nLab Montel topological vector space

Idea

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.

Definition

A Montel space is a topological vector space that is Hausdorff, locally convex, barreled and where every closed bounded subset is compact.

Properties

Proposition

The strong dual $E^'_b$ of a Montel space is a Montel space. Furthermore, on the bounded subsets of $E^'_b$, the strong and weak topologies coincide.

Examples

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 $U \subset \mathbb{R}^n$ the spaces $C^{\infty}(U)$ and $C_0^{\infty}(U)$ are Montel spaces.

Montel's theorem? of classical complex analysis states that the space $\mathcal{O}(U)$ of holomorphic functions on an open set $U \subset \mathbb{C}$ is a Montel space.

References