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Spheres in infinite-dimensional normed spaces are Lipschitz contractible
| Content Provider | Semantic Scholar |
|---|---|
| Author | Benyamini, Yoav Sternfeld, Yaki |
| Copyright Year | 1983 |
| Abstract | Let X be an infinite-dimensional normed space. We prove the following: (i) The unit sphere {x E X: II x II = I) is Lipschitz contractible. (ii) There is a Lipschitz retraction from the unit ball of X onto the unit sphere. (iii) There is a Lipschitz map T of the unit ball into itself without an approximate fixed point, i.e. inf II x TxI 1: 11 x 11 ) I > 0. Introduction. Let X be a normed space, and let Bx= {x E X: Ix I 1} and Sx = {x E X: 1I x II = 1 } be its unit ball and unit sphere, respectively. Brouwer's fixed point theorem states that when X is finite dimensional, every continuous self-map of Bx admits a fixed point. Two equivalent formulations of this theorem are the following. 1. There is no continuous retraction from Bx onto Sx. 2. Sx is not contractible, i.e., the identity map on Sx is not homotopic to a constant map. It is well known that none of these three theorems hold in infinite-dimensional spaces (see e.g. [1]). The natural generalization to infinite-dimensional spaces, however, would seem to require the maps to be uniformly-continuous and not merely continuous. Indeed in the finite-dimensional case this condition is automatically satisfied. In this article we show that the above three theorems fail, in the infinite-dimensional case, even under the strongest uniform-continuity condition, namely, for maps satisfying a Lipschitz condition. More precisely, we prove THEOREM. Let X be an infinite-dimensional normed space. Then (1) The unit sphere Sx is Lipschitz contractible. (2) There is a Lipschitz retraction from Bx onto Sx. (3) There is a Lipschitz map T: Bx -Bx without an approximate fixed point, i.e. inf{IIx TxII: x E Bx} = d > 0. The first study of Lipschitz maps without approximate fixed points, and Lipschitz retractions from Bx onto Sx, was done by K. Goebel [3]. B. Nowak [5] proved the theorem for several classical Banach spaces. Our work was greatly influenced by the work of Nowak. Actually, the general scheme of the proof as well as two of the three Received by the editors November 1, 1982. 1980 Mathematics Subject Classification. Primary 47H 10, 46B20. 1 The first author was partially supported by the Fund for the Promotion of Research at the Technion -Israel Institute of Technology. 01983 American Mathematical Society 0002-9939/82/0000-1337/$02.50 |
| Starting Page | 439 |
| Ending Page | 445 |
| Page Count | 7 |
| File Format | PDF HTM / HTML |
| DOI | 10.1090/S0002-9939-1983-0699410-7 |
| Volume Number | 88 |
| Alternate Webpage(s) | http://www.ams.org/journals/proc/1983-088-03/S0002-9939-1983-0699410-7/S0002-9939-1983-0699410-7.pdf |
| Alternate Webpage(s) | https://doi.org/10.1090/S0002-9939-1983-0699410-7 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
