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On completion of this couse, students have acquired a deep knowledge of functional analysis by autonomous studies. Moreover they are familiar with relevant mathematical literature and can judge its mathematical content. They are able to examine problems of functional analysis in a creative and methodically correct manner and to present the result of those examinations in written form and orally as to be understood well by experts.

The research seminar offers the possibility to present own research results, or explore and present  a current  topic of functional analysis.

This semester the focus lies on convexity properties of Banach spaces in connection with the existence of fixed points and the behavior of linear and metric projections.

Continuous assessment (based on regular written and oral contributions by the participants).

As applications of convexity and differentiability properties of Banach spaces e.g. parts of the following publications can be discussed.

• A. Pinkus. The alternating algorithm in a uniformly convex and uniformly smooth
• Banach space. J. Math. Anal. Appl., 421(1):747–753, 2015.
• F. Schöpfer, T. Schuster, and A. K. Louis. An iterative regularization
  method for the solution of the split feasibility problem in Banach spaces. Inverse
  Problems, 24(5):055008, 20, 2008.
• S. Reich and S. Sabach. A projection method for solving nonlinear
  problems in reflexive Banach spaces. J. Fixed Point Theory Appl., 9(1):101–116,
  2011.
• T. Schuster, B. Kaltenbacher, B. Hofmann, and K. S. Kazimierski.
  Regularization methods in Banach spaces. Walter de Gruyter GmbH & Co. KG, Berlin,
  2012.

In addition, the following classical characterisations of Hilbert spaces among Banach spaces can be presented:

• W. J. Stiles: Closest-point maps and their products. Nieuw Archiev voor Wiskunde (3), XIII, 19-29 (1965)
• R. A. Hirschfeld: On best approximation in normed vector spaces, II, Nieuw Archief voor Wieskunde (3), VI (1958), 99-107
• S. Kakutani: Some characterizations of Euclidean space, Jap. J. Math. vol. 16 (1939), 93-97

For the necessary background on convexity and differentiablity properties of Banach spaces, parts of the following text books will be used:

• F. Albiac and N. J. Kalton. Topics in Banach space theory. Springer, New York,  2017
• K. Goebel and S. Reich. Uniform convexity, hyperbolic geometry, and
  nonexpansive mappings. Marcel Dekker, Inc., New York, 1984.