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If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell_p$ are not topologically isomorphic unless $p=2$.

Maybe I would have to use the Rademacher's functions.

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    A relevant MathOverflow question: http://mathoverflow.net/questions/79713/lp-mathbbr-vs-lq-mathbbr/79892#798922012-01-07
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    Ok, I edited it better.2012-01-07
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    Trivial remark: you certainly need to exclude $p=2$ since all separable Hilbert spaces are isomorphic.2012-01-07
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    You should formulate your questions more clearly. As formulated now, the question can be understood in two ways: A. Fix some $p\in\langle 1,\infty)$. Does it hold that $L^p$ and $\ell_p$ are isomorphic? B. Choose $p,q\in\langle 1,\infty)$, $p\ne q$. Are $\ell_p$ and $\ell_q$ isomorphic? Are $L_p$ and $L_q$ isomorphic? (The MO link posted in Jonas comment and t.b's comment confirm, that people read both these interpretations in your question.)2012-01-07

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