Corollary 2

— Let M be a finite dimensional vector subspace of E, a locally convex Hausdorff space. Then there exists a closed vector subspace N of E, which is the topological complement of M in E.

There exists a topological complement to M in E if, and only if, the identity mapping of M on itself can be extended to a continuous linear mapping of E on M, which mapping is then necessarily a continuous projector (GT, III, § 6.2, corollary). Now, this follows from the remark above since M is isomorphic to a space $\mathbf{R}^n$ (I, p. 13, th. 2).