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#1 2009-02-11 09:41:55
#2 2009-02-11 10:27:25
- Ricky
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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..."
#3 2009-02-11 12:44:31
- JaneFairfax
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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?
#4 2009-02-11 13:59:29
- Ricky
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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..."
#5 2009-02-11 14:15:55
- JaneFairfax
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Re: Metrizable topological spaces
So the norms for the above examples would be
#6 2009-02-11 14:46:13
- Ricky
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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..."
#7 2009-02-11 22:17:27
- JaneFairfax
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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.
