Montel topological vector space
A Montel space is a topological vector space such that the strong dual has a compactness property that infinite dimensional normed spaces cannot have: The weak and strong topologies coincide.
Important examples are the spaces of distributions.
As a counterexample take an infinite Hilbert space and a sequence of orthonormal vectors, this will converge to zero in the weak topology but does not converge in the strong topology.
A Montel space is a topological vector space that is Hausdorff, locally convex, barreled and where every closed bounded subset is compact.
The strong dual of a Montel space is a Montel space. Furthermore, on the bounded subsets of , the strong and weak topologies coincide.
A normed space is a Montel space iff it is finite dimensional, because only then the closed unit ball is compact.
For an open subset the spaces and are Montel spaces.
Montel's theorem? of classical complex analysis states that the space of holomorphic functions on an open set is a Montel space.
Revised on January 15, 2011 07:59:28
by Toby Bartels