Topological Equivalence of Discontinuous Norms

We show that for every p > 0 there is an autohomeomorphism h of the countable infinite product of lines R N such that for every r > 0, h maps the Hilbert cube [ r , r ] N precisely onto the "elliptic cube" o~ {X ~ R N : ~ i = l I xi]p ~rP)" T h i s m e a n s that the supremum n o r m and, for instance, the Hilbert norm (p ---2) are topologically indistinguishable as functions on R N. The result is obtained by means of the Bing Shrinking Criterion. * Research s u p p o r t e d in pa r t by a g ran t f rom N S F E P S C o R A l a b a m a . ** Current address: Divisie der W i s k u n d e en Informat ica , Vrije Univers i te i t , De Boele laan 1081a, 1081 HV A m s t e r d a m , T he Nether lands . Received July 20, 2000