Define, Norms have the same topology

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polymer
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Define, Norms have the same topology

Postby polymer » Sun May 27, 2012 7:00 pm UTC

I've been googling "norms have the same topology" with only vague descriptions. Suddenly my analysis professors started using this terminology. I'm pretty sure it means all sets open with respect to some norm are open with respect to all norms. Is this correct and is there more to it then that?

Dark Avorian
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Re: Define, Norms have the same topology

Postby Dark Avorian » Sun May 27, 2012 7:09 pm UTC

I'm pretty sure that just means that the canonical topology induced by the open balls using two given norms are identical. For example, all of the p-norm topolgies are equivalent to the canonical Euclidean norm topology.
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jestingrabbit
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Re: Define, Norms have the same topology

Postby jestingrabbit » Sun May 27, 2012 7:33 pm UTC

Dark Avorian wrote:I'm pretty sure that just means that the canonical topology induced by the open balls using two given norms are identical. For example, all of the p-norm topolgies are equivalent to the canonical Euclidean norm topology
on finite dimensional spaces.

A condition equivalent to this is that every ball in one norm has two balls in the other norm, one that contains it, and the other that is contained by it.
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