Fall 2010

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**(12/15)**If you haven't emailed me your paper already, please try to do so by the end of this week. If I owe you any homework solutions, let me know. Otherwise: HAPPY HOLIDAYS AND ENJOY YOUR WINTER BREAK!**(11/27)**The last homework (#7) has been posted. It is due on the last day of class, December 8 (not 6, as it previously said, by mistake). Btw, the last day of instruction this semester is December 9.**(11/22)**As you may know, I'm teaching Math 213B: Introduction to Riemannian geometry next semester, but I would like to move the class to an earlier time also on MW (currently the class is scheduled for MW 16:30-17:45). If you are thinking of taking this class, please let me know of your time constraints as soon as you can. Two top candidates for a new time slot are 12 PM or 3 PM, but 10:30 or 1:30 also may end up fitting in my schedule.**(11/18)**I just posted a LaTeX template for the expository paper called paper.tex. (It was actually written for Math 231A, but it'll do for 231B.) There you'll find a few examples of how to write math formulas, definitions, theorems, etc. in LaTeX. There is also a file called ref.bib, which contains all the references. You need to use BiBTeX to include the references into the paper, but this should be the last thing to worry about (first write the paper, then take care of the references). When the file paper.tex is compiled using LaTeX, the final product looks like this.You will find most information you would ever want to know about LaTeX at the LaTeX project page. Their page of guides is especially helpful.

**(11/9)**Homework 6 has been posted and is due around November 24.**(10/19)**I posted homework 5, due November 3, more or less.**(10/5)**I just posted homework 4. It's due somewhere in the vicinity of October 20.**(9/21)**Homework 3 has been assigned and is due (roughly) on October 4.**(9/7)**I just posted homework 2, due roughly on Monday, September 20.**(8/27)**Homework 1 has been posted and is due on September 8. Please let me know if you have any questions.**(8/24)**Welcome! If you are thinking about potentially taking this class but are not sure about it, please talk to me as soon as you can.

- Functional analysis is in a sense the calculus of infinitely many
variables. More accurately, it is a branch of analysis which studies
infinite dimensional (topological or normed) vector spaces and maps
between them. Functional analysis provides an ideal setting for the
synthesis of analysis and algebra. Its historical roots are in the
study of spaces of and transformations on functions, such as the
Fourier transform, and various differential and integral
operators. The need to study such transformations originated in
physics, both classical and (especially!) quantum, as well as other
branches of mathematics, such as the calculus of variations,
differential equations, and so on. In this course we will focus on
*linear*functional analysis, which studies linear maps.

- Math 231A or
**instructor consent**. **Note**: To make the course more accessible, I will not assume familiarity with measure theory. I'm willing to accept anyone with a basic background in analysis and linear algebra, and a certain level of mathematical maturity. Physics and engineering students are welcome.

Barbara D. MacCluer: Elementary Functional Analysis,
Springer, GTM vol. 253, 2009 |

- Green sheet

Banach and Hilbert spaces. Linear functionals. Orthonormal
bases. Bounded linear operators. Adjoints of Hilbert and Banach space
operators. The Big Three: the Hahn-Banach Theorem, the Principle of
Uniform Boundedness, the Open Mapping and Closed Graph
Theorems. Quotient spaces. Compact operators: the spectral theorem for
compact, self-adjoint operators, the invariant subspace problem, the
Fredholm alternative. Banach and C^{*} algebras: ideals and
homomorphisms, the Gelfand-Neimark and Gelfand-Mazur theorems,
spectral radius, the Gelfand transform, the continuous functional
calculus, Fredholm operators. The spectral theorem for normal
operators. Applications of functional analysis in quantum mechanics,
differential and integral equations, etc. (time permitting).

# | Date | Reading |
---|---|---|

1 | 8/25 | Introduction and 1.1 Normed linear spaces |

2 | 8/30 | 1.2 Orthogonality |

3 | 9/1 | 1.3 Hilbert space geometry |

4 | 9/6 | (Labor Day, no class) |

5 | 9/8 | 1.4 Linear functionals |

6 | 9/13 | 1.5 Orthonormal bases |

7 | 9/15 | 2.1 Bounded linear operators |

8 | 9/20 | 2.1 Adjoints of Hilbert space operators |

9 | 9/22 | 2.2 Adjoints of Banach space operators |

10 | 9/27 | 3.1 The Hahn-Banach theorem |

11 | 9/29 | 3.2 The principle of uniform boundedness |

12 | 10/4 | 3.3 Open mapping and Closed graph theorems |

13 | 10/6 | 3.4 Quotient spaces |

14 | 10/11 | Discussion, problems, questions, etc. |

15 | 10/13 | 4.1 Finite-dimensional spaces and 4.2 Compact operators |

16 | 10/18 | 4.3 A preliminary spectral theorem |

17 | 10/20 | 4.4 The invariant subspace problem and 4.5 Introduction to the spectrum |

18 | 10/25 | 4.6 The Fredholm alternative (and applications) |

19 | 10/27 | Discussion, problems, questions, etc. |

20 | 11/1 | 5.1 Banach and C^{*} algebras: First examples |

21 | 11/3 | 5.2 Results on spectra |

22 | 11/8 | 5.3 Ideals and homomorphisms |

23 | 11/10 | 5.4 Commutative Banach algebras |

24 | 11/15 | 5.5. Weak topologies |

25 | 11/17 | 5.6 The Gelfand transform |

26 | 11/22 | 5.7 The continuous functional calculus |

27 | 11/24 | 5.8 Fredholm operators |

28 | 11/29 | 6.1 Normal operators as multiplication operators |

29 | 12/1 | 6.2 Spectral measures |

30 | 12/6 | Functional analysis and quantum mechanics |

31 | 12/8 | Discussion, problems, questions, etc./presentations |

- Based on homework (50%), and a class paper and presentation (50%). There will be no exams.

- None yet.

- There will be regular homework assignments, taken mostly from the textbook.

# | Due date | Assignment |
---|---|---|

1 | 9/8 | Exercises 1.2, 1.3, 1.7, and 1.8 |

2 | 9/20 | Ex. 1.13, 1.16, 1.20, 1.21, and 1.28 |

3 | 10/4 | Ex. 2.2, 2.6, 2.8, and 2.10 |

4 | 10/20 | Ex. 2.15, 2.16, 2.23, and 3.8 |

5 | 11/3 | Ex. 3.14, 3.17, 3.23, and 3.27 |

6 | 11/24 | Ex. 4.6, 4.9, 4.10(b), 4.16, and 4.19 |

7 | 12/8 | Ex. 5.11, 5.16, 5.25, 5.28. |

- Each student is expected to write a short paper (in LaTeX) on a
topic related to functional analysis. Here are some potential
topics:
- Applications to quantum mechanics
- Applications of Fredholm's integral equations to potential theory
- Applications to the virating string problem
- Survey of fixed point theorems
- Introduction to von Neumann algebras
- Introduction to calculus of variations
- History of functional analysis

Slobodan N. Simić

Last modified: Wed Dec 15 15:56:41 PST 2010