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Invertibility in infinite-dimensional spaces


Authors: Chia-Chuan Tseng and Ngai-Ching Wong
Journal: Proc. Amer. Math. Soc. 128 (2000), 573-581
MSC (1991): Primary 46B20, 57N20, 57N50
DOI: https://doi.org/10.1090/S0002-9939-99-05076-5
Published electronically: July 6, 1999
MathSciNet review: 1628416
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Abstract: An interesting result of Doyle and Hocking states that a topological $n$-manifold is invertible if and only if it is a homeomorphic image of the $n$-sphere $S^n$. We shall prove that the sphere of any infinite-dimensional normed space is invertible. We shall also discuss the invertibility of other infinite-dimensional objects as well as an infinite-dimensional version of the Doyle-Hocking theorem.


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

  • 1. C. Bessaga and T. Dobrowolski, Affine and homeomorphic embedding into $\ell^2$, Proc. Amer. Math. Soc. 125 (1997), 259-268. MR 97e:57022
  • 2. C. Bessaga and V. L. Klee, Two topological properties of topological linear spaces, Israel J. Math. 2 (1964), 211-220. MR 31:5055
  • 3. C. Bessaga and A. Pe{\l}czy\'{n}ski, Selected Topics in infinite-dimensional topology, Polish Scientific Publishers, Warszawa, 1975. MR 57:17657
  • 4. M. Brown, A proof of the generalized Schoenflies theorem, Bull. Amer. Math. Soc. 66 (1960), 74-76. MR 22:8470b
  • 5. H. Corson and V. Klee, Topological classification of convex sets, Proc. Symp. Pure Math. 7 - Convexity, Amer. Math. Soc., Providence, R. I., 1963, 37-51. MR 28:4328
  • 6. T. Dobrowolski and H. Toru\'{n}czyk, Separable complete ANR's admitting a group structure are Hilbert manifolds, Topology and its Applications 12 (1981), 229-235. MR 83a:58007
  • 7. T. Dobrowolski, Every infinite-dimensional Hilbert space is real-analytically isomorphic with its unit sphere, J. Funct. Analy. 134 (1995), 350-362. MR 96m:46030
  • 8. P. H. Doyle and J. G. Hocking, A characterization of Euclidean n-spaces, Mich. Math. J., 7 (1960), 199-200. MR 22:12515
  • 9. -, Invertible spaces, Amer. Math. Monthly, 68 (1961), 959-965. MR 24:A1711
  • 10. W. J. Gray, On the metrizability of invertible spaces, Amer. Math. Monthly 71 (1964), 533-534. MR 28:5424
  • 11. D. W. Henderson, Open subsets of Hilbert space, Compositio Math. 21 (1969), 312-318. MR 40:4975
  • 12. -, Infinite-dimensional manifolds are open subsets of Hilbert space, Topology 9 (1970), 25-33. MR 40:3581
  • 13. S. K. Hildebrand and R. L. Poe, The separation axioms for invertible spaces, Amer. Math. Monthly 75 (1968), 391-392. MR 37:2170
  • 14. V. L. Klee, Topological equivalence of a Banach space with its unit cell, Bull. Amer. Math. Soc. 67 (1961), 286-290. MR 23:A2733
  • 15. N. Levine, Some remarks on invertible spaces, Amer. Math. Monthly 70 (1963), 181-183. MR 26:4322
  • 16. P. E. Long, L. L. Herrington, and D. S. Jankovic, Almost-invertible spaces, Bull. Korean Math. Soc. 23 (1986), 91-102. MR 88k:54041
  • 17. J. van Mill, Domain invariance in infinite-dimensional linear spaces, Proc. Amer. Math. Soc. 101 (1987), 173-180. MR 88k:57023
  • 18. S. A. Naimpally, Function spaces of invertible spaces, Amer. Math. Monthly 73 (1966), 513-515. MR 33:3269
  • 19. H. Toru\'{n}czyk, Characterizing Hilbert space topology, Fund. Math. CXI (1981), 247-262. MR 82i:57016
  • 20. -, A correction of two papers concerning Hilbert manifolds, Fund. Math. CXXV (1985), 89-93. MR 87m:57017

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Additional Information

Chia-Chuan Tseng
Affiliation: Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan, Republic of China

Ngai-Ching Wong
Affiliation: Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan, Republic of China
Email: wong@math.nsysu.edu.tw

DOI: https://doi.org/10.1090/S0002-9939-99-05076-5
Keywords: Invertible spaces, spheres, infinite-dimensional topology, infinite-dimensional manifolds
Received by editor(s): June 20, 1997
Received by editor(s) in revised form: April 14, 1998
Published electronically: July 6, 1999
Additional Notes: This work was partially supported by the National Science Council of Republic of China. Grant Number: NSC 83-0208-M-110-0171, 87-2115-M-110-002.
Communicated by: Dale Alspach
Article copyright: © Copyright 1999 American Mathematical Society

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