Exercise 6, § 1

Let E be the space $\mathbf{C}^{\mathbf{N}}$ endowed with the topology associated with the sequence of semi-norms $(p_n)_{n\in\mathbf{N}}$ given by $p_n(x) =|x_n|$ for every element $x= (x_n)_{n\in\mathbf{N}}$ of E.

a) The space E is a Fréchet space. b) The set $\mathscr{L}^c(E)$ of compact endomorphisms of E is not closed in $\mathscr{L}(E)$.