Let $p,q \in (1,\infty)$ with $p\neq q$. Are the Banach spaces $L^p(\mathbb{R})$, $L^q(\mathbb{R})$ isomorphic?
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44$\begingroup$ This is a perfectly reasonable question for a non expert to ask. Can anyone who voted to close prove that $L_4(0,1)$ is not isomorphic to $L_6(0,1)$? In practice, it is quite difficult to decide whether 2 Banach spaces are isomorphic. $\endgroup$– Bill JohnsonCommented Nov 1, 2011 at 14:59
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6$\begingroup$ @Bill: Quick Google search for "Lp and Lq are not isomorphic" gives this book books.google.com/…, page 180. $\endgroup$– user6976Commented Nov 1, 2011 at 15:18
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12$\begingroup$ Sure, Mark, and it is in other books as well (going back to Banach's classic). You will not find it in basic texts, though, and the result is certainly not obvious. I have been asked this exact question by famous people who work in a different part of functional analysis. $\endgroup$– Bill JohnsonCommented Nov 1, 2011 at 15:52
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11$\begingroup$ @Bill: the key here is not that it is in a book, but that a 1 minute Google search is enough to answer that question. $\endgroup$– user6976Commented Nov 1, 2011 at 16:17
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7$\begingroup$ I started a meta thread about this: tea.mathoverflow.net/discussion/1197/… $\endgroup$– Matthew DawsCommented Nov 3, 2011 at 13:04
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1 Answer
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The proof Fabian alludes to in the book reference Mark gave is a modern one using the notions of cotype and type. One way to prove that a Banach space $X$ is not isomorphic to a Banach space $Y$ is to exhibit a property that is preserved under isomorphisms that $X$ has but $Y$ does not. Type and cotype are examples of such properties. The (best) type and cotype of $L_p$ are calculated in many books. I suggest you look at Theorem 6.2.14 in the book of Albiac and Kalton. From the statement you see that if $p\not=q$, then $L_p$ and $L_q$ either have different (best) type or different (best) cotype.
Type and cotype depend only on the collection of finite dimensional subspaces of a space (we call such a property a local property). So you cannot use either to prove, e.g., that $L_p$ is not isomorphic to $\ell_p$ when $p\not= 2$. One way of proving this is to show that $\ell_2$ embeds isomorphically into $L_p$ but not into $\ell_p$ when $p\not=2$. These facts you can also find in Albiac-Kalton. You can also use infinite dimensional techniques to prove that $L_p$ and $L_q$ are not isomorphic when $p\not=q$. Banach knew this result through infinite dimensional considerations--the concepts of type and cotype came on the scene only 40 years ago.
You will also find in Albiac-Kalton a discussion of when $L_p$ or $\ell_p$ embeds isomorphically into $L_q$. That is more complicated and in fact Banach did not know everything. He called the question the problem of the linear dimension of $L_p$ spaces, IIRC.
answered Nov 3, 2011 at 3:45
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3$\begingroup$ @Bill: Out of curiosity, what was Banach's original argument for proving $L_p$ is not isomorphic to $L_q$? $\endgroup$ Commented Nov 3, 2011 at 8:49
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3$\begingroup$ Look at Chapter XII in Banach's book, Yemon, where he discusses the linear dimension of the
$L_p$and$\ell_p$spaces. He shows that they are of incomparable linear dimension except possibly that$\ell_q$ or$L_q$embeds into$L_p$when $p<q<2$ or $2<q<p$$^1$. Of course, we now know that $\ell_q$ and $L_q$ do not embed into$L_p$when $2<q<p$ but do embed even isometrically when $p<q<2$. If you have Oeuvres vol. II version, there is a nice update to Banach's book written in 1979 by Pelczynski. 1. Well, except that$L_2$isometrically embeds into all$L_p$spaces. $\endgroup$ Commented Nov 3, 2011 at 16:03 -
3$\begingroup$ How do you make a line break in a comment? $\endgroup$ Commented Nov 3, 2011 at 16:03
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1$\begingroup$ From linear dimension results in Banach's book, you can check the non isomorphism results even though the linear dimension problem was not completely solved until much later. BTW: When Banach says that the linear dimension of $X$ is less than the linear dimension of $Y$, he means that $X$ embeds isomorphically into $Y$. $\endgroup$ Commented Nov 3, 2011 at 16:06
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2$\begingroup$ Completely tangential: software doesn't like line breaks in comments. You can cheat, however, by using an empty math environment. $$ $$ Like this. (Credit goes to Will Jagy for showing me it.) $\endgroup$ Commented Nov 28, 2011 at 13:19