Invertibility in infinite-dimensional spaces

Tseng, Chia-Chuan; Wong, Ngai-Ching

doi:10.1090/S0002-9939-99-05076-5

Invertibility in infinite-dimensional spaces
HTML articles powered by AMS MathViewer

by Chia-Chuan Tseng and Ngai-Ching Wong PDF

Proc. Amer. Math. Soc. 128 (2000), 573-581 Request permission

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

Czesław Bessaga and Tadeusz Dobrowolski, Affine and homeomorphic embeddings into $l^2$, Proc. Amer. Math. Soc. 125 (1997), no. 1, 259–268. MR 1389505, DOI 10.1090/S0002-9939-97-03832-X
Czeslaw Bessaga and Victor Klee, Two topological properties of topological linear spaces, Israel J. Math. 2 (1964), 211–220. MR 180825, DOI 10.1007/BF02759736
Czesław Bessaga and Aleksander Pełczyński, Selected topics in infinite-dimensional topology, Monografie Matematyczne, Tom 58. [Mathematical Monographs, Vol. 58], PWN—Polish Scientific Publishers, Warsaw, 1975. MR 0478168
Marston Morse, A reduction of the Schoenflies extension problem, Bull. Amer. Math. Soc. 66 (1960), 113–115. MR 117694, DOI 10.1090/S0002-9904-1960-10420-X
Harry Corson and Victor Klee, Topological classification of convex sets, Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, R.I., 1963, pp. 37–51. MR 0161119
T. Dobrowolski and H. Toruńczyk, Separable complete ANRs admitting a group structure are Hilbert manifolds, Topology Appl. 12 (1981), no. 3, 229–235. MR 623731, DOI 10.1016/0166-8641(81)90001-8
Tadeusz Dobrowolski, Every infinite-dimensional Hilbert space is real-analytically isomorphic with its unit sphere, J. Funct. Anal. 134 (1995), no. 2, 350–362. MR 1363804, DOI 10.1006/jfan.1995.1149
P. H. Doyle and J. G. Hocking, A characterization of Euclidean $n$-space, Michigan Math. J. 7 (1960), 199–200. MR 121781
P. H. Doyle and J. G. Hocking, Invertible spaces, Amer. Math. Monthly 68 (1961), 959–965. MR 131864, DOI 10.2307/2311802
William J. Gray, On the metrizability of invertible spaces, Amer. Math. Monthly 71 (1964), 533–534. MR 162225, DOI 10.2307/2312594
David W. Henderson, Open subsets of Hilbert space, Compositio Math. 21 (1969), 312–318. MR 251748
David W. Henderson, Infinite-dimensional manifolds are open subsets of Hilbert space, Topology 9 (1970), 25–33. MR 250342, DOI 10.1016/0040-9383(70)90046-7
S. K. Hildebrand and R. L. Poe, The separation axioms for invertible spaces, Amer. Math. Monthly 75 (1968), 391–392. MR 226581, DOI 10.2307/2313431
Victor Klee, Topological equivalence of a Banach space with its unit cell, Bull. Amer. Math. Soc. 67 (1961), 286–290. MR 125431, DOI 10.1090/S0002-9904-1961-10589-2
Norman Levine, Some remarks on invertible spaces, Amer. Math. Monthly 70 (1963), 181–183. MR 146802, DOI 10.2307/2312889
Paul E. Long, Larry L. Herrington, and Dragan S. Janković, Almost-invertible spaces, Bull. Korean Math. Soc. 23 (1986), no. 2, 91–102. MR 888219
Jan van Mill, Domain invariance in infinite-dimensional linear spaces, Proc. Amer. Math. Soc. 101 (1987), no. 1, 173–180. MR 897091, DOI 10.1090/S0002-9939-1987-0897091-4
S. A. Naimpally, Function spaces of invertible spaces, Amer. Math. Monthly 73 (1966), 513–515. MR 195064, DOI 10.2307/2315476
H. Toruńczyk, Characterizing Hilbert space topology, Fund. Math. 111 (1981), no. 3, 247–262. MR 611763, DOI 10.4064/fm-111-3-247-262
H. Toruńczyk, A correction of two papers concerning Hilbert manifolds: “Concerning locally homotopy negligible sets and characterization of $l_2$-manifolds” [Fund. Math. 101 (1978), no. 2, 93–110; MR0518344 (80g:57019)] and “Characterizing Hilbert space topology” [ibid. 111 (1981), no. 3, 247–262; MR0611763 (82i:57016)], Fund. Math. 125 (1985), no. 1, 89–93. MR 813992, DOI 10.4064/fm-125-1-89-93