Corollary 1 (Schauder)

Let E be a seminormed space, F a Banach space, $E'$ and $F'$ their respective strong duals, and $u$ a continuous linear mapping of E into F. The following properties are equivalent:

(i) The linear mapping $u$ of E into F is compact;

(ii) The linear mapping $^tu$ of $F'$ into $E'$ is compact;

(iii) The linear mapping $^tu$ of $F'_c$ into $E'$ is continuous.

The equivalence of (i) and (iii) results from prop. 9, a) and b); the implication (iii)$\Rightarrow$(ii) follows from the fact that the identity mapping of $F'$ into $F'_c$ is compact (cor. of prop. 7 of III, p. 7).

To conclude, let us prove that condition (ii) implies (i). Suppose that the linear mapping $^tu: F'\rightarrow E'$ is compact. It results from the implication (i)$\Rightarrow$(ii), applied to $^tu$, that $^{tt}u$ is a compact linear mapping of $E''$ into $F''$. Let us denote by $v$ the canonical linear mapping of E into $E''$; it is continuous. Since F is identified with a closed vector subspace of $F''$ and since $u$ coincides with $^t(^tu)\circ v$ on E, it results from remark 5 of III, p. 3 that the linear mapping $u$ is compact.