### Functional Analysis (MATH3402)

The University of Queensland (Semester 1, 2011)

Lecturers:

Weeks 1-7: Stephan Tillmann (Course Coordinator)
Weeks 8-13: Lorenz Schabrun
Tutor:

Weeks 1-5: Colin MacLaurin
Weeks 6-7: Stephan Tillmann
Weeks 8-13: Lorenz Schabrun

### about

This foundational course provides a basis for further studies in
modern analysis, geometry, topology, differential equations and
quantum mechanics. It introduces the idea of a metric space with a
general distance function, and the resulting concepts of convergence,
continuity, completeness, compactness and connectedness. The subject
also introduces Hilbert spaces: infinite dimensional vector spaces
(typically function spaces) equipped with an inner product that allows
geometric ideas to be used to study these spaces and linear maps
between them.

The course is the pre-requisite for Functional Analysis (MATH4404),
and a recommended pre-requisite for Advanced Analysis (MATH4401).

### pre-requisites:

Calculus: Bounded and monotone sequences. Sequences and series of real
functions. Intermediate and mean value theorems, iterative
procedures. Taylor's Theorem and error estimates. Criteria for
integrability. Vector functions, continuity and differentials. Implicit
and Inverse Function Theorems and their applications. Multiple integrals.

Linear Algebra: Vector spaces, bases, linear functions, image and kernel,
diagonalisation of matrices.

The formal prerequisites are: MATH2400 and (MATH2301 or MATH2000)

### outline

Week 1: Metric spaces (Contraction mapping principle)
Week 2: Complete metric spaces (Nesting principle, Compactness)
Week 3: Topological spaces (Hausdorff property, subspace topology, separation axioms)
Week 4: Continuous maps (Product topology, connectedness)
Week 5: Normal spaces (Urysohn's lemma and Tietze's theorem)
Week 6: Compact spaces (Heine-Borel theorem, Bolzano-Weierstrass property)
Week 7: Compactness theorems (Tychonoff and Arzela-Ascoli theorems)
Week 8: Inequalities
Week 9: Normed spaces (Completeness; bounded linear operators; operator norm; continuity)
Week 10: Euclidean and Hilbert spaces (parallelogram law/polarization identity; Riesz representation theorem)
Week 11: L^{p}-spaces
Week 12: Orthonormal systems (Gram-Schmidt; Bessel's and Parseval's identities)
Week 13: Applications
### learning activities

Tu 08:00-09:50 67-342 (lecture)
Tu 15:00-15:50 43-105 (tutorial)
We 12:00-12:50 67-342 (lecture)
### consultation hours

Tu 10:00-10:50 67-710
We 11:00-11:50 67-710
### problem sets (pdf)

The problem sets will be posted as the semester progresses.

Problem Set 1 (Metric spaces)
Problem Set 2 (Metric spaces)
Problem Set 3 (Topological spaces)
Problem Set 4 (Topological spaces)
Problem Set 5 (Product spaces)
Problem Set 6 (Compactness)
Problem Set 7 (Function spaces)
Problem Set 8 (Normed spaces)
Problem Set 9 (Normed spaces)
### assignments (pdf)

Each assignment will be given out in the lecture and posted here 8 days
before the due date. Partial solutions will be posted on blackboard.

Each assignment is worth 5% of the final mark.

Assignment 1 (due Thursday, 10 March)
Assignment 2 (due Thursday, 24 March)
Assignment 3 (due Thursday, 7 April)
Assignment 4 (due Thursday, 21 April)
Assignment 5 (due Thursday, 12 May)
Assignment 6 (due Thursday, 26 May)
### final exam

The final exam is worth 70% of the final mark.

22 June at 14:30 in 67-442
(10 min perusal; 120 min duration)
### resources

Here are the
lecture notes and
tutorial sheets
from previous years.

You may wish to use these tutorial sheets as extra
practice material. I suggest you only look at the worked solutions
once you've spent a considerable amount of time trying to solve them.
If you are stuck, you should first consult the lecture notes, ask a friend or the tutor for help or
explanations, or consult a book. Only after trying all of this, have a
look at the solutions provided.

### general references

The following books have been placed on reserve in the Dorothy Hill Physical Sciences & Engineering Library in Hawken (50):

"Linear Analysis" by Bela Bollobas (Cambridge, 1999) (has been requested)
"A course in functional analysis" by John Conway (Springer-Verlag, 1985)
"Real Analysis: Modern Techniques and Their Application" by Gerald
B. Folland (Wiley, 1999)
"Counterexamples in analysis" by Bernard R. Gelbaum and John M.H. Olmsted (Holden-Day, 1964)
"Introductory Functional Analysis with Applications" by Erwin Kreyszig (Wiley, 1989)
"Topology" by James R. Munkres (Prentice Hall, 2000)
"Foundations of topology" by Wayne Patty (Jones and Bartlett
Publishers, 2009)
"Introduction to metric and topological spaces" by
W. A. Sutherland (Oxford, 1975)