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The Size of the Unit Sphere

Published online by Cambridge University Press:  20 November 2018

Robert Whitley
Robert Whitley*
Affiliation:
University of Maryland, College Park, Md.
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Banach (1, pp. 242-243) defines, for two Banach spaces X and Y, a number (X, Y) = inf (log (‖L‖ ‖L-1‖)), where the infimum is taken over all isomorphisms L of X onto F. He says that the spaces X and Y are nearly isometric if (X, Y) = 0 and asks whether the concepts of near isometry and isometry are the same; in particular, whether the spaces c and c0, which are not isometric, are nearly isometric. In a recent paper (2) Michael Cambern shows not only that c and c0 are not nearly isometric but obtains the elegant result that for the class of Banach spaces of continuous functions vanishing at infinity on a first countable locally compact Hausdorff space, the notions of isometry and near isometry coincide.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 450 - 455
Copyright
Copyright © Canadian Mathematical Society 1968

Footnotes

The author was supported by National Science Foundation grant GP 5424.

References

1. Banach, S., Opérations linéaires (New York, 1955).Google Scholar
2. Cambern, M., A generalized Banach-Stone theorem, Proc. Amer. Math. Soc, 17 (1966), 396–400.Google Scholar
3. Dunford, N. and Schwartz, J., Linear operators, Vol. I (New York, 1958).Google Scholar
4. Gillman, L. and Jerison, M., Rings of continuous functions (New York, 1960).10.1007/978-1-4615-7819-2CrossRefGoogle Scholar
5. Taylor, A., Introduction to functional analysis (New York, 1958).Google Scholar