This is a functional analysis course, using Rudin's classic book
`Functional Analysis' as the text. Math 502 is a prerequisite for
the course. The course is presently scheduled to meet MWF 12:20 -
1:10 in 115 Osmond Laboratory.

We'll begin the course by reviewing some of the fundamental results of Banach space theory and generalizing them to other locally convex topological vector spaces. This should amount to about a third of the course. We will spend another part of the course discussing the theory of operators on Hilbert space, including the spectral theorem and the rudiments of C*-algebra theory; and a third discussing distribution theory. I hope to show how the seemingly abstract techniques of functional analysis can be brought to bear on concrete problems such as those arising from linear partial differential equations.

There will be some scope for varying the advanced topics included in
the course according to the interests of the class.

The department requires Math 502 as a prerequisite for this course.
The outline of the functional analysis portion of Math 502 is as follows.

The Hahn-Banach Theorem and Open Mapping Theorem. The Uniform Boundedness
Principle. Hilbert Spaces. Dual Spaces. Duals of Lp spaces. Selections
from the following topics as time allows: Weak and weak* topology. Banach-Alaoglu
Theorem. Elementary operator theory through the spectral theorem for compact
normal operators.

I will assume some familiarity with this material (not total mastery!)
Detailed lecture
notes and outlines from the course given by Professor Arnold (Spring
1997) are available on the Penn State MathNet. At least, you will
need to be comfortable with the contents of Chapters 4 and 5 of Rudin's
*Real
and Complex Analysis.*

Homework will be assigned regularly. You will be told which
parts of your homework will be graded, but you are encouraged to attempt
it all.

Course grades will be calculated on the basis of homework, a 1-hour midterm, and and a final interview. An A grade on the final will normally secure an overall A on the course. The exams will be scheduled later.

My office is 427 McAllister. I will be available for discussion
of the course in my office after the class (1:10 - 2:00) on Mondays and
Fridays. If you want to contact me about the course at any other
time please send email to roe@math.psu.edu.

This is the plan of the course as I envisage it at present.
`Pages' refers to reading from Rudin's book. A star indicates that
I expect the lecture also to contain material which is not in the book.

Lecture |
Date |
Topics |
Pages |

1 | 1/11/99 | Introduction. Topological vector spaces, examples. Seminorms, Frechet spaces. Overview of the course. | 1-9 |

2 | 1/13/99 | Topological topics: induced and coinduced topologies, product topology, Tychonoff theorem. | 367-371 |

3 | 1/15/99 | Hausdorffness of TVS. Uniform structure, associated topologies, equicontinuity, Ascoli's theorem. | 367-371* |

4 | 1/18/99 | NO LECTURE OWING TO WEATHER | |

5 | 1/20/99 | Linear functionals. Finite dimensional TVS. Boundedness | 13-24 |

6 | 1/22/99 | Local convexity and seminorms. Quotient spaces. | 24-36 |

7 | 1/25/99 | Menagerie of examples | |

8 | 1/27/99 | Baire category theorem. Banach-Steinhaus theorem, applications | 40-46, 51 |

9 | 1/29/99 | Open mapping theorem, closed graph theorem | 47-51, 110 |

10 | 2/1/99 | Dominated extension and Hahn-Banach theorems. Application to duality. Weak and weak-star topologies, Alaoglu's theorem. | 55-67 |

11 | 2/3/99 | More of the same | |

12 | 2/5/99 | Convex hulls, Krein-Milman theorem, bipolar theorem. | 68-71 |

13 | 2/8/99 | Weak integration and weak holomorphicity | 73-81 |

14 | 2/10/99 | Duality in Banach spaces: dual transformation, dual of an exact sequence, closed range theorems | 88-97 |

15 | 2/12/99 | More of the same | |

16 | 2/15/99 | An example: the dual of C(X) | |

17 | 2/17/99 | Further topics on duality (as time permits) | |

18 | 2/19/99 | Approximation theorems: Runge,Bishop, Stone-Weierstrass | 115-117* |

19 | 2/22/99 | Application to Haar measure | 120-125 |

20 | 2/24/99 | Review session | |

21 | 2/26/99 | Midterm Exam | |

22 | 3/1/99 | Test function spaces | 6.1-6.6 |

23 | 3/3/99 | Distributions: definition and examples | 6.7-6.9 |

24 | 3/5/99 | Operations on distributions | 6.10-6.15 |

25 | 3/15/99 | Topology on distributions. Support; distributions of point support | 6.16-6.25 |

26 | 3/17/99 | Convolution | 6.29-6.37 |

27 | 3/19/99 | The Schwartz Kernels Theorem | |

28 | 3/22/99 | Fourier transforms and inversion | 7.1-7.7 |

29 | 3/24/99 | L^2 theory of the Fourier transform ; Heisenberg's Uncertainty Principle | 7.9* |

30 | 3/26/99 | Temperate distributions; relation of Fourier transform and Fourier series | 7.10-7.19 |

31 | 3/29/99 | A characterization of the sine function | |

32 | 3/31/99 | Distributions and PDE: fundamental solutions | 8.1-8.5 |

33 | 4/2/99 | Distributions and PDE: examples | |

34 | 4/5/99 | Local solvability: Malgrange-Ehrenpreis theorem | |

35 | 4/7/99 | Banach algebras: properties of invertibles, the spectrum | 10.1-10.14 |

36 | 4/9/99 | Commutative Banach algebras: Gelfand duality theory | 11.1-11.5 |

37 | 4/12/99 | Examples and the spectral radius theorem | 10.13, 11.13 |

38 | 4/14/99 | Commutative C*-algebras | 11.14-11.19 |

39 | 4/16/99 | The spectral theorem for bounded normal operators | 12.22-12.26 |

40 | 4/19/99 | The spectral theorem for compact normal operators | 12.28-12.31 |

41 | 4/21/99 | Theory of unbounded operators | 13.1-13.10 |

42 | 4/23/99 | Self-adjoint operators and the Cayley Transform | 13.11-13.19 |

43 | 4/26/99 | Spectral theorem for unbounded selfadjoint operators | 13.29ff |

44 | 4/28/99 | A Schrodinger equation example | |

45 | 4/30/99 | Where next? |