From: bruck@math.usc.edu (Ronald Bruck)
Newsgroups: sci.math
Subject: Re: Question on equivalent metrics
Date: 11 Oct 1998 12:39:11 -0700

In article <6vqvue$c81$1@nnrp1.dejanews.com>,  <schmera@spawar.navy.mil> wrote:
:I have a simple question on equivalent metric topologies:
:
:Are the following two definitions equivalent?
:
:Let X be a nonempty set with two metrics d1 and d2.
:
:Def1: d1 and d2 are equivalent if there exist c and C pos. real numbers
:         s.t.    c*d1(x, y)  <=  d2(x,y)  <=  C*d1(x, y)  for all  x, y in X.
:
:Def2: d1 and d2 are equivalent if they generate the same topology.
:
:(I mean the standard metric topology based on open balls.)
:
:It is clear to me that Def1 => Def2.
:
:What about the other way?
:Does Def2 => Def1? How does one find c and C?

No.  Consider the real line, with d1(x,y) = |x-y|; and 
d2(x,y) = sqrt(|x-y|).  Both are metrics, both generate the
same topology, yet neither is <= a constant times the other.
There are problems at infinity and at 0.

Of course, d2 is not induced from a Banach space norm.  Two
NORMS ||.||_1 and ||.||_2 induce the same topology iff they're
equivalent in the sense there exist c1, c2 such that

   c1 ||.||_1 <= ||.||_2 <= c2 ||.||_1

(look at the unit ball in one norm, which must contain a ball of
some positive radius r > 0 around 0 in the other norm; i.e.

  ||z||_2 <= r ==> ||z||_1 <= 1,  for example.

Then scale.)

--Ron Bruck
==============================================================================

From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Newsgroups: sci.math
Subject: Re: Question on equivalent metrics
Date: 12 Oct 1998 16:12:50 -0400

In article <6vqvue$c81$1@nnrp1.dejanews.com>,  <schmera@spawar.navy.mil> wrote:
:I have a simple question on equivalent metric topologies:
:
:Are the following two definitions equivalent?
:
:Let X be a nonempty set with two metrics d1 and d2.
:
:Def1: d1 and d2 are equivalent if there exist c and C pos. real numbers
:         s.t.    c*d1(x, y)  <=  d2(x,y)  <=  C*d1(x, y)  for all  x, y
:in X.
:
:Def2: d1 and d2 are equivalent if they generate the same topology.
:
:(I mean the standard metric topology based on open balls.)
:
:It is clear to me that Def1 => Def2.
:
:What about the other way?
:Does Def2 => Def1? How does one find c and C?
:
:Any help is appreciated.
:I apologize if this question is too easy.
:

No need to apologize; it is a legitimate question (or exercise).

Test the definitions on the two following metrics on real numbers:

d1(x,y) = abs(x-y)

d2(x,y) = abs(x^3 - y^3)

Good luck, ZVK(Slavek).