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
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 3 Let , . 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 4 Let , be independent standard Gaussian vectors. If is -Lipschitz on Euclidean space, then
Jensen again gives
Now . So . For all ,
So our assumption gives
and this yields a bound on variance
Therefore . By independence,
2.5. Type and cotype
Definition 5 Let be a normed space. Say that has type if there exists a constant such that for all sets of vectors ,
Say that has cotype if 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.
Example 1 If , 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.