locally convex topological vector space

Definition Let V be a topological vector spaceMathworldPlanetmath over a subfieldMathworldPlanetmath of the complex numbersMathworldPlanetmathPlanetmath (usually taken to be or ). If the topology of V has a basis where each member is a convex set, then V is a locally convex topological vector space [1].

Though most vector spaces occurring in practice are locally convex, the spaces Lp for 0<p<1 are examples of spaces which are not locally convex.


  • 1 G.B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.
Title locally convex topological vector space
Canonical name LocallyConvexTopologicalVectorSpace
Date of creation 2013-03-22 13:44:03
Last modified on 2013-03-22 13:44:03
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 9
Author mathcam (2727)
Entry type Definition
Classification msc 46A03
Classification msc 46-00