The type and cotype of a Banach space measure how far it is from being a Hilbert space. The definition is based on the observation, due to John von Neumann, that a Banach space is a Hilbert space if and only if it satisfies the parallelogram identity. Recall that this states that in a Hilbert space,
This can be thought of as a way of improving the triangle inequality, which relates to and , by finding an equality relating to and .
To measure the type and cotype of a Banach space, one takes the parallelogram identity and finds out how bad it gets. Slightly more precisely, one tries to see what happens if one looks merely for an inequality, perhaps with a constant. Since the equality can break in one of two ways, this leads to two notions.
To define the type and cotype of a Banach space, we start with a finite family of vectors, say . Then we look for constants and such that the following inequalities are true:
Here, the left-hand side is the average value over all choices of (so in the original parallelogram identity we have divided each side by ).
The smallest constant making the first inequality true for all finite sequences of vectors is the type constant of the space.
The smallest constant making the second inequality true for all finite sequences of vectors is the cotype constant of the space.
Either of these is allowed to be infinite. A space is said to be type if its type constant is finite. Similarly, it is said to be cotype if its cotype constant is finite.
If we consider all Banach spaces that are either of type or cotype then we find that these split into the two obvious classes with Hilbert spaces sitting plum in the middle. Not only are Hilbert spaces the intersection of these classes, but also a continuous linear operator from a space of type into a space of cotype factors through a Hilbert space. (This follows from a generalization/extension of Grothendieck’s inequality.)
As type and cotype are isomorphism invariants, they can be used to distinguish between some Banach spaces. See isomorphism classes of Banach spaces for more.
In the inequalities for type and cotype it is possible to replace the by a natural number and define the notions of type and cotype . Taken as a whole, these provide more information and thus give a finer classification of Banach spaces. In particular:
We also have the following properties: