Proposition 12

Let E and F be Fréchet spaces. Endow their duals $E'$ and $F'$ with the weak topology, the topology of compact convergence, or the topology of bounded convergence. Let $u\in \mathscr{L}(E; F)$. In order that $u$ be a Fredholm mapping of E into F, it is necessary and sufficient that $^tu$ be a Fredholm mapping of $F'$ into $E'$.

The condition is necessary (III, p. 43, n$^o3$). Let us prove that it is sufficient. If $^tu$ is a Fredholm mapping, it is a strict morphism (III, p. 42, prop. 2), hence $u$ is a strict morphism with closed image (TVS, IV, p. 28, th. 1 and p. 29, cor. 3). In view of the canonical isomorphisms Coker($^tu$)$\rightarrow$ Ker($u$)$'$ and Ker($^tu$)$\rightarrow$ Coker($u$)$'$ (TVS, IV, p. 27, prop. 2), the vector spaces Ker($u$) and Coker($u$) are finite-dimensional and $u$ is a Fredholm mapping (prop. 2 of III, p. 42).