This page is inspired by the following question, which appeared on MathOverflow.
More generally, one can ask:
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 is not isomorphic to a Banach space is to exhibit a property which is preserved under isomorphisms that has but does not. For example, among the spaces for , is the only nonseparable space, and is the only separable space with a nonseparable dual. Thus and are not isomorphic to each other or to any with .
To distinguish among the with finer properties are needed. Type and cotype are examples of such properties. The (best) type and cotype of are standard calculations: if then has type and cotype (and no better), and if then has type and cotype (and no better). See for example in Theorem 6.2.14 of AK06. From that, one can see that if , then and 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 , is not isomorphic to . One way of proving this is to show that for , embeds isomorphically into but not into (see also AK).