Exercise 10, § 1

Let $p$ and $r$ be real numbers such that $1\leqslant p < r$. Let $u$ be a continuous linear mapping from $\ell^r_K(\mathbf{N})$ into $\ell^p_K(\mathbf{N})$.

We denote by $r_n$ (resp. $p_n$) the projection

onto $\ell^r_K(\mathbf{N})$ (resp. onto $\ell^p_K(\mathbf{N})$). We denote by $\|\cdot \|_p$ (resp. $\|\cdot \|_r$) the norm on $\ell^p_K(\mathbf{N})$ (resp. on $\ell^r_K(\mathbf{N})$).

a) Prove that for every $n\in \mathbf{N}$, one has $\|p_nu(1-r_m)\| \rightarrow 0$ when $m\rightarrow +\infty$, where 1 denotes the identity morphism of $\ell^r_K(\mathbf{N})$.

b) For $n,m\in \mathbf{N}$, let $u_{n,m}= (1-p_n)u(1-r_m)$. Let $\alpha = (\alpha_n)_{n\in\mathbf{N}}$ be a sequence of strictly positive real numbers such that $\|\alpha \|_r= 1$ and $\alpha$ does not belong to $\ell^p_K(\mathbf{N})$, and let $(\varepsilon_n)$ be a sequence of strictly positive real numbers such that the series $\sum\varepsilon_n\alpha_n$ converges.

If $u$ is not compact, prove that there exist $\delta >0$, strictly increasing sequences of integers $(n_i)_{i\in\mathbf{N}}$ and $(m_i)_{i\in\mathbf{N}}$ and two sequences $(x_i)_{i\in\mathbf{N}}$ of elements of $\ell^r_K(\mathbf{N})$ and $(y_i)_{i\in\mathbf{N}}$ of elements of $\ell^p_K(\mathbf{N})$, with the following properties:

(i) One has $\|x_i\|_r= 1$ for every $i\in \mathbf{N}$;

(ii) The sets $S_i=\{n\in \mathbf{N}|x_{i,n}\not = 0\}$ are pairwise disjoint and the sets $T_i=\{n\in \mathbf{N}|y_{i,n}\not = 0\}$ are pairwise disjoint;

(iii) One has $\|y_i\|_p\geqslant \delta$;

(iv) One has $\|u(x_i)\|_q> \delta -2\varepsilon_i$.

(Proceed by induction and use the formulas

for $n$ and $m$ in $\mathbf{N}$.)

c) Deduce by contradiction that $u$ is compact (« Pitt’s theorem »). d) Prove that the canonical injection of $\ell^p_K(\mathbf{N})$ into $\ell^r_K(\mathbf{N})$ is not compact.

e) Let $p$ and $r$ be real numbers such that $1\leqslant p\leqslant r\leqslant +\infty$. Prove that $\ell^r_K(\mathbf{N})$ is isomorphic to $\ell^p_K(\mathbf{N})$ if and only if $p=r$.