The continuous extension theorem might fail if the topological vector space (TVS) is not locally convex. For example, for the Lebesgue space is a complete metrizable TVS (an F-space) that is locally convex (in fact, its only convex open subsets are itself and the empty set) and the only continuous linear functional on is the constant function . Since is Hausdorff, every finite-dimensional vector subspace is linearly homeomorphic to Euclidean space or (by F. Riesz's theorem) and so every non-zero linear functional on is continuous but none has a continuous linear extension to all of However, it is possible for a TVS to not be locally convex but nevertheless have enough continuous linear functionals that its continuous dual space separates points; for such a TVS, a continuous linear functional defined on a vector subspace have a continuous linear extension to the whole space.
The continuous extension theorem might fail if the topological vector space (TVS) is not locally convex For example for the Lebesgue space is a complete metrizable TVS (an F-space) that is locally convex (in fact its only convex open subsets are itself and the empty set) and the only continuous linear functional on is the constant function  Since is Hausdorff every finite-dimensional vector subspace is linearly homeomorphic to Euclidean space or (by F Riesz's theorem) and so every non-zero linear functional on is continuous but none has a continuous linear extension to all of However it is possible for a TVS to not be locally convex but nevertheless have enough continuous linear functionals that its continuous dual space separates points; for such a TVS a continuous linear functional defined on a vector subspace have a continuous linear extension to the whole space