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I have the following question: does the Lorentz spaces $\Lambda(w,1)$ have nontrivial cotype and admit an unconditional basis(or even a symmetric basis)?Thank you.

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I think that your question about cotype is answered in S. Reisner, Ann. Inst. Fourier, Grenoble 31 (1981), 239-255 (see Theorem 2).

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