Timeline for Can $L^p(\mathbb{R})$ and $ L^q(\mathbb{R})$ be isomorphic?
Current License: CC BY-SA 4.0
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| Nov 17, 2018 at 15:18 | history | edited | Bill Johnson | CC BY-SA 4.0 |
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| Nov 17, 2018 at 15:16 | comment | added | Bill Johnson | The hard case is $2<p<q<\infty$, where the spaces have the same type and $\ell_p$ has better cotype than $\ell_q$. You can use Kadec-Pelczynski after taking ultra products. "All" you need to know is that ultra products of $L_p$ spaces is again an abstract $L_p$ space, and then that an abstract $L_p$ space is isometrically isomorphic to $L_p(\mu)$ for some measure $\mu$. Or you can localize Kadec-Pelczynski--this was done by Pelczynski and Rosenthal in a paper titled "Localization Techniques in $L^p$ spaces". | |
| Nov 16, 2018 at 1:24 | comment | added | user58955 | @BillJohnson Does type/co-type argument give anything about embeddability of finite-dimensional subspaces? For instance, when $p>2$ and $p\neq q$, I want to show that any finite-dimensional subspace of $\ell_p$ cannot be embedded with a constant distortion into $\ell_q$. The case of $p>q$ would follow from a co-type argument. Is there a quick argument for the case of $p<q$? | |
| Nov 28, 2011 at 13:19 | comment | added | Willie Wong | 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.) | |
| Nov 3, 2011 at 16:06 | comment | added | Bill Johnson | 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 $Y$. | |
| Nov 3, 2011 at 16:03 | comment | added | Bill Johnson | How do you make a line break in a comment? | |
| Nov 3, 2011 at 16:03 | comment | added | Bill Johnson |
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.
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| Nov 3, 2011 at 8:49 | comment | added | Yemon Choi | @Bill: Out of curiosity, what was Banach's original argument for proving $L_p$ is not isomorphic to $L_q$? | |
| Nov 3, 2011 at 3:45 | history | answered | Bill Johnson | CC BY-SA 3.0 |