Remark 2

Let E be a locally convex space. Let N be the closure of $\{0\}$ in E and P an algebraic supplementary subspace of N in E. Then P is a topological supplementary subspace of N in E, and is isomorphic to the locally convex space $E/N$. The space N is an approximation space. By Remark 1, in order that E be an approximation space, it is necessary and sufficient that the separated locally convex space $E/N$ be one.