Proposition 4

Let E and F be separated locally convex spaces, $u: E\rightarrow F$ a Fredholm mapping, and $\widehat{u}:\widehat{E}\rightarrow \widehat{F}$ the extension of $u$ to the completions. Then $\widehat{u}$ is a Fredholm mapping and we have Ker($\widehat{u}$) $=$ Ker($u$) and ind($\widehat{u}$) $=$ ind($u$).

Let us adopt the notation of condition (iv) of Prop. 2 of III, p. 42. Since the vector spaces $E_2$ and $F_2$ are finite-dimensional, they are complete. The completion of $E_1$ (resp. $F_1$) is identified with the closure of $E_1$ in $\widehat{E}$ (resp. of $F_1$ in $\widehat{F}$), and $\widehat{E}$ (resp. $\widehat{F}$) is the topological direct sum of $\widehat{E}_1$ and $E_2$ (resp. $\widehat{F}_1$ and $F_2$). The linear mapping $\widehat{u}$ defines by restriction an isomorphism of $\widehat{E}_1$ onto $\widehat{F}_1$ and vanishes on $E_2$. The proposition then follows from implication (iv)$\Rightarrow$(i) of loc. cit.