( See also: [ vignettes ] ... [ functional analysis ] ... [ intro to modular forms ] ... [ representation theory ] ... [ Lie theory, symmetric spaces ] ... [ buildings notes ] ... [ number theory ] ... [ algebra ] ... [ complex analysis ] ... [ real analysis ] ... [ homological algebra ] )

[ambient page updated 17 Sep '19] ... [ home ] ... [ garrett@math.umn.edu ]


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 a 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.


2019-2020

Fall 2019 : MWF, 10:10-11:00, Vincent 2. Text will be notes posted here, supplemented by whatever books or other notes you like.

Fall grades will be determined by 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.

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 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, and printed out.

Office hours: Mon 12:20-1:15, Wed 12:20-2:00. Send email anytime!

Notes, examples, discussions

  1. Overview/preview: extending/refining notions of "function", "integral", "derivative" and "limit"
  2. Metrics and topologies on vector spaces [ updated 17 Sep '19]
  3. Basic inequalities
  4. A classical attempt at an extended notion of function: measurable functions. Integrating measurable functions: Fatou, Lebesbue, Fubini-Tonelli
  5. Riesz-Markov-Kakutani, and a more conceptual idea of what "integration" is
  6. Extended example: Banach spaces Cj[a,b] of differentiable functions, Frechet spaces C\infty[a,b] of smooth functions.
  7. Extended example: Fourier series
  8. Hilbert spaces, Cauchy-Schwartz-Bunyakowsy inequality
  9. Further fundamental inequalities: Minkowski, Holder, ... Lp spaces
  10. ...

Exam and homework-example schedule, fall 2019:

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 hmwk 03
Nov 11 Nov 13 Nov 15 exam 03
Nov 18 Nov 20 Nov 22
Nov 25 Nov 27 hmwk 04 Thanksgiving Nov 29
Dec 02 Dec 04 Dec 06 exam 04
Dec 09 Dec 11 last class

2018-2019

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...

Exam and homework-example schedule, spring 2019:

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

2017-2018

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:


2016-2017

Tentative outline:

  1. Preview [ updated 15 Sep '16]
  2. Review of structure of real numbers, metric spaces, completeness, compactness, topology
  3. Introduction to integration
  4. Spaces of functions, convergence of Fourier series
  5. [partial draft] Fourier transforms: functions on lines [ updated 11 Sep '18]
  6. More measure theory, generalized functions (distributions), etc
  7. Distributions on Rn [ updated 26 Feb '17]
  8. Operators on Hilbert spaces
  9. ...
  10. ...

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