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Real Analysis 2017-18

Fall 2017: MWF, 10:10-11:00, Vincent 311. Text will be notes posted here. Also, any standard text or notes will be adequate back-up for much of the course content.

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!

Grades will be determined by three in-class midterms, on Friday 29 September, Friday 3 November, and Friday 8 December .

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. Given the limitations of marking-up electronic documents, please submit paper print-outs of homeworks.

Office hours: MWF 9:30-10:00 and MWF 12:10-12:40. Send email anytime!

Tentative partial outline:

  1. Preview
  2. Review of structure of real line, compactness, continuity, metric spaces, topology, Riemann integrals of continuous functions, uniform pointwise limits
  3. (Lebesgue) measurable sets, measurable functions, closure under pointwise limits, monotone and dominated convergence, Fatou's lemma
  4. ...

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 27 Nov '16]
  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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