# Math 175: Elementary Functional Analysis

Spring quarter 2011/2012

### Instructor: Richard Bamler

office 382-N
rbamler@math.stanford.edu
650-723-2975
office hours: Mon 2:00 pm - 3:00 pm, Thu 1:00 pm - 3:00 pm or by appointment
### Course Assistant: Iurie Boreico

office 380-T
boreicoATmath.stanford.edu
office hours: Tues 11:00 am - 12:00 pm, Wed 4:00 pm - 6:00 pm, Thu 11:00 am - 12:00 pm, Fri 1:30 pm - 4:30 pm

**Classes:** MoWeFr 11:00 am - 11:50 am

**Room:** Building 380, Room 380-X

**Course book:** N. Young, *"An introduction to Hilbert space"* (1988)

### Organization of the course

Homeworks will be assigned weekly, they can be downloaded from the course website and are due each Monday, unless otherwise specified
There will be one midterm, on Wednesday 5/2 from 5:15 pm - 6:15 pm in L103 (Littlefield Center)
The final will be in 380-X on Wednesday 6/13, 8:30 am - 11:30 am
The final grade will be based on the homework assignments (30%), the midterm (30%) and the final (40%).
The lowest homework score will be dropped (more precisely, since problem sets might have different maximal scores, the homework, in which the most points are lost, will be given a full score).

### Manuscript

Please be aware that the manuscript is not a replacement for you own notes since it might contain a lot of typos and mistakes.
I normally find them while teaching and write it correctly on the blackboard, but I don't update them in the manuscript.
I also don't stick completely to these notes (i.e. I sometimes leave out/add material in class).
I strongly recommend to use the course book instead to review past classes!

Chapter I

Chapter II

Chapter III

Chapter IV

Chapter V

### MATH 175 website from last year

Here you can find the old website of the course MATH 175 taught by Leon Simon in spring 2010.
Feel free to use the problem sets for additional practice.

Leon Simon's old midterm (solution) Note that problem 3 would not be a typical exam question for this course since we haven't introduced some of the terminology.

Leon Simon's old final exam (solution)

### Remarks on the midterm and recommendations how to study for it

The exam problems will be much lighter than the ones from the past problem sets. They should require only one small idea and then it should be possible to write them up in a short paragraph. The exam will be designed in such a way that you will most likely not run out of time, i.e. there won't be a big difference if you work on it for 45min instead of 1h. The questions will test whether you have understood the basic concepts taught in class and whether you are able to apply them in various settings. I will repeat certain rarely used definitions, so you will not have to learn anything by heart.

Please be aware: the purpose of the problems in the problem set is to either apply the theory from class in many different situations, to provide counterexamples, or to prove things that are too technical to do in class.
The exam questions, however, are designed to test your understanding of the material in the most time-efficient way.

In order to study for the midterm, I suggest the following:
**This is actually the most important method:** go through your notes and check whether you have understood everything from class. Whenever there is a Lemma or Theorem, cover its proof and see if you can reconstruct it in your mind or on a piece of scrap paper. If you were not able to reconstruct it, try again on the next day (start to study early!).
go through the past problem sets and do the same as mentioned above. Cross-check with the solutions posted online.
read the corresponding chapters in the book (the material is sometimes presented slightly differently there and some things are more elaborate). Try to solve the problems in the book which have not been on the problem sets.
try to solve Leon Simon's midterm and problem sets, you will find additional midterms online by googling (be aware that some courses take a different approach and therefore some things might look unfamiliar).
if you have any questions, communicate with your classmates pass by my office or send me an email. I will try to reply as quickly as possible.

### Schedule

The following schedule is subject to change!
#### Week 1: 04/02 - 04/06

__Class 1 (Mon):__ Overview
__Class 2 (Wed):__ Review of the Lebesgue integral
__Class 3 (Fri):__ Review of the Lebesgue integral
Problem Set 1 (due Mon 4/9)
#### Week 2: 04/09 - 04/13

__Class 4 (Mon):__ Review of the Lebesgue integral / L^1-space
__Class 5 (Wed):__ Review of the Lebesgue integral / Lebesgue-measurability
__Class 6 (Fri):__ Review of the Lebesgue integral / L^2-space
Problem Set 2 (due Mon 4/16)
#### Week 3: 04/16 - 04/20

__Class 7 (Mon):__ Inner product spaces
__Class 8 (Wed):__ Normed spaces
__Class 9 (Fri):__ Closest point property
Problem Set 3 (due Mon 4/23)
#### Week 4: 04/23 - 04/27

__Class 10 (Mon):__ Orthogonal expansions
__Class 11 (Wed):__ Orthogonal expansions
__Class 12 (Fri):__ Orthogonal expansions
Problem Set 4 (due **Wed** 5/2)
#### Week 5: 04/30 - 05/04

__Class 13 (Mon):__ Fourier Series
__Class 14 (Wed):__ Fourier Series ; **(midterm between 5:15 pm - 6:15 pm in L103 (Littlefield Center))** solution
__Class 15 (Fri):__ Fourier Series / Linear operators and linear functionals
Problem Set 5 (due **Wed 5/9**)
#### Week 6: 05/07 - 05/11

__Class 16 (Mon):__ Linear operators and linear functionals / Dual spaces
__Class 17 (Wed):__ Dual spaces / Invertible operators
__Class 18 (Fri):__ Compact linear operators / Adjoints
Problem Set 6 (due Mon, 5/14)
#### Week 7: 05/14 - 05/18

__Class 19 (Mon):__ Adjoints / Spectra
__Class 20 (Wed):__ Spectra / Compact operators
__Class 21 (Fri):__ Compact operators
Problem Set 7 (due Mon, 5/21)
#### Week 8: 05/21 - 05/25

__Class 22 (Mon):__ Compact operators / Spectral Theorem
__Class 23 (Wed):__ Spectral Theorem
__Class 24 (Fri):__ Sturm-Liouville systems
Problem Set 8 (due Wed, 5/30)
#### Week 9: 05/28 - 06/01

__Class 25 (Wed):__ Sturm-Liouville systems
__Class 26 (Fri):__ Sturm-Liouville systems
#### Week 10: 06/04 - 06/08

__Class 27 (Mon):__ Sturm-Liouville systems
__Class 28 (Wed):__ Sturm-Liouville systems / Dirichlet problem
#### Final exam: Wednesday 6/13, 8:30 am - 11:30 am

solution

*last modified: 06/06/12*