Exercise 15, § 1

Let $E_0$ be the complex vector space of functions $f:\mathbf{N}^*\times \mathbf{N}^*\rightarrow \mathbf{C}$. For $k\in \mathbf{N}^*$, let $\alpha_k\in E_0$ be the function such that

$\{m^k$ if $1\leqslant n\leqslant k-1$

$n^k$ if $n\geqslant k$.

Let E be the vector subspace of $E_0$ formed by the functions $f$ such that $p_k(f)$ is finite for every $k\in \mathbf{N}^*$, where $p_k$ is defined by

a) The space E endowed with the topology defined by the norms $(p_k)_{k\in\mathbf{N}^*}$ is a Fréchet space and a Montel space. Its dual $E'$ identifies with the subspace of $E_0$ formed by the functions $g$ such that, for at least one $k\geqslant 1$, one has

sup$_{m,n}\alpha_k(m, n)^{-1}|g(m, n)|<+\infty$

(cf. EVT, IV, p. 63, exercise 8).

b) We consider the linear mapping $u$ from E into $\ell^1(\mathbf{N}^*)$ defined by

The linear mapping $u$ is continuous.

c) Let us identify $\ell^{\infty}(\mathbf{N}^*)$ with the dual of $\ell^1(\mathbf{N}^*)$. The transpose of $u$ is the mapping $v:\ell^{\infty}(\mathbf{N}^*)\rightarrow E'\subset E_0$ determined by $(x_n)\mapsto f$ where $f(n, m) =x_m$ for all $(n, m)\in \mathbf{N}^*\times \mathbf{N}^*$.

d) The mapping $u$ induces an isomorphism $E/$ Ker($u$)$\rightarrow \ell^1(\mathbf{N}^*) ($cf. loc. cit.).

e) The mapping $u$ is not compact.

f) The transpose $v$ of $u$ is compact for the topology of convergence on the bounded parts of $\ell^{\infty}(\mathbf{N}^*)$.

g) The mapping $v$ is an example of a compact linear mapping between separated locally convex spaces whose transpose is not compact.