Type and cotype

Summary

Type and cotype measure how far a Banach space is from a Hilbert space. They compare random sign sums $\sum_i \pm x_i$ with the sizes of the $x_i$. A space of type $p$ makes such sums small, and a space of cotype $q$ makes them large. A theorem of Kwapień says that a Banach space has type $2$ and cotype $2$ exactly when it is isomorphic to a Hilbert space. These notions organise much of the local theory of Banach spaces, around results such as Dvoretzky's theorem.