Proposition 5

Let E be a locally convex space, F a separated topological vector space and $u$ a compact linear mapping of E into F. Then there exist a Banach space G, a continuous linear mapping $v$ of E into G and a compact linear mapping $w$ of G into F such that $u=w\circ v$.

Let V be a neighbourhood of 0 in E such that $u(V)$ is relatively compact in F. Let $p$ be a continuous seminorm on E such that V contains the unit ball of $p$ (EVT, II, p. 26, cor.). Denote by $E_p$ the seminormed space obtained by endowing E with the seminorm $p$. The mapping $u$ is a compact linear mapping of $E_p$ into F. The separated completion G of $E_p$ is a Banach space. Denote by $v$ the canonical linear mapping of $E_p$ into G. According to Prop. 4, there exists a compact linear mapping $w: G\rightarrow F$ such that $u=w\circ v$. On the other hand, $v$ is a continuous mapping of E into G because the topology of E is finer than that of $E_p$.