Definition:Locally Convex Space/Hausdorff

Definition
Let $\struct {X, \mathcal P}$ be a locally convex space. We say that $\struct {X, \mathcal P}$ is Hausdorff :


 * for all $x \in X$ with $x \ne 0$ there exists $p \in \mathcal P$ such that $\map p x \ne 0$.

Also see

 * Locally Convex Space is Hausdorff iff induces Hausdorff Topology justifies the use of the term "Hausdorff" by showing that the standard topology on a Hausdorff locally convex space is Hausdorff. Coonversely, if the standard topology is Hausdorff, then the space is Hausdorff as a locally convex space.