Proposition 12

Let E be a locally convex space, I a set, and for each $i\in I$, let $F_i$ be a locally convex space and $v_i: E\rightarrow F_i$ a continuous linear mapping with dense image. Suppose that for every neighbourhood U of 0 in E, there exist $i\in I$ and a neighbourhood V of 0 in $F_i$ such that $\overset{-1}{v_{i}}(V)\subset U$. If the $F_i$ are approximation spaces, then E is also one.

Let A be a precompact subset of E and U a neighbourhood of 0 in E. By hypothesis there exist $i\in I$ and a continuous semi-norm $p$ on $F_i$ such that U contains $(p\circ v_i)^{-1}([0,1])$. Put $F = F_i,v=v_i$ and $B =v$(A), and suppose that F is an approximation space. The set B is precompact in F. There therefore exist (lemma 2) an integer $n\geqslant 1$, elements $y_1, . . . , y_n$ of F and continuous linear forms $f_1, . . . , f_n$ on F, such that one has, for every $y\in B$,

$j=1$

Since B is bounded (EVT, III, p. 3, prop. 2), there exists a real number $M>0$ such that $|f_j(y)|\leqslant M$ for every $j$ such that $1\leqslant j\leqslant n$ and every $y\in B$. Moreover, since $v(E)$ is dense in F, there exists, for each integer $j$ such that $1\leqslant j\leqslant n$, an element $x_j$ of E such that $p(y_j-v(x_j))\leqslant (2nM)^{-1}$. The linear mapping

belongs to $\mathscr{L}^f(E)$. For every $x\in A$, one has

whence $p(v(x-u(x)))\leqslant \frac{1}{2}+\frac{nM}{2nM}= 1$, and therefore $x-u(x)\in U$. The proposition follows.