Type and cotype of a Banach space information

In functional analysis, the type and cotype of a Banach space are a classification of Banach spaces through probability theory and a measure, how far a Banach space from a Hilbert space is.

The starting point is the Pythagorean identity for orthogonal vectors ${\displaystyle (e_{k})_{k=1}^{n}}$ in Hilbert spaces

${\displaystyle \left\|\sum _{k=1}^{n}e_{k}\right\|^{2}=\sum _{k=1}^{n}\left\|e_{k}\right\|^{2}.}$

This identity no longer holds in general Banach spaces, however one can introduce a notion of orthogonality probabilistically with the help of Rademacher random variables, for this reason one also speaks of Rademacher type and Rademacher cotype.

The notion of type and cotype was introduced by French mathematician Jean-Pierre Kahane.