## Instructor:Richard MelroseMIT, 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 SukOffice hour:- Tues, 9:3-10:30, 2-091. |

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.

- 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 -- L
^{1}. Photos:25 Sept - Lecture 7: 27 Sept:- Completeness of L
^{1}. Photos:27 Sept - Lecture 8: 2 Oct:- Definition and completeness of L
^{2}. 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 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)

Debnaith and Mikusiński Introduction to Hilbert Spaces with Applications.

- 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:-
- (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].
- (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]
- (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.
- (No.9) Prove that the functions f_n(x)=x^n, n (a non-negative integer) are linearly independent.
- (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