Proposition 3

Let $E_1$, E, F, $F_1$ be topological vector spaces, F and $F_1$ being supposed separated. Let $v: E_1\rightarrow E,u: E\rightarrow F$ and $w: F\rightarrow F_1$ be linear mappings. If $v$ and $w$ are continuous and $u$ is compact, then $w\circ u\circ v$ is compact.

By hypothesis, there exists a neighbourhood V of 0 in E such that $u(V)$ is relatively compact in F. Put $U =\overset{-1}{v}(V)$. Then U is a neighbourhood of 0 in $E_1$ and its image by $w\circ u\circ v$ is contained in $w(u$(V)), hence is relatively compact in $F_1$. Consequently, the linear mapping $w\circ u\circ v$ is compact.

Let E be a separated topological vector space; according to Props. 1 and 3, $\mathscr{L}^c(E)$ is a two-sided ideal of the algebra $\mathscr{L}(E)$. When E is a Banach space, $\mathscr{L}(E)$ is a Banach algebra and $\mathscr{L}^c(E)$ is a closed two-sided ideal of $\mathscr{L}(E)$ (cor. to Prop. 2). It is a proper ideal if E is of infinite dimension (cf. Remark 3, p. 2).