# topology of locally convex spaces is generated by seminorms

Theorem - Let $V$ be a topological vector space over $\mathbb{R}$ or $\mathbb{C}$. 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.

Title topology of locally convex spaces is generated by seminorms TopologyOfLocallyConvexSpacesIsGeneratedBySeminorms 2013-03-22 17:43:29 2013-03-22 17:43:29 asteroid (17536) asteroid (17536) 4 asteroid (17536) Theorem msc 46-00 msc 46A03