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Linearization of certain uniform homeomorphisms

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  09 April 2009

Anthony Weston
Anthony Weston
Affiliation:
Department of Mathematics & StatisticsCanisius CollegeBuffalo, NY 14208USA e-mail: [email protected]
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Abstract

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This article concerns the uniform classification of infinite dimensional real topological vector spaces. We examine a recently isolated linearization procedure for uniform homeomorphisms of the form φ: X →Y, where X is a Banach space with non-trivial type and Y is any topological vector space. For such a uniform homeomorphism φ, we show that Y must be normable and have the same supremal type as X. That Y is normable generalizes theorems of Bessaga and Enflo. This aspect of the theory determines new examples of uniform non-equivalence. That supremal type is a uniform invariant for Banach spaces is essentially due to Ribe. Our linearization approach gives an interesting new proof of Ribe's result.

MSC classification

Secondary: 46B20: Geometry and structure of normed linear spaces 46B04: Isometric theory of Banach spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 82 , Issue 1 , February 2007 , pp. 1 - 9
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Aharoni, I., ‘Uniform embeddings of Banach spaces’, Israel J. Math. 27 (1977), 174–179.CrossRefGoogle Scholar
[2]Aharoni, I., Maurey, B. and Mitjagin, B. S., ‘Uniform embeddings of metric spaces and of Banach spaces into Hilbert spaces’, Israel J. Math. 52 (1985), 251–265.CrossRefGoogle Scholar
[3]Bessaga, C., ‘On topological classification of complete linear metric spaces’, Fund. Math. 56 (1965), 251–288.Google Scholar
[4]Bourgain, J., Milman, V. and Wolfson, H., ‘On type of metric spaces’, Trans. Amer. Math. Soc. 294 (1986), 295–317.CrossRefGoogle Scholar
[5]Enflo, P., ‘On the nonexistence of uniform homeomorphisms between L p-spaces’, Ark. Mat. 8 (1968), 103–105.CrossRefGoogle Scholar
[6]Enflo, P., ‘Uniform structures and square roots in topological groups I’, Israel J. Math. 8 (1970), 230–252.CrossRefGoogle Scholar
[7]Enflo, P., ‘Uniform structures and square roots in topological groups II’, Israel. Math. 8 (1970), 253–272.CrossRefGoogle Scholar
[8]Fack, T., ‘Type and cotype inequalities for non commutative L p spaces’, J. Operator Theory 17 (1987), 255–279.Google Scholar
[9]Lennard, C. J., Tonge, A. M. and Weston, A., ‘Roundness and metric type’, J. Math. Anal. Appl. 252 (2000), 980–988.CrossRefGoogle Scholar
[10]Lindenstrauss, J., ‘On nonlinear projections in Banach spaces’, Michigan Math. J. 11 (1964), 263–287.CrossRefGoogle Scholar
[11]Pisier, G., Probabilistic methods in the geometry of Banach spaces Lecture Notes in Math. 1206 (Springer-Verlag, New York, 1986) pp. 167–241.Google Scholar
[12]Prassidis, E. and Weston, A., ‘Uniform Banach groups and structures’, C. R. Math. Acad. Sci. Canada 26 (2004), 25–32.Google Scholar
[13]Ribe, M., ‘On uniformly homeomorphic normed spaces’, Ark. Mat. 14 (1976), 237–244.CrossRefGoogle Scholar
[14]Rudin, W., Functional analysis, McGraw-Hill International Series in Pure and Applied Mathematics, 2nd edition (McGraw-Hill, New York, 1991).Google Scholar
[15]Sims, B., “Ultra”-techniques in Banach space theory, Queen's Papers in Pure and Applied Mathematics 60 (Queens University, Kingston, 1982).Google Scholar
[16]Yost, D., ‘If every doughnut is a teacup, then every Banach space is a Hilbert space’, Seminar on Functional Analysis, Universidad de Murica, Notas de Matematica 1 (1987), 125–148.Google Scholar