Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
See the answer here: math.stackexchange.com/questions/97126/… –  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

1 Answer 1

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.

share|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.