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

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**Ricky****Moderator**- Registered: 2005-12-04
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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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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

Ricky wrote:

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

I think Im 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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**Ricky****Moderator**- Registered: 2005-12-04
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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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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

So the norms for the above examples would be

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

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

Arent norms defined only on vector spaces? In that case, any metric space that isnt a vector space wont 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.

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