# nLab isomorphism classes of Banach spaces

## Topics in Functional Analysis

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

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

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 spaces ${L}_{p}\left(ℝ\right)$ for $p\in \left[1,\infty \right]$, ${L}_{\infty }$ is the only nonseparable space, and ${L}_{1}$ is the only separable space with a nonseparable dual. Thus ${L}_{1}$ and ${L}_{\infty }$ are not isomorphic to each other or to any ${L}_{p}$ with $p\in \left(1,\infty \right)$.

To distinguish among the ${L}_{p}$ with $p\in \left(1,\infty \right)$ finer properties are needed. Type and cotype are examples of such properties. The (best) type and cotype of ${L}_{p}$ are standard calculations: if $p\in \left[1,2\right]$ then ${L}_{p}$ has type $p$ and cotype $2$ (and no better), and if $p\in \left[2,\infty \right)$ 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.

Type and cotype depend only on the collection of finite dimensional subspaces of a space (we call such a property a local property?). So neither can be used to prove, e.g., that for $p\ne 2$, ${L}_{p}$ is not isomorphic to ${\ell }_{p}$. One way of proving this is to show that for $p\ne 2$, ${\ell }_{2}$ embeds isomorphically into ${L}_{p}$ but not into ${\ell }_{p}$ (see also AK).

### 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
Revised on November 8, 2011 17:49:19 by Mark Meckes? (129.22.117.158)