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I guess that for all $1\le p,q<\infty $, such that $p\ne q$ , the spaces $\ell_p$ and $\ell_q$ are not isomorphic, but how do you prove this?

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See the answer here:… – David Mitra Jan 14 '12 at 19:33
I wrote a proof of the result mentioned by David Mitra: "In fact no infinite dimensional subspace of ℓp isomorphically embeds in ℓq for p≠q, 1≤p,q<∞", in great detail, with no need to mention compact maps. It's in portuguese though -- let me know if you're interested, in that case I could translate it. Or maybe you could give Google Translator a try ... – Rafael Feb 26 '12 at 23:52
up vote 7 down vote accepted

To expand on David's comment, since the question he links to is not the same one that you ask: a theorem of Pitt (which is mentioned in the question Linked to) tells us that when $1\leq p < q <\infty$, every bounded linear map from $\ell^q$ to $\ell^p$ is compact. In particular, since the only Banach spaces with compact unit ball are finite-dimensional, there can be no bounded linear bijection between the two spaces.

Showing that naturally occurring Banach spaces are non-isomorphic can be surprisingly difficult; I don't know a simpler approach in this case, although one could probably rig up a direct argument by using ingredients from the proof of Pitt's theorem.

More information and some other non_-isomorphism results can be found in a MathOverflow answer of Bill Johnson.

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In fact no infinite dimensional subspace of $\ell_p$ isomorphically embeds in $\ell_q$ for $p\ne q$, $1\le p,q<\infty$. – David Mitra Jan 14 '12 at 19:45

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