Definition 1

Let E be a topological vector space. A vector subspace of E is said to be direct if it admits a topological supplement.

In order that a vector subspace F of E be direct, it is necessary and sufficient that there exist in $\mathscr{L}(E)$ a projector with image F. If $p$ is such a projector, E is the topological direct sum of F and the kernel of $p$. The kernel of $1_E-p$ is F; if E is separated, F is therefore closed.

Let F be a direct vector subspace of E and let $E_1$ be a vector subspace of E containing F; then F is a direct vector subspace of $E_1$.

Every closed vector subspace of finite codimension in E is direct (TVS, I, p. 15, prop. 3), and every vector subspace F of E contained in the closure of $\{0\}$ is direct (for, there exists a projector with image F, and such a projector is necessarily continuous).