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A rigid space homeomorphic to Hilbert space

Authors: Nguyen To Nhu and Paul Sisson
Journal: Proc. Amer. Math. Soc. 126 (1998), 85-95
MSC (1991): Primary 46A16, 54F65; Secondary 46C05, 54G15
MathSciNet review: 1459139
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Abstract: A rigid space is a topological vector space whose endomorphisms are all simply scalar multiples of the identity map. This is in sharp contrast to the behavior of operators on $\ell _{2}$, and so rigid spaces are, from the viewpoint of functional analysis, fundamentally different from Hilbert space. Nevertheless, we show in this paper that a rigid space can be constructed which is topologically homeomorphic to Hilbert space. We do this by demonstrating that the first complete rigid space can be modified slightly to be an AR-space (absolute retract), and thus by a theorem of Dobrowolski and Torunczyk is homeomorphic to $\ell _{2}$.

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Additional Information

Nguyen To Nhu
Affiliation: Institute of Mathematics at Hanoi, P.O. Box 631, Bo Ho, Hanoi, Vietnam
Address at time of publication: Department of Mathematics, New Mexico State University, Las Cruces, New Mexico 88003

Paul Sisson
Affiliation: Department of Mathematics, Louisiana State University - Shreveport, One University Place, Shreveport, Louisiana 71115

Received by editor(s): January 23, 1996
Communicated by: James West
Article copyright: © Copyright 1998 American Mathematical Society