"uniformly equivalent" metric
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From: Graven Water on 31 Mar 2010 10:22 What does it mean for two metrics on a space to be uniformly equivalent? I found a statement in a book: "all metrics on a compact metrizable space are uniformly equivalent". I thought if d and e are two metrics on a space X, d and e are _uniformly equivalent_ if there are K and M such that d(x,y) <= K e(x,y) all x,y in X e(x,y) <= M d(x,y) all x,y in X. But then the book's statement doesn't seem to be true! Laura
From: John Coleman on 31 Mar 2010 11:18 On Mar 31, 10:22 am, p...(a)grex.org (Graven Water) wrote: > What does it mean for two metrics on a space to be uniformly equivalent? > For every \epsilon > 0 there exists a \delta > 0 such that d(x,y) < \delta => e(x,y) < \epsilon e(x,y) < \delta => d(x,y) < \epsilon > I found a statement in a book: "all metrics on a compact metrizable space > are uniformly equivalent". > > I thought if d and e are two metrics on a space X, d and e are > _uniformly equivalent_ if there are K and M such that > d(x,y) <= K e(x,y) all x,y in X > e(x,y) <= M d(x,y) all x,y in X. That sounds more like Lipschitz equivalence > But then the book's statement doesn't seem to be true! > > Laura
From: Graven Water on 31 Mar 2010 12:36 OK, I found out the following isn't the actual definition. Laura > I thought if d and e are two metrics on a space X, d and e are > _uniformly equivalent_ if there are K and M such that > d(x,y) <= K e(x,y) all x,y in X > e(x,y) <= M d(x,y) all x,y in X.
From: Dave L. Renfro on 31 Mar 2010 13:31
Graven Water (Laura?) wrote (in part): > What does it mean for two metrics on a space to be > uniformly equivalent? John Coleman has answered your question already, but if you're interested in more about this topic, I posted quite a bit on it back in 2006: "Lipschitz, uniformly, and topologically equivalent metrics" sci.math post made on 4 October 2006 http://groups.google.com/group/sci.math/msg/9e825cd2be094cd7 Dave L. Renfro |