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Real Analysis 2018-19

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.

Prerequisites for 8601: strong understanding of a year of undergrad real analysis, such as our 5615H-5616H or equivalent, including 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.

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 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...

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
Mar 04 Mar 06 Mar 08
Mar 11 Mar 13 Oct 26
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

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 ]

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