nLab
linearly compact module

Linearly compact vector spaces, modules, rings, objects

Motivation

Linearly compact vector spaces were introduced in the development of the idea of duality. The (algebraic) linear dual of the discrete infinite-dimensional vector space is of larger cardinality so the original space is not isomorphic to the dual of its dual. But if a natural formal topology (which comes say from the filtration of the original space by its finite-dimensional subspaces; the formal topology on the dual is equivalent to consider the dual cofiltration) is given to the algebraic dual, then it makes sense to take the space of continuous linear functional and we recover the original vector space. More precisely this amounts to an embedding of the category of (discrete) vector spaces into the category of linearly compact vector spaces, the latter category has a duality which extends the duality for finite-dimensional vector spaces.

A standard reference for the basics is the Dieudonné‘s book on formal groups.

The next definition is copied from Tom Leinster’s note that’s listed below.

Definition

A linearly compact vector space over a field kk is a topological vector space VV over kk such that:

More generally, a linearly compact module MM over a topological ring RR is a Hausdorff linearly topologized (meaning there is a basis of neighborhoods of 0 consisting of open submodules) such that every family of closed cosets with the finite intersection property (meaning finite subfamilies have nonempty intersections) has nonempty intersection.

Examples

Every pseudocompact module is linearly compact.

Literature

Linearly compact rings and modules are treated in chapter VII, linear compactness and semisimplicity, in

A similar concept: profinite kk-modules - is treated in

The definitions of linearly compact subcategories and linearly compact objects in (co)Grothendieck categories can be found in the chapter on duality in