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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. – t.b. Nov 10 '11 at 18:21
    
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, chapter 7. – t.b. Nov 10 '11 at 18:29
    
Thank you, @t.b. – Walter A. Nov 10 '11 at 19:58

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