locally convex topological vector space

Definition Let $V$ be a topological vector space over a subfield of the complex numbers (usually taken to be $\mathbb{R}$ or $\mathbb{C}$). 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 $L^{p}$ for $0 are examples of spaces which are not locally convex.

References

• 1 G.B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.
Title locally convex topological vector space LocallyConvexTopologicalVectorSpace 2013-03-22 13:44:03 2013-03-22 13:44:03 mathcam (2727) mathcam (2727) 9 mathcam (2727) Definition msc 46A03 msc 46-00