 | Feb 21 | ||||
Feb 24 | Feb 26 | Feb 28 hmwk 06 due | ||||
Mar 02 | Nar 04 | Mar 06 exam 06 | ||||
Mar 09 | Mar 11 | Mar 13 | ||||
Mar 16 | Mar 18 | Mar 20 | ||||
Mar 23 | Mar 25 | Mar 27 | ||||
Mar 30 | Apr 01 | Apr 03 hmwk 07 due | ||||
Apr 06 | Apr 08 | Apr 10 exam 07 | ||||
Apr 13 | Apr 15 | Apr 17 | ||||
Apr 20 | Apr 22 | Apr 24 hmwk 08 due | ||||
Apr 27 | Apr 29 | May 01 exam 08 | ||||
May 04 last class |
Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
Sept 04 | Sept 06 | |||||
Sept 09 diagnostic exam | Sept 11 | Sept 13 | ||||
Sept 16 | Sept 18 | Sept 20 | ||||
Sept 23 | Sept 25 | Sept 27 hmwk 01 due | ||||
Sept 30 | Oct 02 | Oct 04 exam 01 | ||||
Oct 07 | Oct 09 | Oct 11 | ||||
Oct 14 | Oct 16 | Oct 18 hmwk 02 | ||||
Oct 21 | Oct 23 | Oct 25 exam 02 | ||||
Oct 28 | Oct 30 | Nov 01 | ||||
Nov 04 | Nov 06 | Nov 08 | ||||
Nov 11 | Nov 13 hmwk 03 | Nov 15 | ||||
Nov 18 | Nov 20 exam 03 | Nov 22 | ||||
Nov 25 | Nov 27 hmwk 04 | Thanksgiving | Nov 29 | |||
Dec 02 | Dec 04 hmwk 04 | Dec 06 | ||||
Dec 09 | Dec 11 last class, exam 04 |
Spring 2019 : MWF, 10:10-11:00, Vincent 206. Text will be notes posted here, supplemented by any standard text or notes that suit your taste.
Sources: The measure-and-integration we will cover is standard, and can be found in many sources, in addition to the write-ups here. Similarly, basics about Hilbert spaces and Banach spaces can be found in many places. The notes that will appear here will de-emphasize pathologies (except as cautionary tales). Looking at a variety of sources is recommended, as a way to avoid getting caught up in the idiosyncrasies of any particular source. At the same time, many of these sources are ridiculously expensive... Also, many have not been updated to reflect progress in mathematics over the last 80+ years! Although measure-and-integration succeeded in addressing certain issures, it by far did not completely succeed, and a good part of the progress in analysis 1915-2015 aims at greater success than measure-and-integration alone can achieve.
Spring grades will be determined by three in-class midterms , scheduled as in the table below. 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.
In Spring 2019, classes end Monday, May 6, 2019.
There will be regular homework assignments preparatory to exams, on which I'll give you feedback about mathematical content and writing style. The homeworks will not directly 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.
Office hours: MWF 11:15-12:20. Send email anytime!
Cumulative notes are or will be also grouped into an evolving single PDF. There is a Fall 2017 version already available.
Partial outline: ... resembling last year's...
Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
Jan 23 | Jan 25 | |||||
Jan 28 | Jan 39 | Feb 01 | ||||
Feb 04 | Feb 06 | Feb 08 hmwk 4 due | ||||
Feb 11 | Feb 13 | Feb 15 | ||||
Feb 18 | Feb 20 | Feb 22 hmwk 5 due | ||||
Feb 25 | Feb 27 | Mar 01 exam 5 | ||||
Stony Brook | Mar 04 | Mar 06 | Mar 08 | |||
Mar 11 | Mar 13 | Mar 15 | ||||
Mar 18 spring break | Mar 20 | Mar 22 | ||||
Mar 25 | Mar 27 | Mar 29 hmwk 6 due | ||||
Apr 01 | Apr 03 | Apr 05 exam 6 | ||||
Apr 08 | Apr 10 | Apr 12 | ||||
Apr 15 | Apr 17 | Apr 19 | ||||
Apr 22 | Apr 24 | Apr 26 hmwk 7 due | ||||
Apr 29 | May 01 | May 03 exam 7 | ||||
May 06 last class |
Cumulative notes are also grouped into a single pdf for easier searching for keywords, etc. This will be updated as more notes are added... [ updated 04 Dec '17]
Tentative partial outline:
Tentative outline: