A rigid space homeomorphic to Hilbert space

Nhu, Nguyen; Sisson, Paul

doi:10.1090/S0002-9939-98-04462-1

A rigid space homeomorphic to Hilbert space
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by Nguyen To Nhu and Paul Sisson

Proc. Amer. Math. Soc. 126 (1998), 85-95

DOI: https://doi.org/10.1090/S0002-9939-98-04462-1

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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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