Type and cotype of a Banach space

In functional analysis, the type and cotype of a Banach space are a classification of Banach spaces through probability theory and a measure for how far a Banach space is away from being a Hilbert space.
The starting point is the Pythagorean identity for orthogonal vectors in Hilbert spaces
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.

Definition

Let

be a Banach space,
be a sequence of independent Rademacher random variables, i.e. and for and.

The notation means that we integrate with respect to the variable.

Type

is of type for if there exists a finite constant such that
for all finite sequences. The sharpest constant is called type constant and denoted as.

Cotype

is of cotype for if there exists a finite constant such that
respectively
for all finite sequences. The sharpest constant is called cotype constant and denoted as.

Remarks

By taking the -th resp. -th root one gets the equation for the Bochner norm.

Properties

Every Banach space is of type .
A Banach space is of type and cotype if and only if the space is also isomorphic to a Hilbert space.

If a Banach space:

is of type then it is also type.
is of cotype then it is also of cotype.
is of type for, then its dual space is of cotype with . Further it holds that

Examples

The spaces for are of type and cotype, this means is of type, is of type and so on.
The spaces for are of type and cotype.
The space is of type and cotype.