Introduction to Functional Analysis: 18.102 -- Spring 2014

• Lectures in 4-163, Tuesday and Thursdays 1-2:30PM.
• Office hour in E17-340 Wednesday 2-3.
• Kestutis Cesnavicius will have an office hour Fridays, 2:45-3:45 in E17-306.
• Please ask questions!
• Comments and remarks:
  • The question for the final are up -- see under Final below.
  • There will be no further problem sets.
  • Professor Tomasz Mrowka will give the remaining lectures, starting tomorrow, Tuesday May 6.
  • Corrected version of Problems 7, thanks Sookyo.
  • Gigliola Staffilani will give the lecture on 1st May.
  • Problem set 7 is (finally) up.
  • No individual acknowledgements of problems set 6 have gone out -- I should get to these later today. 19 April, 2014.
• Old remarks
• Syllabus May change a little with time but my best guess as to where I will go.
• Lecture schedule and informal notes -- this briefly describes what I expect to talk about/did talk about with pointers for more details.
  1. Lecture 1: 4 February. Metric, normed and Banach spaces. Completion.
  2. Lecture 2: 6 February. Bounded operators
  3. Lecture 3: 11 February.
  4. Lecture 4: 13 February.
     18 February is an MIT Monday.
  5. Lecture 5: 20 February.
  6. Lecture 6: 25 February.
  7. Lecture 7: 27 February.
  8. Lecture 8: 4 March.
  9. Lecture 9: 6 March.
  10. Lecture 10: 11 March.
  11. Lecture 11: 13 March.
  12. Lecture 12: 18 March.
  13. Lecture 13: 20 March.
  14. Lecture 14: 1 April
  15. Lecture 15: 3 April.
  16. Lecture 16: 8 April.
  17. Lecture 17: 10 April.
  18. Lecture 18: 15 April.
  19. Lecture 19: 17 April.
  20. Lecture 20: 24 April.
  21. Lecture 21: 29 April.
  22. Lecture 22: 1 May.
  23. Lecture 23: 6 May.
  24. Lecture 24: 8 May.
  25. Lecture 25: 13 May.
  26. Lecture 26: 15 May.
• Graders:
  • Kestutis Cesnavicius
  • Felipe Hernandez
• 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 7AM. 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 7AM, so try to send it the preceding Friday),
  1. First problem set Problems1.pdf Due 15 Feb.
  2. Second problem set Problems2.pdf Due 22 Feb.
  3. Third problem set Problems3.pdf Due 1 Mar.
  4. Fourth problem set Problems4.pdf Due 14 Mar.
  5. Fifth problem set Problems5.pdf Due 28 March.
  6. Sixth problem set Problems6.pdf Due 18 Apr.
  7. Seventh problem set Problems7.pdf Due 2 May; v2 -- the eigenvalues A in Prob5 were wrong.
• Two in-class tests on 6 March and 3 April in in 4-163. These are on days when I will be out of town.
• First test will be selected from these Test 1 Problems
• Second test will be selected from these Test 2 Problems
• Final exam 9AM-12Noon, 21 May 2014, Walker Gym Final Problems
• 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.

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 changes to some parts of the notes.
• Lecture notes (6 May, 2014, typo in Hahn-Banach fixed)
  Broken out into chapters:
  Normed and Banach spaces -- Chapter 1
  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