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.

Title	topology of locally convex spaces is generated by seminorms
Canonical name	TopologyOfLocallyConvexSpacesIsGeneratedBySeminorms
Date of creation	2013-03-22 17:43:29
Last modified on	2013-03-22 17:43:29
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	4
Author	asteroid (17536)
Entry type	Theorem
Classification	msc 46-00
Classification	msc 46A03