Exercise 25, § 1

a) Let $N\geqslant 1$ be an integer. We denote by $G_N$ (resp. $H_N$) the set $\{-1,1\}^N$ (resp. the set $\{-1,2\}^N$). We denote by $\mu_N$ (resp. $\nu_N$) the positive measure of total mass 1 on $G_N$ (resp. on $H_N$) such that, for every $(t_i)_{1\leqslant i\leqslant N}$ in $G_N$ (resp. in $H_N$), one has $\mu_N((t_i)_{1\leqslant i\leqslant N}) =_2^{1_N}$, resp.

1 $^j$ 2 $^{N-j}$

3 3 where $j$ is the number of indices $i$ such that $t_i= 2$.

There exists a real number $c >0$ such that, for every family $(a_i)_{1\leqslant i\leqslant N}$ of complex numbers, one has

$\mu_N(\{(t_i)_{1\leqslant i\leqslant N}\in G_N||\sum^Na_it_i|> c$(log $N$)$^{1/2}(\sum^N|a_i|^2)^{1/2}\})<\frac{c}{N^3}$,

$i=1i=1$

and

$\nu_N(\{(t_i)_{1\leqslant i\leqslant N}\in H_N||\sum^Na_it_i|> c$(log $N$)$^{1/2}(\sum^N|a_i|^2)^{1/2}\})<\frac{c}{N^3}$.

$i=1i=1$

(One may suppose that $a_i\in \mathbf{R}$ for every $i$ and that $\sum|a_i|^2= 1$; for the first inequality, prove and use the upper bound

$\int_{G_N}$ exp$(\lambda |\sum_{i=1}^Na_it_i|)d\mu_N\leqslant 2$ exp($\lambda^2$)

for every real $\lambda >0$; argue similarly for the second upper bound by using the inequality $\frac{1}{3}(e^{2x}+ 2e^{-x})\leqslant e^{2x^2}$ for every $x\in \mathbf{R}.)$

b) For $k\in \mathbf{N}$, let $A_k=\mathbf{Z}/(3\cdot 2^k)\mathbf{Z}$. There exists a real number $c'>0$, independent of $k$, and a function $f_k:\widehat{A}_k\rightarrow \{-1,2\}$ such that

and, for every $x\in A_k$, one has

(With the aid of a), establish the existence of a function $f_k$ satisfying the second condition, then modify $f_k$ to obtain the first.)

We denote by $B_k$ (resp. $C_k$) the set of characters $\chi$ of $A_k$ such that $f_k(\chi ) = 2$ (resp. $f_k(\chi ) =-1$). One has Card(B$_k$) $= 2^k$ and Card(C$_k$) $= 2^{k+1}$.

For every $k\geqslant 1$, we fix a bijection $\iota_k: B_k\rightarrow C_{k-1}$.

We denote by A the sum set of the $A_k$ for $k\in \mathbf{N}$; we endow A with the discrete topology, and we denote by $E =\mathscr{C}_b(A)$.

We fix a family $\varepsilon = (\varepsilon_k)_{k\geqslant\mathbf{N}}$ of functions $\varepsilon_k: B_k\rightarrow \{-1,1\}$. We then define functions $e_{k,\chi}\in E$, for $k\in \mathbf{N}$ and $\chi \in B_k$, by

$\iota_k(\chi )(x)$ if $k\geqslant 1$ and $x\in A_{k-1}$,

$e_{k,\chi}(x) =\varepsilon_k(\chi )\chi (x)$ if $x\in A_k$

0 otherwise.

We denote by $E_{\varepsilon}$ the closed subspace of E generated by the functions $(e_{k,\chi})_{k\in\mathbf{N},\chi\in B_k}$. It is a complex Banach space of countable type. The rest of the exercise is intended to prove that, for a suitable choice of $\varepsilon$, the space $E_{\varepsilon}$ does not have the approximation property; it does not admit a Banach basis.

c) For $k\in \mathbf{N}$ and $\chi \in B_k$, there exists a continuous linear form $\lambda_{k,\chi}\in E'_{\varepsilon}$ such that, for every $f\in E_{\varepsilon}$,

$\lambda_{k,\chi}(f) =$ Card(A1 $_k$)$\sum\varepsilon_k(\chi )\chi (x^{-1})f(x)$.

$x\in A_k$

For $k\geqslant 1$ and $f\in E_{\varepsilon}$, one has

$\lambda_{k,\chi}(f) =$ Card(A2 $_k$)$\sum\iota_k(\chi )(x^{-1})f(x)$.

$x\in A_k$

d) For every $k\in \mathbf{N}$, let $\beta_k$ be the continuous linear form on $\mathscr{L}(E_{\varepsilon})$ defined by

$\chi \in B_k$

One has $\beta_k(1_{E_{\varepsilon}}) = 1$.

e) Let $u\in \mathscr{L}(E_{\varepsilon})$ and $k\in \mathbf{N}$. One has

$\beta_{k+1}(u)-\beta_k(u) =$ Card(A1 $_k$)$\sum u(\varphi_{k,x})(x)$

$x\in A_k$

where, for $x\in A_k$, we put

$\chi \in B_{k+1}\chi \in B_k$

One has $|\varphi_{k,x}(y)|\leqslant c'(k+ 1)^{1/2}2^{-k/2}$ for $(x, y)\in A^2_k$.

f) There exists a real number $c''\geqslant c'$ such that, for every $k\geqslant 1$, there exists a function $\varepsilon_k: B_k\rightarrow \{-1,1\}$ satisfying $|\varphi_{k,x}(y)|\leqslant c''3(k+ 1)^{1/2}2^{-k/2}$ for $(x, y)\in A_k\times A_{k-1}$. (Apply the first inequality of a).) We henceforth suppose that $\varepsilon = (\varepsilon_k)$ satisfies these properties.

g) For every $k\in \mathbf{N}$ and $x\in A_k$, one has $\|\varphi_{k,x}\|\leqslant c''(k+ 1)^{1/2}2^{-k/2}$.

h) Let K be the subset of $E_{\varepsilon}$ whose elements are $e_{0,\chi_0}$, where $\chi_0$ is the unique element of $B_0$, and the functions $(k+ 1)^2\varphi_{k,x}$ for $k\in \mathbf{N}$ and $x\in A_k$. The set K is relatively compact in $E_{\varepsilon}$, and one has

$|\beta_{k+1}(u)-\beta_k(u)|\leqslant$ 1 $_2$ sup$\|u(x)\|$

for every $k\in \mathbf{N}$ and every $u\in \mathscr{L}(E_{\varepsilon})$.

i) The sequence $(\beta_k)_{k\in\mathbf{N}}$ converges in the dual of the space $\mathscr{L}(E_{\varepsilon})$ endowed with the topology of simple convergence. Let $\beta$ denote its limit. For every $u\in \mathscr{L}(E_{\varepsilon})$, one has $|\beta (u)|\leqslant 3$ sup$_{x\in K}\|u(x)\|$.

j) One has $\beta (1_{E_{\varepsilon}}) = 1$ and $\beta (u) = 0$ if $u$ is of finite rank.

k) The space $E_{\varepsilon}$ does not have the approximation property. l) Let $p\in ]2,+\infty [$. Endow A with the counting measure and denote by $E^p_{\varepsilon}$ the closed subspace of $L^p(A)$ generated by the $(e_{k,\chi})_{k\in\mathbf{N},\chi\in B_k}$. The space $E^p_{\varepsilon}$ does not have the approximation property. (Verify that the cardinal of the support of $\varphi_{k,x}$ is $\leqslant 21\cdot 2^k$ and deduce that $\|\varphi_{k,x}\|_p= O((k+ 1)^{1/2}2^{-k(p-2)/2p})$ for $k\in \mathbf{N}$ and $x\in A_k.$)