Topological equivalence of discontinuous norms

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”$\left\{ {x \in R^N :\sum {_{i = 1}^\infty \left| {x_i } \right|^p \leqslant r^p } } \right\}$. 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.