Proposition 9

Let E be a locally convex space, F a separated locally convex space and $u$ a continuous linear mapping of E into F.

a) If the mapping $u$ is compact, then its transpose $^tu$ is a compact linear mapping of $F'_c$ into $E'_c$ and a continuous linear mapping of $F'_c$ into $E'_b$;

b) Suppose that the space E is a semi-normed space and that the space F is quasi-complete. If the transpose $^tu$ is a continuous mapping of $F'_c$ into $E'_b$, then the linear mapping $u$ is compact.

Suppose first that the linear mapping $u$ is compact. Then there exists a neighbourhood V of 0 in E whose image under $u$ is contained in a compact subset C of F. By definition (TVS, II, p. 47, def. 2 and p. 68, def. 1), the polar $C^{\circ}$ of C is a neighbourhood of 0 in $F'_c$ and one has $^tu(C^{\circ})\subset V^{\circ}$. Now $V^{\circ}$ is an equicontinuous subset of $E'$, closed for the weak topology, hence compact in $E'_c$ (TVS, III, p. 17, cor. 2 and prop. 5). This proves that $^tu$ is a compact linear mapping of $F'_c$ into $E'_c$.

Retain the preceding hypotheses and let U be a neighbourhood of 0 in $E'_b$. It contains the polar $A^{\circ}$ of a bounded subset A of E. Since the linear mapping $u$ is compact, the set $u(A)$ is relatively compact in F (remark 1 of III, p. 2), and $(^tu)^{-1}(A^{\circ}) =u(A)^{\circ}$ is a neighbourhood of 0 in $F'_c$. This proves that $^tu$ is a continuous linear mapping of $F'_c$ into $E'_b$.

Let us now place ourselves under the assumptions of b) and denote by B the unit ball of E. The polar $B^{\circ}$ of B is the unit ball of $E'_b$ and one has $(^tu)^{-1}(B^{\circ}) =u(B)^{\circ}$. If $^tu$ is a continuous mapping of $F'_c$ into $E'_b$, the set $u(B)^{\circ}$ is therefore a neighbourhood of 0 in $F'_c$; it therefore contains the polar $C^{\circ}$ of a compact subset C of F. By the bipolar theorem (EVT, II, p. 49, cor. 3), the set $u(B)$ is contained in the closed convex envelope of $C\cup \{0\}$; this is compact because the space F is quasi-complete (EVT, III, p. 8). This proves that the linear mapping $u$ is compact.