# Topological Vector Spaces/Direct sums

## ExercisesEdit

- Let be a TVS. Prove that all finite-dimensional subspaces of have a topological complement if and only if for every , there exists so that .

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