Prof. N. Kajino, Maßtheorie und Funktionalanalysis

SS 2012

Themen des Proseminars ,,Maßtheorie und Funktionalanalysis“
1. Stone-Weierstrass theorem and its applications [4, 6, 7, 17, 20]
2. Baire’s category theorem and its applications [3, 20, 24, 27, 28]
3. Lp -spaces and their duals [1, 3, 6, 19, 24]
4. Weak topologies on Banach spaces and applications [3, 4, 20, 28]
5. Fixed point theorems and their applications [4, 7, 9, 27]
6. Pointwise convergence and non-convergence of Fourier series [5, 21, 29]
7. Hausdorff measure and Hausdorff dimension with applications to fractal geometry [8, 12, 15, 23]
8. Linear functionals on continuous functions and Riesz(-Markov-Kakutani) representation theorem [4, 13, 19, 24]
 Advanced related topic: Daniell-Stone integrals and their applications [6]9. Absolutely continuous functions and Lebesgue’s differentiation theorem [6, 19,23]
10. Fourier transform and its applications to differential equations [5, 19, 20, 21, 29]
11. Sobolev spaces and Sobolev embedding theorem [1, 3, 9]
12. Harmonic functions and the Dirichlet problem for Laplace’s equation [2, 10, 11]
13. Optimal transport problem [14, 25, 26]

References[1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd ed., Pure and Applied
Math. 140, Academic Press / Elsevier, Amsterdam, 2003.
[2] R. F. Bass, Probabilistic Techniques in Analysis, Springer-Verlag, New York Berlin - Heidelberg, 1995.
[3] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York - Dordrecht - Heidelberg - London, 2011.
[4] J. B. Conway, A Course in Functional Analysis, 2nd ed., Graduate Texts in Math.
96, Springer-Verlag, New York - Berlin - Heidelberg, 1990.
[5] A. Deitmar, A First Course in Harmonic Analysis, 2nd ed., Springer-Verlag, NewYork - Berlin - Heidelberg, 2005.
1[6] R. M. Dudley, Real Analysis and Probability, Cambridge studies in advanced
math. 74, Cambridge University Press, Cambridge, 2002.
[7] K. Evers, Mengentheoretische Topologie, 2011.
http://www.math.uni-rostock.de/~evers/Topologie/top.pdf[8] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, 2nded., Wiley, Chichester, 2003.
[9] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second
Order, 2nd ed., Grundlehren der mathematischen Wissenschaften 224, SpringerVerlag, New York - Berlin - Heidelberg, 1983.
[10] L. L. Helms, Potential Theory, Springer-Verlag, Dordrecht - Heidelberg - London- New York, 2009.
[11] F. John, Partial Differential Equations, 4th ed., Springer-Verlag, New York, 1982.
[12] J. Kigami, Analysis on Fractals, Cambridge Tracts in Math. 143, Cambridge University Press, Cambridge, 2001.
[13] S. Lang, Real and Functional Analysis, 3rd ed., Graduate Texts in Math. 142,
Springer-Verlag, New York - Berlin - Heidelberg, 1993.
[14] S. T. Rachev and L. Rüschendorf, Mass Transportation Problems — Volume I:
Theory, Springer-Verlag, New York - Berlin - Heidelberg, 1998.
[15] C. A. Rogers, Hausdorff Measures, Cambridge University Press, Cambridge,
1998.
[16] W. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, NewYork, 1976.
[17] W. Rudin, Analysis, 4. Auflage (German translation of [16]), Oldenbourg Wissenschaftsverlag, München, 2008.
[18] W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987.
[19] W. Rudin, Reele und Komplexe Analysis, 2. Auflage (German translation of [18]),
Oldenbourg Wissenschaftsverlag, München, 2009.
[20] W. Rudin, Functional Analysis, 2nd ed., McGraw-Hill, New York, 1991.
[21] E. M. Stein and R. Shakarchi, Princeton Lectures in Analysis — I Fourier Analysis: An Introduction, Princeton University Press, Princeton - Oxford, 2003.
[22] E. M. Stein and R. Shakarchi, Princeton Lectures in Analysis — II Complex Analysis, Princeton University Press, Princeton - Oxford, 2003.
[23] E. M. Stein and R. Shakarchi, Princeton Lectures in Analysis — III Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press,Princeton - Oxford, 2005.
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[24] E. M. Stein and R. Shakarchi, Princeton Lectures in Analysis — IV Functional
Analysis: Introduction to Further Topics in Analysis, Princeton University Press,Princeton - Oxford, 2011.
[25] C. Villani, Topics in Optimal Transportation, Graduate Studies in Math. 58,
American Mathematical Society, 2003.
[26] C. Villani, Optimal Transport: Old and New, Grundlehren der mathematischenWissenschaften 338, Springer-Verlag, New York - Berlin - Heidelberg, 2009.
[27] D. Werner, Funktionalanalysis, 7. Auflage, Springer-Verlag, New York - Berlin Heidelberg, 2011.
[28] K. Yosida, Functional Analysis, 6th ed., Grundlehren der mathematischen Wissenschaften 123, Springer-Verlag, New York - Berlin - Heidelberg, 1980.
[29] A. Zygmund, Trigonometric Series, 3rd ed., Cambridge University Press, Cambridge, 2002.

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