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ON A CHARACTERISTIC PROPERTY OF FINITE-DIMENSIONAL BANACH SPACES*

Published online by Cambridge University Press:  10 March 2011

ANTONÍN SLAVÍK
ANTONÍN SLAVÍK*
Affiliation:
Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic e-mail: slavik@karlin.mff.cuni.cz
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Abstract

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This paper is inspired by a counter example of J. Kurzweil published in [5], whose intention was to demonstrate that a certain property of linear operators on finite-dimensional spaces need not be preserved in infinite dimension. We obtain a stronger result, which says that no infinite-dimensional Banach space can have the given property. Along the way, we will also derive an interesting proposition related to Dvoretzky's theorem.

Keywords

47A3047A6346B0715A45
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 3 , September 2011 , pp. 443 - 449
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

REFERENCES

1.Albiac, F. and Kalton, N. J., Topics in Banach space theory (Springer, New York, NY, 2006).Google Scholar
2.Dvoretzky, A., Some results on convex bodies and Banach spaces, in Proc. Int. Symp. Linear Spaces (Jerusalem, 1960) (Jerusalem Academic Press, Jerusalem, 1961) 123–160.Google Scholar
3.Jarník, J. and Kurzweil, J., A general form of the product integral and linear ordinary differential equations, Czech. Math. J. 37 (1987), 642–659.Google Scholar
4.Kadets, M. I. and Snobar, M. G., Some functionals over a compact Minkowski space, Math. Notes 10 (1971), 694–696 (translated to English from Mat. Zametki 10 (1971), 453–457).CrossRefGoogle Scholar
5.Schwabik, Š., The Perron product integral and generalized linear differential equations, Časopis Pěst. Mat. 115 (1990), 368–404.CrossRefGoogle Scholar
6.Slavík, A. and Schwabik, Š., Henstock-Kurzweil and McShane product integration; descriptive definitions, Czech. Math. J. 58 (2008), 241–269.CrossRefGoogle Scholar