nLab isomorphism classes of Banach spaces (changes)

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This page is inspired by the following question, which appeared on MathOverflow.

Let $p,q \in (1,\infty)$ with $p\neq q$. Are the Banach spaces $L^p(\mathbb{R})$, $L^q(\mathbb{R})$ isomorphic?

More generally, one can ask:

Given two Banach spaces, $X$ and $Y$, when are they isomorphic?

The following started out as an adapted version of Bill Johnson’s answer to the MathOverflow question.

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. For example, among the spacesType$L_p(\mathbb{R})$ and forcotype$p \in [1,\infty]$ , are examples of such properties. The(best) type$L_\infty$ is the only nonseparable space, andcotype$L_1$ of is the only separable space with a nonseparable dual. Thus$L_p$ L_p L_1 are and standard calculations, for example in Theorem 6.2.14 ofAK06$L_\infty$ . From are that, not one isomorphic can to see each that other if or to any$p{L}_p\ne q$ p L_p \ne q , then with$L_{p}p\in (1,\mathrm{\infty })$ L_p p \in (1,\infty) . and$L_q$ either have different (best) type or different (best) cotype.

Type To and distinguish cotype among depend only on the collection of finite dimensional subspaces of a space (we call such a property alocal property?$L_p$ ). So with neither can be used to prove, e.g., that for$p<span><del\; class="diffmod">\; \ne </del><ins\; class="diffmod">\; \in </ins></span>2(1,\mathrm{\infty })$ p \ne \in 2 (1,\infty) , finer properties are needed.$L_p$Type is and not isomorphic to$\ell_p$cotype . One are way examples of proving such this properties. is The to show that for$p \ne 2$(best) type , and$\ell_2$cotype embeds of isomorphically into$L_p$ but are not standard into calculations: if${\ell }_{p}p\in [1,2]$ \ell_p p \in [1,2] (see then alsoAK$L_p$ ). has type$p$ and cotype $2$ (and no better), and if $p \in [2,\infty)$ then $L_p$ has type $2$ and cotype $p$ (and no better). See for example in Theorem 6.2.14 of AK06. From that, one can see that if $p \ne q$, then $L_p$ and $L_q$ either have different (best) type or different (best) cotype.

References

• AK06 Albiac, Fernando and Kalton, Nigel. Topics in Banach space theory. Graduate Texts in Mathematics, 233. Springer, New York, 2006. xii+373 pp. ISBN: 978-0387-28141-4; MR2192298