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?
How do you prove that $\ell_p$ is not isomorphic to $\ell_q$?
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1I 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 ... – 2012-02-26
1 Answers
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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5In fact no infinite dimensional subspace of $\ell_p$ isomorphically embeds in $\ell_q$ for $p\ne q$, 1\le p,q<\infty. – 2012-01-14