**Type and cotype of Banach spaces**

**1. Plan of course **

- Gaussian random vectors (Kahane, Landau-Shepp-Fernique). Type and cotype.
- Factorization theorems (Grothendieck).
- Grothendieck’s inequality, Krivine’s theorem on ‘s in Banach spaces.
- -convexity (Pisier).
- Dvoretzky theorem in cotype spaces.

**2. Gaussian random vectors **

** 2.1. Elementary properties **

Recall that the characteristic function of a standard Gaussian is

It follows that any linear combination of independant Gaussian vectors is a Gaussian, its norm only depends on and on the variance. So the infinite dimensional Gaussian vector gives rise to a linear isometric embedding of Hilbert space into . Also, orthogonal Gaussian vectors are independent.

** 2.2. Bernoulli variables **

Khintchin’s theorem is a Bernoulli version of the isometric embedding of Hilbert space.

Theorem 1 (Khintchin)Let . There are constants and such that, if are independant Bernoulli variables, then for all ,

*Proof:* In case is an even integer. Expand

Expanding leads to a similar expression, up to binomial coefficients. This leads to an upper bound, with .

In general, use Laplace transform and inequality . If , then, for ,

It follows that

and

The lower bound follows from the upper bound and Hölder’s inequality. For instance,

The sharp constants are known, Haagerup.

** 2.3. Kahane’s inequalities **

Theorem 2 (Kahane)Let be a normed space. There exist constants , such that for every set of vectors ,

*Proof:* We use the following fact. Let be a closed convex subset of , let be a -valued random vector with a symmetric distribution. Then if ,

This implies that if and are independent and symmetric,

Introduce the martingale and prove a maximal inequality (a bit finer than Doob’s inequality)

For this, introduce the stopping time . Note that and are independent. Both have symmetric distributions, so inequality applies.

Assume that . The principle implies that if , then all have norm . Let and . If , then and , so

This shows that the tail of the distribution of decays exponentially, and gives an estimate on .

** 2.4. The Landau-Shepp-Fernique Theorem **

It is a Gaussian analogue of Kahane’s inequality.

Theorem 3Let , . There exists a constant with the following effect. Let be a Gaussian vector in a finite dimensional Banach space . Assume that . Then

*Proof:* (Inspired from Fernique’s proof). Let and be independent copies of . Let

Note that and are independent. If is a differentiable function on ,

By convexity of , Jensen’s inequality gives

Lemma 4Let , be independent standard Gaussian vectors. If is -Lipschitz on Euclidean space, then

Jensen again gives

This implies

Now . So . For all ,

So our assumption gives

and this yields a bound on variance

Therefore . By independence,

** 2.5. Type and cotype **

Definition 5Let be a normed space. Say that hastypeif there exists a constant such that for all sets of vectors ,

Say that hascotypeif there exists a constant such that for all sets of vectors , the reverse inequality holds, i.e.

If all are colinear, this boils down to a -dimensional inequalityn which holds iff . So type always belongs to and similarly cotype always belongs to .

Note that type implies type for . Every Banach space has type and cotype , by the triangle inequality.

In the inequality, thanks to Kahane, one can replace the -norm on the left-hand side by an norm.

Using Khintchin,

Example 1If , has type and cotype . If , has type and cotype .

These values are sharp. Indeed, embeds in . If is the standard basis of , then

and this shows that type has to be .

** 2.6. Properties of type and cotype **

- Type and cotype pass to subspaces.
- Type passes to quotients.
- If has type , then has cotype .

Tomorrow, I will discuss ultraproducts, a tool to construct large Banach spaces reflecting properties of much smaller spaces.

** References **