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Are there uncountably many not Lipschitz equivalent norms on the space of real sequences with only finitely many non-zero elements? Thanks. (If so, how might one find/construct them?) Thanks.

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    All norms is going to be hard... However, the $\ell^p$-norms $\|x\| = \left(\sum |x_n|^p\right)^{1/p}$ already provide you with an uncountable family of distinct, inequivalent norms.2011-11-10
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    My previous comment was written before you edited the body of your question to clarify the meaning of *"equivalent"*, but it applies to Lipschitz equivalence as well. See [Benyamini-Lindenstrauss](http://books.google.com/books?id=lXZ95EKwjYUC), chapter 7.2011-11-10

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