nLab
type (functional analysis)

Type and Cotype in Functional Analysis

Idea

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,

x+y 2+xy 2=2x 2+2y 2. {\|x + y\|^2} + {\|x - y\|^2} = 2{\|x\|^2} + 2{\|y\|^2}.

This can be thought of as a way of improving the triangle inequality, which relates x±y\|x \pm y\| to x\|x\| and y\|y\|, by finding an equality relating x±y\|x \pm y\| to x\|x\| and y\|y\|.

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.

Definition

To define the type and cotype of a Banach space, we start with a finite family of vectors, say {x 1,,x n}\{x_1,\dots,x_n\}. Then we look for constants T 2T_2 and C 2C_2 such that the following inequalities are true:

Average ±±x i 2 T 2 2x i 2, Average ±±x i 2 C 2 2x i 2 \begin{aligned} \Average_\pm {\left\|\sum \pm x_i\right\|^2} &\le T_2^2 \sum {\|x_i\|^2}, \\ \Average_\pm {\left\|\sum \pm x_i\right\|^2} &\ge C_2^{-2} \sum {\|x_i\|^2} \end{aligned}

Here, the left-hand side is the average value over all choices of ±\pm (so in the original parallelogram identity we have divided each side by 22).

Definition

The smallest constant T 2T_2 making the first inequality true for all finite sequences of vectors is the type 22 constant of the space.

The smallest constant C 2C_2 making the second inequality true for all finite sequences of vectors is the cotype 22 constant of the space.

Either of these is allowed to be infinite. A space is said to be type 22 if its type 22 constant is finite. Similarly, it is said to be cotype 22 if its cotype 22 constant is finite.

Properties

If we consider all Banach spaces that are either of type 22 or cotype 22 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 22 into a space of cotype 22 factors through a Hilbert space. (This follows from a generalization/extension of Grothendieck’s inequality.)

More precisely:

  1. If a Banach space is of type 22 and of cotype 22 (ie both constants are finite) then it is a Hilbert space. This is due to Kwapien.
  2. Any bounded linear operator from a Banach space of type 22 to a Banach space of cotype 22 factors through a Hilbert space.

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.

Generalisations

In the inequalities for type 22 and cotype 22 it is possible to replace the 22 by a natural number pp and define the notions of type pp and cotype pp. Taken as a whole, these provide more information and thus give a finer classification of Banach spaces. In particular:

  1. For 1p21 \le p \le 2, the Lebesgue space, L pL_p, has type pp and cotype 22.
  2. For 2p<2 \le p \lt \infty, the Lebesgue space, L pL_p, has type 22 and cotype pp.

We also have the following properties:

  1. For r<pr \lt p, type pp implies type rr.
  2. For r>pr \gt p, cotype pp implies cotype rr.
  3. Both type and cotype pass to subspaces.
  4. Type passes to quotients, cotype does not in general but if the space has some type strictly larger than one then cotype does pass to quotients.
  5. Type dualises to cotype, in that if XX has type pp then X *X^* has cotype pp' (where p 1+p 1=1p^{-1} + {p'}^{-1} = 1), but cotype does not dualise to type unless the space has some type strictly larger than one.

Revised on November 10, 2011 05:52:25 by Yemon Choi (128.233.81.71)