18.102: Introduction to Functional Analysis

Instructor:

Richard Melrose
MIT, Department of Mathematics
Room 2-174
Email: rbm at math.mit.edu
Office hours: Wed, 2-3pm, 2-174. Or by appointment.

Time and Place:

Lecture:- Tuesday and Thursday, 1-2:30pm.
Room 2-102

Grader:

Tonghoon Suk
Office hour:- Tues, 9:3-10:30, 2-091.

Tonghoon Suk will have a special Office Hour, 11-12, Wed 10 Oct, 2-091

Outline of the Course

I will follow the text fairly closely, at least at the beginning. My aim is to cover the first four chapters quite completely and with some applications from the later chapters. I will also add some material for which there will be notes.

Material covered

Lecture 1: 6 Sept:- Linear (vector) spaces, examples. Dependence. Bases. Norms. Metrics. Banach spaces. More examples -- sequence spaces. Photos:6 Sept
Lecture 2: 11 Sept:- Linear maps and operators. Photos:11 Sept (Thanks to Tasos Sidiropoulos for some of these)
Lecture 3: 13 Sept:- Holder and Minkowski inequalities. `Little l^p spaces' Photos:13 Sept
Lecture 4: 18 Sept:- Dual of l^p, step functions.Photos:18 Sept
Lecture 5: 20 Sept:- Lebesgue integrability, including the discovery that one of the preliminary lemmas in Edition 2 is simply wrong!Photos:20 Sept
Lecture 6: 25 Sept:- The space of integrable functions -- L¹. Photos:25 Sept
Lecture 7: 27 Sept:- Completeness of L¹. Photos:27 Sept
Lecture 8: 2 Oct:- Definition and completeness of L². Photos:2 Oct
Lecture 9: 4 Oct:- Hilbert spaces. Convexity.Photos:4 Oct
Lecture 10: -- in class Test, 11 Oct:- See Below.
Lecture 11: 16 Oct:- Riesz representation Theorem completed. Applications.
Lecture 12: 18 Oct:- Uniform boundedness, weak convergence, orthonormal sequences, orthonormal bases.Photos:18 Oct
Lecture 20: 15 Nov:- Spectral theorem for compact self-adjoint operators completed, Schmidt classes.Photos:15 Nov
Lecture 24: 4 Dec:- Spectral theorem for compact operators completed, approximate orthonormal bases.Photos: 4 Dec
Lecture 25: 6 Dec:- Lebesgue integral on n-space, Hilbert-Schmidt operatorsPhotos: 6 Dec
Lecture 26: 11 Dec:- Brief survey: Sobolev spaces, Dirichlet problem, weak derivatives, distributions.

Notes

I have written up a few notes on the contents of Chapter 2 of the text -- let me know if you find them useful.
Notes on integration: Last altered 3 Oct 2007
Here are some more notes on Baire's theorem and related things -- including the Open Mapping Theorem.
Notes on Hilbert space: Last altered 5 Nov 2007
Notes on Dirichlet problems: Last altered 15 Nov 2007 (2nd version, lightly proofread)

Prerequisites

18.100B.

Textbook

Debnaith and Mikusiński Introduction to Hilbert Spaces with Applications.

Homework and Exams

There will be 8 problem sets due on Thursdays, two midterm exams and a final exam (Tuesday afternoon, 18 December). All homework should either be submitted to me by email (it does not have to be in TeX, scanned written pages are fine) or on paper of reasonable quality, so it goes through the scanner! If you need decent paper I am happy to supply some.

Homework 1. Due Thursday, 13th Sept, 1PM. Text -- unfortunately I am working from the 2nd edition and many of you have the third. The problems are Chapter 1, problems 3, 7, 8, 9, 11. Here they are as a pdf file Problems1 and here they are written out:-
1. (No.3) Prove that a subspace of a vector space is a vector space itself. [The point of course is to write this out carefully but succinctly].
2. (No.7) Show that any vector of (3 dimensional real Euclidean space) is a linear combination of (the) vectors (1,0,0), (1,1,0) and (1,1,1). [Once again, try to keep it brief and as clear as you can]
3. (No.8) Prove that every quadruple of (i.e. set of four) vectors in (3 dimensional real Euclidean space) is linearly dependent. [Meaning show there is a non-trivial linear relation between them.
4. (No.9) Prove that the functions f_n(x)=x^n, n (a non-negative integer) are linearly independent.
5. (No.11) Prove that each of the spaces of continuous, of k-times continuously differentiable, and of infinitely differentiable functions on n-dimensional Euclidena space is infinite dimensional. [Try the case n=1 first, the other cases follow from this.]
  Solutions and comments Problems1 solved
Homework 2, due September 20. Problems2
Homework 3, due September 27. Problems3 (corrected, second version)
Homework 4, due October 4. Problems4
Test on October 11 in class. Here are the problems from which the test will be taken!
pre-Test1 (Second revised version -- Hint in Q6 expanded) N.B. This is not a practice, this is the test! If you think there are errors, let me know!
Homework 5, due October 23. Problems5
Homework 6, due November 8. Problems6
Test 2 -- Take home, due November 20 at 2:30. Test2