nLab convergence vector space

Idea

A convergence vector space is a generalisation of a topological vector space where the topological structure is replaced by a convergence structure.

Definition

Definition

A convergence vector space is a vector space, say $V$, together with the structure of a convergence space, say $\Lambda$, which is compatible with the vector space structure in that both addition and scalar multiplication are jointly continuous.

Some authors may impose the condition that the convergence structure be separated. Any topological vector space defines a convergence vector space.

Multilinear Mappings

Convergence structures become particularly useful when considering evaluation mappings in functional analysis. For a locally convex topological vector space $E$ with continuous dual $E^*$, the evaluation mapping $E \times E^* \to \mathbb{F}$ is continuous with respect to the product topology if and only if $E$ is normable. This proves to be somewhat of a limitation when one works outside the category of normable LCTVS. However, there is a convergence structure on $E \times E^*$ which makes the evaluation mapping continuous.

Definition

Let $E$ and $F$ be convergence vector spaces and $n \in \mathbb{N}$. Let $\mathcal{L}^n(E,F)$ be the space of continuous mappings $E^n \to F$ which are linear in each argument (see continuous multilinear operator). The continuous convergence structure on $\mathcal{L}^n(E,F)$ is the coarsest convergence structure which makes the evaluation map $\mathcal{L}^n(E,F) \times E^n \to F$ continuous.

We write $\mathcal{L}_c^n(E,F)$ for the resulting convergence space.

Lemma

The convergence space $\mathcal{L}_c^n(E,F)$ is a convergence vector space. It satisfies the following universal property for a convergence space $X$ then a mapping $X \to \mathcal{L}_c^n(E,F)$ is continuous if and only if the associated map $X \times E^n \to F$ is continuous.