topology of locally convex spaces is generated by seminorms

Theorem - Let $V$ be a topological vector space over $ℝ$ or $ℂ$. Then $V$ is locally convex (http://planetmath.org/LocallyConvexTopologicalVectorSpace) if and only if the topology of $V$ is a family of seminorms.

Moreover, $V$ is Hausdorff and locally if and only if the topology of $V$ is a family of seminorms.