Theorem 4

Let E be a Fréchet space.

a) Suppose that E is an approximation space. Then, for every semi-normed space F, the space $\mathscr{L}^f(F; E)$ is dense in $\mathscr{L}^c(F; E)$ for the topology of bounded convergence;

b) Conversely, suppose that for every Banach space F, every compact linear mapping of F into E belong to the closure of $\mathscr{L}^f(F; E)$ for the topology of bounded convergence. Then E is an approximation space.

Let F be a semi-normed space. The closure of $\mathscr{L}^f(F; E)$ in $\mathscr{L}(F; E)$ is contained in $\mathscr{L}^c(F; E)$ (Prop. 1 and 2 of III, p. 4). It is equal to $\mathscr{L}^c(F; E)$ if E is an approximation space by Cor. to Prop. 11. This proves assertion a).

Suppose that the hypothesis of b) is satisfied. Let $\varepsilon >0$ be a real number. Let A be a compact subset of E and $p$ a continuous semi-norm on E. Let L be a compact convex balanced subset of E such that A is a compact subset of the normed space $E_L$ (Lemma 4). The canonical injection $j: E_L\rightarrow E$ is compact and $E_L$ is a Banach space (TVS, III, p. 8, Cor.). Hence by hypothesis there exist an integer $n\geqslant 1$, elements $e_1, . . . , e_n$ of E, and continuous linear forms $\ell_1, . . . , \ell_n$ on $E_L$, such that the mapping $v$ of $E_L$ into E defined by $v(x) =\sum^n_{i=1}\ell_i(x)e_i$ satisfies $p(x-v(x))\leqslant \frac{\varepsilon}{2}$ for $x\in A$. The image of $^tj: E'\rightarrow (E_L)'$ is dense in $(E_L)'$ for the weak topology (TVS, IV, p. 6, Prop. 5). On $(E_L)'$ the topology of compact convergence is compatible with the duality between $(E_L)'$ and $E_L$ (TVS, IV, p. 3, Example). Therefore $^tj(E')$ is dense in $(E_L)'$ for the topology of compact convergence (TVS, IV, p. 1, Prop. 1), and there exist continuous linear forms $f_1, . . . , f_n$ on E such that

for $x\in A$ and $1\leqslant i\leqslant n$. The endomorphism $u:x\mapsto \sum^n_{i=1}f_i(x)e_i$ of E belong to $\mathscr{L}^f(E)$, and for every $x\in A$, one has

It follows from this that $1_E$ belong to the closure of $\mathscr{L}^f(E)$ for the topology of compact convergence. But the latter coincides with the topology of precompact convergence since E is complete. Hence E is an approximation space.