Definition 1

Let E be a topological vector space and F a separated topological vector space. A linear mapping $u$ from E into F is said to be compact if there exists a neighbourhood V of 0 in E such that $u(V)$ is a relatively compact subset of F.

We denote by $\mathscr{L}^c(E; F)$ the set of compact linear mappings from E into F; we also denote by $\mathscr{L}^c(F)$ the set $\mathscr{L}^c(F; F)$.