Lemma 2

Let E be a locally convex space. Then E is an approximation space if and only if, for every precompact subset C of E and every neighbourhood U of 0 in E, there exist an integer $n\geqslant 0$, elements $e_1, . . . , e_n$ of E, and continuous linear forms $f_1, . . . , f_n$ on E, such that $x-\sum^n_{i=1}f_i(x)e_i$ belongs to U for every $x\in C$.

The definition of the topology of precompact convergence shows that the condition is sufficient. Conversely, suppose that E is an approximation space. Let P be a topological complement in E of the closure of $\{0\}($cf. remark 2). Then the set A of the $u\in \mathscr{L}^f(E)$ such that $u(E)\subset P$ is dense in $\mathscr{L}^f(E)$. Since P is separated, every element $u$ of A is of the form $x\mapsto \sum^n_{i=1}f_i(x)e_i$, where $n\in \mathbf{N},e_1, . . . , e_n$ are elements of E and $f_1, . . . , f_n$ continuous linear forms on E (EVT, I, p. 14, th. 2). This proves that the condition is necessary.