$L^p(\mathbb{R})$ vs.$L^q(\mathbb{R})$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T05:26:30Z http://mathoverflow.net/feeds/question/79713 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79713/lp-mathbbr-vs-lq-mathbbr $L^p(\mathbb{R})$ vs.$L^q(\mathbb{R})$ Lost 2011-11-01T13:37:28Z 2011-11-03T03:45:34Z <p>Let $p,q \in (1,\infty)$ with $p\neq q$. Are the Banach spaces $L^p(\mathbb{R})$, $L^q(\mathbb{R})$ isomorphic?</p> http://mathoverflow.net/questions/79713/lp-mathbbr-vs-lq-mathbbr/79892#79892 Answer by Bill Johnson for $L^p(\mathbb{R})$ vs.$L^q(\mathbb{R})$ Bill Johnson 2011-11-03T03:45:34Z 2011-11-03T03:45:34Z <p>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 which 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.</p> <p>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.</p> <p>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.</p>