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## #1 2009-02-10 10:41:55

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

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## #2 2009-02-10 11:27:25

Ricky
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Registered: 2005-12-04
Posts: 3,791

### Re: Metrizable topological spaces

In fact, all norms on Euclidean space are equivalent.  That is, they all define the same topology.  This fact is quite useful for topology because it means you can take whatever norm is easiest.

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #3 2009-02-10 13:44:31

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

### Re: Metrizable topological spaces

Ricky wrote:

In fact, all norms on Euclidean space are equivalent.  That is, they all define the same topology.

I think Im missing something here. I thought norm and metric were different things? A norm on a space
is a mapping
whereas a metric on
is a mapping
. Would you mind clarifying a little?

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## #4 2009-02-10 14:59:29

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

### Re: Metrizable topological spaces

Every norm defines a metric by:

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #5 2009-02-10 15:15:55

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

### Re: Metrizable topological spaces

So the norms for the above examples would be

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## #6 2009-02-10 15:46:13

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

### Re: Metrizable topological spaces

Correct.  And not every metric space has a norm.  Can you name one that doesn't?

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #7 2009-02-10 23:17:27

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

### Re: Metrizable topological spaces

Arent norms defined only on vector spaces? In that case, any metric space that isnt a vector space wont be normal.

with the discrete metric?

A norm on

has to satisfy the property that
for all scalars
and
. This fails for the discrete metric when
and
.

Discrete spaces are described by Sutherland as pathological examples of metric spaces.

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