Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

An infinite-dimensional pre-Hilbert space not homeomorphic to its own square


Author: Roman Pol
Journal: Proc. Amer. Math. Soc. 90 (1984), 450-454
MSC: Primary 57N20; Secondary 46C99, 54F45
DOI: https://doi.org/10.1090/S0002-9939-1984-0728367-6
MathSciNet review: 728367
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Given an arbitrary infinite-dimensional separable complete linear metric space $ X$, there exists a direct sum decomposition $ X = {V_0} \oplus {V_1}$ such that each summand $ {V_i}$ intersects every linearly independent Cantor set in $ X$ (this decomposition can be considered as a linear analogue to the classical Bernstein's decomposition into totally imperfect sets).

Theorem. Each summand $ V$ of such a decomposition is not homeomorphic to its own square, and if $ T:V \to V$ is a linear bounded operator, then either the kernel or the range of $ T$ is finite-dimensional.

In the case of $ X = {l_2}$ this provides an example of a space $ V$ with the properties stated in the title, which answers a well-known question, cf. Arhangelskiĭ [A, Problem 21] and Geoghegan [G, Problem (LS 12)].


References [Enhancements On Off] (What's this?)

  • [A] A. V. Arhangelskiĭ, Structure and classification of topological spaces, Uspehi Mat. Nauk 33 (1978), 29-84. MR 526012 (80i:54005)
  • [B-P] C. Bessaga and A. Pełczyński, Selected topics in infinite-dimensional topology, PWN, Warsaw, 1975.
  • [Ch] J. P. R. Christensen, Topology and Borel structure, North-Holland, Amsterdam, 1974. MR 0348724 (50:1221)
  • [D] C. Dellacherie, Un cours sur les ensembles analytiques, Analytic Sets, Academic Press, London, 1980.
  • [G] R. Geoghegan, Open problems in infinite-dimensional topology, Topology Proc. 4 (1979). MR 583711 (82a:57015)
  • [K$ _{1}$] K. Kuratowski, Topology, Vols. I and II, PWN, Warsaw, 1966 and 1968. MR 0217751 (36:840)
  • [K$ _{2}$] -, Applications of the Baire category method to the problem of independent sets, Fund. Math. 81 (1973), 65-72. MR 0339092 (49:3855)
  • [M$ _{1}$] J. Mycielski, Independent sets in topological algebras, Fund. Math. 55 (1964), 139-147. MR 0173645 (30:3855)
  • [M$ _{2}$] -, Almost every function is independent, Fund. Math. 81 (1973), 43-48. MR 0339091 (49:3854)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 57N20, 46C99, 54F45

Retrieve articles in all journals with MSC: 57N20, 46C99, 54F45


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1984-0728367-6
Keywords: Linear metric spaces, homeomorphisms, Bernstein's sets
Article copyright: © Copyright 1984 American Mathematical Society

American Mathematical Society