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You are not logged in. #1 20090211 09:41:55#2 20090211 10:27:25
Re: Metrizable topological spacesIn 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 20090211 12:44:31
Re: Metrizable topological spaces
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 20090211 13:59:29
Re: Metrizable topological spacesEvery 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 20090211 14:15:55
Re: Metrizable topological spacesSo the norms for the above examples would be #6 20090211 14:46:13
Re: Metrizable topological spacesCorrect. 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 20090211 22:17:27
Re: Metrizable topological spacesAren’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. 