Math 328
Real Analysis
Winter 2010
Instructor
Roland Speicher
Office: Jeffery Hall, Room 506
Telephone: 533-2388
E-mail: speicher@mast.queensu.ca
Office hour: Wed 12:30-1:30 and by appointment
The lectures are in Jeffery Hall, Room 116
time slot: 21
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Monday, 2:30-3:30 pm
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Tuesday, 4:30-5:30 pm
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Thursday, 3:30-4:30 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, 10am-12noon.
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 sup-norm),
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
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Sets and Cardinality
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Properties of R: Completeness and Bolzano-Weierstrass
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Properties of C[0,1]: sup-norm and Completeness
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Compactness in complete normed vector spaces: Heine-Borel versus
Arzela-Ascoli
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Finite dimensional normed vector spaces
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Inner product and Hilbert spaces
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Fixed point theorems: contraction principle versus compactness
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Metric spaces
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Connected and path connected metric spaces
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Compact metric spaces
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Completeness of Hausdorff metric: construction of fractals
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Completion of metric spaces and abstract integration
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Sets of measure zero and interpreation of L^p-spaces as function
spaces
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The Stone-Weierstrass Theorem
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More on Hilbert spaces and Fourier Series
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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
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K.R.Davidson and A.P. Donsig: Real Analysis with Real Applications.
Prentice Hall
This book is available
online
Another useful book might be:
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J.E. Marsden and M.J. Hoffman: Elementary Classical Analysis.
Freeman
Marking Scheme
assignments: 50%
midterm: 10%
final exam: 40%