## Notes of Bernard Maurey’s lecture nr 1

Type and cotype of Banach spaces

1. Plan of course

1. Gaussian random vectors (Kahane, Landau-Shepp-Fernique). Type and cotype.
2. Factorization theorems (Grothendieck).
3. Grothendieck’s inequality, Krivine’s theorem on ${\ell_p^n}$‘s in Banach spaces.
4. ${K}$-convexity (Pisier).
5. Dvoretzky theorem in cotype ${2}$ spaces.

2. Gaussian random vectors

2.1. Elementary properties

Recall that the characteristic function of a standard Gaussian is

$\displaystyle \begin{array}{rcl} \mathop{\mathbb E}(e^{it\cdot X})=e^{-\frac{|t|^2}{2}}. \end{array}$

It follows that any linear combination of independant Gaussian vectors is a Gaussian, its ${L_p}$ norm only depends on ${p}$ and on the variance. So the infinite dimensional Gaussian vector gives rise to a linear isometric embedding of Hilbert space into ${L^p}$. 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 ${0. There are constants ${A_p}$ and ${B_p}$ such that, if ${\epsilon_j}$ are independant Bernoulli variables, then for all ${a\in\ell_2 ({\mathbb N})}$,

$\displaystyle \begin{array}{rcl} A_p \sqrt{\sum_{j}a_j^2}\leq (\mathop{\mathbb E}(|\sum_{j}a_j\epsilon_j|^p))^{1/p}\leq B_p \sqrt{\sum_{j}a_j^2}. \end{array}$

Proof: In case ${p=2n}$ is an even integer. Expand

$\displaystyle \begin{array}{rcl} \mathop{\mathbb E}(|\sum_{j}a_j\epsilon_j|^p)&=&\sum_{j_1 ,\ldots,j_n}a_{j_1}\cdots a_{j_n}\mathop{\mathbb E}(\epsilon_{j_1}\cdots \epsilon_{j_n})\\ &=&\sum_{k_1 +\cdots+k_p=2n}a_{j_1}^{2k_1}\cdots a_{j_n}^{2k_p} \end{array}$

Expanding ${(\sum_{j}a_j^2)^n}$ leads to a similar expression, up to binomial coefficients. This leads to an upper bound, with ${B_{2n}\leq\sqrt{2n}}$.

In general, use Laplace transform and inequality ${\mathop{\mathbb E}(e^{s\epsilon})=\cosh(s)\leq e^{s^2 /2}}$. If ${\sum_{j}a_j^2 =1}$, then, for ${S=\sum a_j \epsilon_j}$,

$\displaystyle \begin{array}{rcl} \mathop{\mathbb E}(e^{sS})\leq e^{s^2 /2}. \end{array}$

It follows that

$\displaystyle \begin{array}{rcl} \mathop{\mathbb P}(S\geq t)\leq e^{s^2 /2 -st},\quad \mathop{\mathbb P}(|S|\geq t)\leq 2\,e^{-t^2 /2}, \end{array}$

and

$\displaystyle \begin{array}{rcl} \mathop{\mathbb E}(|S|^p)\leq \int_{0}^{\infty}2p t^{p-1}e^{-t^2 /2}\,dt. \end{array}$

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

$\displaystyle \begin{array}{rcl} \|a\|_2 =\|S\|_2 \leq \|S\|_1^{1/3}\|S\|_4^{2/3}\leq B_4^{2/3}\|a\|_2^{2/3}\|S\|_1^{1/3} \end{array}$

$\Box$

The sharp constants are known, Haagerup.

2.3. Kahane’s inequalities

Theorem 2 (Kahane) Let ${E}$ be a normed space. There exist constants ${a_p}$, ${b_p}$ such that for every set of vectors ${x_j \in E}$,

$\displaystyle \begin{array}{rcl} a_p \|\sum\epsilon_j x_j\|_2 \leq \|\sum\epsilon_j x_j\|_p \leq b_p \|\sum\epsilon_j x_j\|_2 . \end{array}$

Proof: We use the following fact. Let ${C}$ be a closed convex subset of ${C}$, let ${V}$ be a ${V}$-valued random vector with a symmetric distribution. Then if ${x\notin C}$,

$\displaystyle \begin{array}{rcl} \mathop{\mathbb P}(x+V \notin C)\geq \frac{1}{2}. \end{array}$

This implies that if ${U}$ and ${V}$ are independent and symmetric,

$\displaystyle \begin{array}{rcl} \mathop{\mathbb P}(|U|>t)\leq 2\mathop{\mathbb P}(|U+V|>t). \end{array}$

Introduce the martingale ${M_k=\sum_{j=1}^{k}\epsilon_j x_j}$ and prove a maximal inequality (a bit finer than Doob’s inequality)

$\displaystyle \begin{array}{rcl} \mathop{\mathbb P}(\sup|M_k|>t)\leq 2\mathop{\mathbb P}(|M_n|>t). \end{array}$

For this, introduce the stopping time ${T=\inf\{k\,;\,|M_k|>t\}}$. Note that ${M_T}$ and ${Y=M_n -M_T}$ are independent. Both have symmetric distributions, so ${U+V}$ inequality applies.

Assume that ${\mathop{\mathbb E}(|\sum\epsilon_j x_j|^2)=1}$. The ${U+V}$ principle implies that if ${\mathop{\mathbb P}(|M_n|>4)\leq \frac{1}{16}}$, then all ${x_j}$ have norm ${\leq 2}$. Let ${T=\inf\{k\,;\,|M_k|>4\}}$ and ${Y=M_n -M_T}$. If ${|M_n|>8+t}$, then ${|M_T|>4}$ and ${|Y|>t}$, so

$\displaystyle \begin{array}{rcl} \mathop{\mathbb P}(|M_n|>8+t)&\leq&\mathop{\mathbb P}(|M_T|>4)\mathop{\mathbb P}(|Y|>t)\\ &\leq&2\mathop{\mathbb P}(|M_T|>4)\mathop{\mathbb P}(|Y|>t)\\ &\leq&4\mathop{\mathbb P}(|M_T|>4)\mathop{\mathbb P}(|Y|>t)\\ &\leq&\frac{1}{4}\mathop{\mathbb P}(|M_n|>t). \end{array}$

This shows that the tail of the distribution of ${|M_n|}$ decays exponentially, and gives an estimate on ${\mathop{\mathbb E}(|M_n|^p)}$. $\Box$

2.4. The Landau-Shepp-Fernique Theorem

It is a Gaussian analogue of Kahane’s inequality.

Theorem 3 Let ${\tau>0}$, ${\alpha<1}$. There exists a constant ${c(\tau,\alpha)}$ with the following effect. Let ${G}$ be a Gaussian vector in a finite dimensional Banach space ${E}$. Assume that ${\mathop{\mathbb P}(|G|>\tau)<\alpha}$. Then

$\displaystyle \begin{array}{rcl} \mathop{\mathbb E}(e^{c|G|^2})\leq \end{array}$

Proof: (Inspired from Fernique’s proof). Let ${X}$ and ${Y}$ be independent copies of ${G}$. Let

$\displaystyle \begin{array}{rcl} X_{\theta}=\cos\theta X+\sin\theta Y. \end{array}$

Note that ${X_{\theta}}$ and ${\frac{d}{d\theta}X_{\theta}}$ are independent. If ${f}$ is a differentiable function on ${E}$,

$\displaystyle \begin{array}{rcl} f(Y)-f(X)=\int_{0}^{\pi/2}\nabla f(X_{\theta})\cdot Y_{\theta}\,d\theta. \end{array}$

By convexity of ${x\mapsto e^{sx}}$, Jensen’s inequality gives

$\displaystyle \begin{array}{rcl} \mathop{\mathbb E}(\exp(s\frac{2}{\pi}(f(Y)-f(X))))\leq \frac{2}{\pi}\int_{0}^{\pi/2}\mathop{\mathbb E}(e^{s\nabla f(X_{\theta})\cdot Y_{\theta}})\,d\theta. \end{array}$

Lemma 4 Let ${X}$, ${Y}$ be independent standard Gaussian vectors. If ${f}$ is ${1}$-Lipschitz on Euclidean space, then

$\displaystyle \begin{array}{rcl} \mathop{\mathbb E}(e^{\frac{2}{\pi}s(f(Y)-f(X))})\leq e^{s^2 /2}. \end{array}$

Jensen again gives

$\displaystyle \begin{array}{rcl} \mathop{\mathbb E}(e^{\frac{2}{\pi}s(f(Y)-\mathop{\mathbb E}(f(X)))})\leq e^{s^2 /2}. \end{array}$

This implies

$\displaystyle \begin{array}{rcl} \mathop{\mathbb P}(|f(Y)-\mathop{\mathbb E}(f(Y))|>t)\leq e^{-(\frac{2}{\pi}t)^2 /2}. \end{array}$

Now ${f(x)=|x|_E}$. So ${|\nabla f|_{E^*}\leq 1}$. For all ${x\in E}$,

$\displaystyle \begin{array}{rcl} |\nabla f(x)\cdot Y_{\theta}|\leq|Y_{\theta}|_E . \end{array}$

So our assumption gives

$\displaystyle \begin{array}{rcl} \mathop{\mathbb P}(|\nabla f(x)\cdot Y_{\theta}|>\tau)<\alpha, \end{array}$

and this yields a bound on variance

$\displaystyle \begin{array}{rcl} \mathop{\mathbb E}(|\nabla f(x)\cdot Y_{\theta}|^2)\leq\kappa(\tau,\alpha). \end{array}$

Therefore ${\mathop{\mathbb E}(\exp(s\frac{2}{\pi}(f(Y)-f(X))))\leq e^{s^2 \kappa^2 /2}}$. By independence,

$\displaystyle \begin{array}{rcl} \mathop{\mathbb E}(\exp(s\frac{2}{\pi}(|Y|)))\leq \end{array}$

$\Box$

2.5. Type and cotype

Definition 5 Let ${X}$ be a normed space. Say that ${X}$ has type ${p}$ if there exists a constant ${T_p}$ such that for all sets of vectors ${x_i \in X}$,

$\displaystyle \begin{array}{rcl} \mathop{\mathbb E}(|\sum_{i}\epsilon_i x_i|)\leq T_p (\sum_{i}|x_i|^p)^{1/p}. \end{array}$

Say that ${X}$ has cotype ${q}$ if there exists a constant ${C_q}$ such that for all sets of vectors ${x_i \in X}$, the reverse inequality holds, i.e.

$\displaystyle \begin{array}{rcl} (\sum_{i}|x_i|^p)^{1/p}\leq C_q \mathop{\mathbb E}(|\sum_{i}\epsilon_i x_i|). \end{array}$

If all ${x_i}$ are colinear, this boils down to a ${1}$-dimensional inequalityn which holds iff ${1\leq p\leq 2}$. So type always belongs to ${[1,2]}$ and similarly cotype always belongs to ${[2,\infty)}$.

Note that type ${p}$ implies type ${r}$ for ${1\leq r\leq p}$. Every Banach space has type ${1}$ and cotype ${\infty}$, by the triangle inequality.

In the ${T_p}$ inequality, thanks to Kahane, one can replace the ${L_1}$-norm on the left-hand side by an ${L_r}$ norm.

Using Khintchin,

Example 1 If ${1\leq r\leq 2}$, ${L^r}$ has type ${p}$ and cotype ${2}$. If ${2\leq r<\infty}$, ${L^r}$ has type ${2}$ and cotype ${r}$.

These values are sharp. Indeed, ${\ell_p}$ embeds in ${L_p (0,1)}$. If ${(e_n)}$ is the standard basis of ${\ell_p}$, then

$\displaystyle \begin{array}{rcl} \mathop{\mathbb E}(|\sum_{i}\epsilon_i e_i|_{\ell_p})=n^{1/p}, \end{array}$

and this shows that type has to be ${\leq p}$.

2.6. Properties of type and cotype

1. Type and cotype pass to subspaces.
2. Type passes to quotients.
3. If ${X}$ has type ${p}$, then ${X^*}$ has cotype ${p'=\frac{p}{p-1}}$.

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

References