×

A rigid space homeomorphic to Hilbert space. (English) Zbl 0889.46004

Summary: 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 \(\ell_{2}\).

MSC:

46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
54F65 Topological characterizations of particular spaces
54G15 Pathological topological spaces
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Czesław Bessaga and Aleksander Pełczyński, Selected topics in infinite-dimensional topology, PWN — Polish Scientific Publishers, Warsaw, 1975. Monografie Matematyczne, Tom 58. [Mathematical Monographs, Vol. 58]. · Zbl 0304.57001
[2] Tadeusz Dobrowolski and Henryk Toruńczyk, On metric linear spaces homeomorphic to \?\(_{2}\) and compact convex sets homeomorphic to \?, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 11-12, 883 – 887 (1981) (English, with Russian summary). · Zbl 0455.46001
[3] James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. · Zbl 0144.21501
[4] N. J. Kalton, N. T. Peck, and James W. Roberts, An \?-space sampler, London Mathematical Society Lecture Note Series, vol. 89, Cambridge University Press, Cambridge, 1984. · Zbl 0556.46002
[5] N. J. Kalton and James W. Roberts, A rigid subspace of \?\(_{0}\), Trans. Amer. Math. Soc. 266 (1981), no. 2, 645 – 654. · Zbl 0484.46004
[6] Nguyen To Nhu, Investigating the ANR-property of metric spaces, Fund. Math. 124 (1984), no. 3, 243 – 254. Nguyen To Nhu, Corrections to: ”Investigating the ANR-property of metric spaces” [Fund. Math. 124 (1984), no. 3, 243 – 254; MR0774515 (86d:54018)], Fund. Math. 141 (1992), no. 3, 297. · Zbl 0573.54009
[7] Nguyen To Nhu and Katsuro Sakai, The compact neighborhood extension property and local equi-connectedness, Proc. Amer. Math. Soc. 121 (1994), no. 1, 259 – 265. · Zbl 0809.54014
[8] Nguyen To Nhu and Ta Khac Cu, Probability measure functors preserving the ANR-property of metric spaces, Proc. Amer. Math. Soc. 106 (1989), no. 2, 493 – 501. · Zbl 0719.54023
[9] Paul Sisson, A rigid space admitting compact operators, Studia Math. 112 (1995), no. 3, 213 – 228. · Zbl 0834.46005
[10] Angus E. Taylor and W. Robert Mann, Advanced calculus, 3rd ed., John Wiley & Sons, Inc., New York, 1983. · Zbl 0584.26001
[11] L. Waelbroeck, A rigid topological vector space, Studia Math. 59 (1976/77), no. 3, 227 – 234. · Zbl 0344.46008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.