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JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Metrizable topological spaces

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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.

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"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..."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Metrizable topological spaces

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?

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Metrizable topological spaces

Every norm defines a metric by:

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"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..."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Metrizable topological spaces

So the norms for the above examples would be

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?

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"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..."

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”.

Or how about

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.