nLab convergence vector space

Convergence vector space


Functional analysis


topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory

Convergence vector space


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



A convergence vector space is a vector space, say VV, together with the structure of a convergence space, say Λ\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 EE with continuous dual E *E^*, the evaluation mapping E×E *𝔽E \times E^* \to \mathbb{F} is continuous with respect to the product topology if and only if EE 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×E *E \times E^* which makes the evaluation mapping continuous.


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

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


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

Last revised on May 23, 2013 at 14:37:28. See the history of this page for a list of all contributions to it.