# Topological Vector Spaces/Direct sums

1. Let ${\displaystyle E}$ be a TVS. Prove that all finite-dimensional subspaces of ${\displaystyle E}$ have a topological complement if and only if for every ${\displaystyle x\notin {\overline {\{0\}}}}$, there exists ${\displaystyle x'\in E'}$ so that ${\displaystyle x'(x)\neq 0}$.