Math 503 Spring 1999

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.


Office Hours

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


Outline and Schedule of Lectures

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*
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?