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Real Analysis 2018-19

Fall 2018: MWF, 10:10-11:00, Vincent 311. Text will be notes posted here, supplemented by any standard text or notes that suit your taste.

Prerequisites for 8601: strong understanding of a year of undergrad real analysis, such as our 5615H-5616H or equivalent. This background includes careful discussion of Riemann integration on Euclidean spaces, the basic topology of Euclidean spaces, and abstract metric spaces. Prerequisites for 8602: 8601 or equivalent.

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.

Fall grades will be determined by four 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 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 Fall 2018, classes begin Tuesday, Sept 4, and end Wednesday, Dec 12, 2018. In Spring 2019, classes begin Tuesday, Jan 22, 2019, and 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.

Tentative partial outline: appearing soon... resembling last year's...

  1. Review examples 00 [ updated 04 Sep '18] due Friday, Sept 14, 2018 ... Discussion 00 [ updated 19 Sep '17]
  2. Metric spaces, completeness, natural spaces of functions
  3. Inequalities: Jensen, Minkowski, Holder, Young, et al
  4. Measure and integral
  5. ...

Exam and homework-example schedule:

assign hmwk
Monday Wednesday Friday
Sept 05 assign hmwk Sept 07
Sept 10 Sept 12 Sept 14: hmwk 1 due
Sept 17 Sept 19 Sept 21: exam 1
Sept 23 Sept 25 Sept 27
Oct 01 Oct 03 Oct 05: hmwk 2 due
Oct 08 Oct 10 Oct 12: exam 2
Oct 15 Oct 17 Oct 19
Oct 22 Oct 24: hmwk 3 due Oct 26
Oct 29 Oct 31: exam 3 Nov 02
Nov 05 Nov 07 Nov 09
Nov 12 Nov 14 Nov 16: hmwk 4 due
Nov 19 Nov 21 Nov 23
Nov 26 Nov 28 Nov 30: hmwk 5 due
Dec 03 Dec 05 Dec 07: exam 4
Dec 10 Dec 12 last class

Real Analysis 2017-18

Spring 2018: MWF, 10:10-11:00, Vincent 207. Text will be notes posted here. Also, any standard text or notes will be adequate back-up for much of the course content.

Prerequisites for 8601: a year of undergrad real analysis, such as our 5615H-5616H or equivalent. This background includes careful discussion of Riemann integration on Euclidean spaces, the basic topology of Euclidean spaces, and abstract metric spaces. Prerequisites for 8602: 8601 or equivalent.

Sources: The measure-and-integration we will cover is standard, and can be found in many sources. 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, for one thing... Also, many have not been updated to reflect progress in mathematics over the last 80+ years!

As in the Fall, Grades will be determined by three in-class midterms, on Friday 23 February, Friday 30 March, and Friday 27 April . Last day of class is 04 May.

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 9:40-10:05 and MWF 12:10-12:40. Send email anytime!

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:


Real Analysis 2016-17

Spring 2017: MWF, 10:10-11:00, Vincent 209. Text will be notes posted here.

Prerequisites: a year of undergrad real analysis, such as our 5615H-5616H or equivalent. This background includes careful discussion of Riemann integration on Euclidean spaces, the basic topology of Euclidean spaces, and abstract metric spaces.

Sources: The measure-and-integration we will cover is standard, and can be found in many sources. 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, for one thing... Also, many have not been updated to reflect progress in mathematics over the last 80+ years!

Spring: Grades will be determined by three in-class midterms, on Friday Feb 17, Friday Mar 24, and Friday Apr 28.

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.

There will be an after-term final exam, Saturday May 13, 1:30 p.m.-3:30 a.m. which is optional for anyone with a B average or better (but, even then, could be used to try to improve one's grade, if desired). The final's weight will be equal to that of 2 midterms.

No late homeworks will be accepted. Homework should be typeset, presumably via (La)TeX. Given the limitations of marking-up electronic documents, please submit paper print-outs of homeworks. Make-up midterms can be negotiated, due to illness, etc., but obviously it's best to avoid getting behind or things getting out of control. University policy prohibits giving Final exams before the last day of classes, and the intricacies of room scheduling for final exams makes it essentially impossible to try to adjust the date or time of our final.

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."