Topological equivalence of discontinuous norms |
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| Authors: | Jan J. Dijkstra Jan van Mill |
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| Affiliation: | (1) Department of Mathematics, The University of Alabama, Box 870350, 35487-0350 Tuscaloosa, Alabama, USA;(2) Present address: Divisie der Wiskunde en Informatica, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands;(3) Divisie der Wiskunde en Informatica, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands |
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| Abstract: | We show that for everyp>0 there is an autohomeomorphismh of the countable infinite product of linesR Nsuch that for everyr>0,h maps the Hilbert cube [−r, r] N precisely onto the “elliptic cube”  . This means that the supremum norm and, for instance, the Hilbert norm (p=2) are topologically indistinguishable as functions onR N.The result is obtained by means of the Bing Shrinking Criterion. Research supported in part by a grant from NSF-EPSCoR Alabama. |
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