Math 328
Real Analysis
Winter 2010
Instructor
Roland Speicher
Office: Jeffery Hall, Room 506
Telephone: 5332388
Email: speicher@mast.queensu.ca
Office hour: Wed 12:301:30 and by appointment
The lectures are in Jeffery Hall, Room 116
time slot: 21

Monday, 2:303:30 pm

Tuesday, 4:305:30 pm

Thursday, 3:304:30 pm
Announcements
I have marked the final exam. If you want to know your result, send me
an email. You can take a look on your exam if you come to my office,
either by appointment or on Wednesday, May 5, 10am12noon.
Prerequisites
You should have had some introduction to real analysis
(dealing with topology and convergence properties of
R^n and continuous functions with respect to supnorm),
like Math 281.
In particular, you are expected to have some fluency in
the language and methods of mathematics, as, for example,
summarized in Chapter 1 of the book of Davidson/Donsig.
Here
are also a few handwritten notes by myself on this.
Assignements
Assignment 1 (due January 21)
Assignment 2 (due February 8)
Assignment 3 (due March 1)
Assignment 4 (due March 23)
Assignment 5 (due April 6)
Topics of Course

Sets and Cardinality

Properties of R: Completeness and BolzanoWeierstrass

Properties of C[0,1]: supnorm and Completeness

Compactness in complete normed vector spaces: HeineBorel versus
ArzelaAscoli

Finite dimensional normed vector spaces

Inner product and Hilbert spaces

Fixed point theorems: contraction principle versus compactness

Metric spaces

Connected and path connected metric spaces

Compact metric spaces

Completeness of Hausdorff metric: construction of fractals

Completion of metric spaces and abstract integration

Sets of measure zero and interpreation of L^pspaces as function
spaces

The StoneWeierstrass Theorem

More on Hilbert spaces and Fourier Series

if time permits: differentiable functions in several variables, higher
derivatives and Taylor's Theorem, inverse function theorem
Text Materials
No book is required, but the following one is recommended as the the
course will be somewhat based on it

K.R.Davidson and A.P. Donsig: Real Analysis with Real Applications.
Prentice Hall
This book is available
online
Another useful book might be:

J.E. Marsden and M.J. Hoffman: Elementary Classical Analysis.
Freeman
Marking Scheme
assignments: 50%
midterm: 10%
final exam: 40%