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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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.