18.102: Introduction to Functional Analysis

Richard Melrose

This is the home page for the course 18.102 for Spring 2013.

Comment and recent changes:

• Solutions to final up for anyone interested. If you want more than your letter grade you will have to ask me by email. Enjoy summer!
• Non-triviality statement added in preamble because of Q3.
• Serious error in P5 on the Pre-Final corrected. This question is now harder than it was.
• Prefinal revised.
• Questions from which the final will be chosen can be found below -- some further refinement of these is possible over the next couple of days.
• Subject evaluation is in progress -- now is your chance to express your opinion!
• Another typo in the notes fixed, near (3.14) -- thanks Raeez.
• A few changes to the notes, thanks Erik.
• Corrected a nasty typo in P7.5
• Problem set 7 is in place.
• Notes for today's lecture, 25 April, 2013.

Old Changes

Background In case you are wondering whether you know enough about metric spaces.

Piazza-18.102 Not up yet.

Basic facts:

• Lectures: TH 1-2:30 in 6-120. This is a big room, please don't all sit up the back!
• My office hour: W 11-12 in 2-174. Since this does not seem to work for quite a few people, I will try Thursdays 11-12 as well (or maybe instead eventually).
• Hans Liu (who is one of our graders) will have an office hour F 4-5, in 2-251.
• Graders:
• Don't be afraid to email to me at rbm AT math.mit.edu -- it will likely be answered! I am always interested to get some feedback on how hard/easy you are finding things.
• Sources. For the first part of the course I suggest you look at some of the following on-line notes to broaden your viewpoint!
  • I.F. Wilde's notes Only the first four chapters are really relevant for us and I will proceed a little more slowly.
  • The coverage of WWL Chen's notes is a bit closer to what we will do
    The first two chapters should help you to recall some of 18.100.
  • Another useful set of notes are those by T.B. Ward Especially Chapters 1-5, but I will do a bit more on Lebesgue integration.
  • For integration I will use a heavily modified version of (Jan) Mikusinski's approach which you can find in Debnaith and (Piotr) Mikusinski ``An introduction to Hilbert spaces with applications'' (Academic Press)
  • A nice reference for Hilbert spaces is G.F. Simmons ``Introduction to Topology and Modern Analysis''
  • You might like to look at P. Halmos' ``A Hilbert space problem book''
  • A good over-all reference, a little more advanced than this course, is P.D. Lax's book ``Functional Analysis'' (Wiley-Interscience) -- a nice book to have.

Homework, tests and grades

• Problem sets will be due on Saturdays, at 4AM. Solutions must be submitted electronically to me at rbm AT math.mit.edu (not to the grader, that will not work) and dated by then. This does not mean that you need to learn LaTeX (although of course that is probably a good idea). You can write out your solutions and scan-to-pdf (there are several places you can do this as you no doubt know better than me). The weird time is not meant to encourage you to all-nighters but seems the easiest to enforce.
  LATE HOMEWORK will be graded. However, this will probably be done by me, so don't expect too much generosity. How many marks you get for late homework is decided by me based on a secret, highly dubious, formula. Try to avoid this route.
  So, homework will due on, Saturdays (at 4AM, so really the preceding Thursday),
  1. First problem set Problems1.pdf Due 16 Feb.
  2. Second problem set Problems2.pdf Due 23 Feb.
  3. Third problem set Problems3.pdf Due 1 Mar.
  4. Fourth problem set Problems4.pdf Due 15 Mar.
  5. Fifth problem set Problems5v5.pdf 28 March, 2013: Hint added for Problem 5.3 (thanks to Charles and also people who asked). [Due 29 Mar, Error in first question corrected 21 March, 2013 -- thanks to Hajir and Stephen].
  6. Sixth problem set Problems6v4.pdf Due 12 Apr -- an extra week; 12 April, 2013 -- typos in P6.4 fixed, thanks to Victor, Problem 6.5 reworded thanks Weicheng.
  7. Seventh problem set Problems7.pdf Due 3 May. With correction to P7.5, thanks Emilio.
• Two in-class tests on 12 March and 18 April in in 6-120.
  • Problems for Test 1
  • Problems for Test 2 Solutions Fourth version -- A hint added to P2.5. Third version -- notation in P2.5 changed to be closer to what I did in class (thanks Julian), Second version, 15 April, 2013 -- typos fixed in P2.5 (thanks Trevor) and P2.8 (thanks Stephen)
• Grades will be computed by two methods -- the cumulative and the hope-springs-eternal method with the actual grade the greater of the two.
  • First method: Homework 30, Tests 30, Final 40.
  • Second method is based purely on the final. Try not to rely on this.
• The final exam will be on Wednesday, May 22, 9:00 to 12:00 Noon in Walker Memorial.
• The questions of the final will be chosen from the PreFinal
  • v2 Revised 16 May, 2013 (Q3, Q7, Q9 changed) thanks Omer.
  • v3 Revised 17 May, 2013 (Q11, Q13 changed) thanks Stephen.
  • v4 Revised 19 May, 2013 (Q17 changed) thanks Kirin.
  • v5 Revised 19 May, 2013 (Q5 changed significantly) thanks to Trevor.
  • v6 Revised 20 May, 2013 (Q5 wording changed) thanks Morris.
  • v7 Revised 20 May, 2013 (non-trivial added to preamble) thanks to Whan.
• Solutions

Notes for the course

• A version of the notes for you to annotate will go up on nb (under 18.102, but not up yet).
  Note that I will continue making substantial changes to some parts of the notes.
• Lecture notes (5 Feb, 2013)
  Broken out into chapters:
  Normed and Banach spaces -- Chapter 1 Small changes to Hahn-Banach section, 16 May, 2013.
  Log of changes
  Lebesgue integration -- Chapter 2
  Log of changes
  Hilbert space -- Chapter 3
  Log of changes
  Applications -- Chapter 4
  Log of changes
  Problems -- Chapter 5
  Log of changes
  -- I will make further revisions to these lecture during the semester -- so the dates represent the most recent change. I am certainly interested in suggestions, criticisms and especially corrections.
• Syllabus May change a little with time but my best guess as to where I will go.