Exercise 1, § 5

Let $E$ be a locally convex space.

(a) Suppose that every neighbourhood of 0 in $E$ contains a vector subspace of $E$ of finite codimension. Now prove that the space $\mathscr{F}(E)$ of Fredholm endomorphisms of $E$ is not closed in $\mathscr{L}(E)$ endowed with the topology of bounded convergence.

(b) Give an example of an infinite-dimensional Fréchet space such that every neighbourhood of 0 contains a vector subspace of $E$ of finite codimension. (c) Let $E_{\sigma}$ be an infinite-dimensional Banach space endowed with the weakened topology. The set $\mathscr{F}(E_{\sigma})$ is not open in $\mathscr{L}(E_{\sigma})$.