When V is a topological vector space then Ψ(x) can still be defined by the same formula, for every , however several difficulties arise. First, when V is not locally convex, the continuous dual may be equal to { 0 } and the map Ψ trivial. However, if V is Hausdorff and locally convex, the map Ψ is injective from V to the algebraic dual of the continuous dual, again as a consequence of the Hahn–Banach theorem.If V is locally convex but not Hausdorff, the kernel of Ψ is the smallest closed subspace containing {0}.
When V is a topological vector space then Ψ(x) can still be defined by the same formula for every  however several difficulties arise First when V is not locally convex the continuous dual may be equal to { 0 } and the map Ψ trivial However if V is Hausdorff and locally convex the map Ψ is injective from V to the algebraic dual of the continuous dual again as a consequence of the Hahn–Banach theoremIf V is locally convex but not Hausdorff the kernel of Ψ is the smallest closed subspace containing {0}