PMA 445 Functional Analysis 2003-2004

by Dr. P. G. Dixon

 


Important Announcement: Examination Format

Please note that the exam format has changed. The examination will last two and a half hours, as before. The paper will contain 5 questions and the rubric will ask you to do Question 1 and three other questions. Question 1 will contain easier material from throughout the syllabus. All questions will carry the same number of marks and the marks given for parts of questions will be indicated. 

 


Changes in the real world can happen in a potentially unlimited number of ways, and for this reason infinite dimensional spaces are required for their description. Functional Analysis is the subject which lays the foundations for the study of infinite dimensions. The course will begin by introducing normed vector spaces and their dual spaces, look at some examples, and then establish, amongst other things, some of the basic results of the theory: the Hahn-Banach Theorem, the Uniform Boundedness Theorem, the Open Mapping Theorem and the Closed Graph Theorem.

Aims of the Course

  1. To introduce students to the basic ideas and theorems of functional analysis.
  2. To develop students' analysis skills further than in the Real Analysis course.
  3. To show students the value of the use of abstract algebraic/topological structures in obtaining properties of down-to-earth analysis.
  4. To allow students to taste the subject with a view to further work in the subject as postgraduates.
  5. To give students a working knowledge of the basic properties of Banach spaces and their bounded linear operators.
  6. To give students a first view of duality ideas.
  7. To give students an insight into the axiom of choice and its equivalents.

Course organization

The course is two lectures per week in the second semester (Tuesdays at 9.00 and Thursdays at 3.10). Homework will be set each week.

Assessment

One formal 2.5 hour exam. Format: 4 out of 5 questions, with a compulsory Question 1. Note that this differs from the format in previous years.

Prerequisites

PMA215 Metric spaces and PMA344 Real Analysis.

Course content

Normed spaces
Definition and examples of normed and Banach spaces. Hilbert spaces.
Linear Mappings
Bounded linear mappings and functionals; normed spaces of bounded linear mappings; the dual of a normed space.
Linear Mappings
The Axiom of Choice and Zorn's lemma. The Hahn-Banach theorem, and its corollaries. Examples of dual spaces.
Category Theorems
The Uniform Boundedness (or Banach-Steinhaus) theorem.
Finite-dimensional spaces
Special properties of finite dimensional spaces

Recommended Books

C. Goffman and G Pedrick ``First Course in Functional Analysis'', Chelsea (1983) (Available in Blackwell £ 21).

G.F. Simmons, ``Introduction to Topology and Modern Analysis'', Kreiger (1999), £ 40.50 (paperback version temporarily(?) out of print).

W. Rudin, ``Real and Complex Analysis'', McGraw-Hill (1966).

N.J. Young ``An Introduction to Hilbert Space'', Cambridge (1988).

Online Resources

  1. The search engine Google has a page on Functional Analysis.
  2. Functional Analysis Lecture Notes by T. B. Ward, University of East Anglia. The first three chapters of this 33-lecture course cover much of PMA345.
  3. Online notes of Paul Garrett's Functional Analysis course at the University of Minnesota. This would appear to be a (long) graduate course and certainly it goes far beyond PMA345. However, some of the chapters, notably Basics about Banach space and the first bit of Basics about Hilbert spaces look useful.
  4. For revision of some elementary real analysis:
  5. "Interactive Real Analysis" by Bert G. Wachsmuth, Dept. of Math and Computer Science, Seton Hall University, New Jersey, USA.
  6. For revision of linear algebra: "Linear Algebra" by Jim Hefferon, Dept. of Mathematics, Saint Michael's College, Colchester, Vermont, USA.
  7. Brian Davies's Modern Analysis Online page has some interesting links to lecture notes by a variety of authors.

Notes and problem sheets

Follow these links for the complete notes and problem sheets.   A statement about non-examinable sections of the course appears after Problem Sheet 7.

Lecturer's Office Hours, Telephone, E-mail.

Office hours: Tuesdays 13.00-14.00 and Thursdays 12.00-13.00 during the teaching semester up to Easter; my office is K22.
Telephone extension 23775 (prefix by 22 from outside the University and further prefix by 0114 from outside Sheffield).
E-mail: P.Dixon@sheffield.ac.uk

Past Examination Papers

The course was not given in 2000-01 and 2002-03.

The past papers for June 2002 are available in pdf and in postscript form. The solutions are available in pdf form here.

The past papers for June 2000 are available in pdf and in postscript form. The solutions are available in pdf form here.


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