# 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

Tonghoon Suk
Office hour:- Tues, 9:3-10:30, 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 lp spaces' Photos:13 Sept
• Lecture 4: 18 Sept:- Dual of lp, 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)

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.]