Course Description
This is the second part of a graduate course on an introduction to
functional analysis.
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Topics
- Review of fundamental theorems for Banach spaces. Selected applications:
Numerical integration, convergence of Fourier series,
Bramble-Hilbert Lemma,
Lax-Milgram Theorem, Negative norms, fixed-point theorems.
- Locally convex topological spaces. Weak topologies and weak convergence. Selected applications: Dunford-Pettis Theorem on L1 weak compactness,
weak lower semicontinuity, Young measures.
- Distributions with applications to linear partial differential equations.
- Gaussian measures on Banach spaces: basic concepts, stochastic integrals, and nonlinear transformations.
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Textbook and References
For the standard topics, we will follow closely the textbook
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J. B. Conway, A Course in Functional Analysis, 2nd ed., Springer, 1990.
We will also occasionally consult the following references:
- J. Barros-Neto, An Introduction to the Theory of Distributions,
Reprint ed., Krieger, 1981.
- V. I. Bogachev, Gaussian Measures, Amer. Math. Soc., 1998.
- N. Dunford and J. T. Schwartz,
Linear Operators. Part I. General Theory, Wiley-Interscience, 1958.
- R. E. Edwards, Functional Analysis, Dover, 1994.
- W. Rudin, Functional Analysis, 2nd ed., McGraw-Hill, 1991.
- H. H. Schaefer and M. P. Wolff, Topological Vector Spaces,, 2nd ed., Springer, 1999.
- K. Yosida, Functional Analysis,, 6th ed., Springer, 2003.
Lecture notes on special topics and applications will be distributed in the class.
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Homework There will be a few homework assignments.
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Office Hours
By appointment.
Lecture Notes
Last updated by Bo Li on April 4, 2012.
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