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Proceedings of the American Mathematical Society

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Hyperfinite-dimensional subspaces of the nonstandard hull of $ c\sb{0}$

Author: L. C. Moore
Journal: Proc. Amer. Math. Soc. 80 (1980), 597-603
MSC: Primary 46B99; Secondary 03H05
MathSciNet review: 587935
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Abstract: Let $ {\hat c_0}$ be the nonstandard hull of the Banach space $ {c_0}$ formed with respect to an $ {\aleph _1}$-saturated extension. Then $ {\hat c_0}$ is not isometrically isomorphic to any hyperfinite-dimensional subspace of $ {\hat c_0}$ and hence not to any hyperfinite-dimensional Banach space. This gives a negative answer to the question posed by Ward Henson: ``Does every Banach space have a nonstandard hull which is isometrically isomorphic to a hyperfinite-dimensional Banach space?'' As a consequence of the result, no ultrapower of $ {c_0}$ is isometrically isomorphic to an ultraproduct of finite-dimensional Banach spaces.

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Keywords: Nonstandard hulls, ultraproducts, ultrapowers
Article copyright: © Copyright 1980 American Mathematical Society