linearly compact module

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.

A **linearly compact vector space** over a field $k$ is a topological vector space $V$ over $k$ such that:

- The topology is linear: the open affine subspaces form a basis for the topology.
- Any family of closed affine subspaces with the finite intersection property has nonempty intersection.
- The topology is Hausdorff.

More generally, a **linearly compact module** $M$ over a topological ring $R$ 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.

Every pseudocompact module is linearly compact.

- L. S. Pontrjagin,
*Über stetige algebraische Körper*, Ann. of Math. 33 (1932) 163-174 - N. Jacobson?,
*Totally disconnected locally compact rings*, Amer. J. Math. 58 (1936) 433-449;*A note on topological fields*, Amer. J. Math. 59 (1937) 889-894 - N. Jacobson , O. Taussky , Locally compact rings, Proc. Nat. Acad. Sci. U.S.A. 21 (1935) 106-108
- Solomon Lefschetz,
*Algebraic topology*, Amer. Math. Soc. Colloq. Pub. 27, 1942, reprinted 1963 - I. Kaplansky,
*Topological rings*, Amer. J. Math. 69 (1947) 153-183;*Topological methods in valuation theory*, Duke Math. J. 14 (1947) 527-541;*Locally compact rings I*, Amer. J. Math. 10 (1948) 447-459;*II*, Amer. J. Math. 13 (1951) 20-24;*III*, Amer. J. Math. 14 (1952) 929-935;*Topological representations of algebras II*, Trans. Amer. Math. Soc. 68 (1950) 62-75 - Daniel Zelinsky,
*Linearly compact modules and rings*, American Journal of Mathematics**75**, No. 1 (Jan., 1953), pp. 79-90 jstor - O. Goldman, C. H. Sah,
*On a special class of locally compact rings*, J. Algebra 4 (1966) 71-95;*Locally compact rings of special type, J. Algebra 11 (1969) 363-454* - Tom Leinster, Where Do Linearly Compact Vector Spaces Come From?, $n$-Cat Café 2012
- Tom Leinster,
*Codensity and the ultrafilter monad*, arxiv/1209.3606 - Jean Dieudonné,
*Introduction to the theory of formal groups*, Dekker, New York 1973. - J. Dieudonné,
*Linearly compact spaces and double vector spaces over sfields*, Amer. J. Math. 73, No. 1 (Jan., 1951), pp. 13-19 jstor

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

- S. Warner,
*Topological rings*, North-Holland Math. Studies 178, 1993

A similar concept: *profinite $k$-modules* - is treated in

- A. Yekutieli,
*On the Structure of Behaviors*, Linear Algebra and its Applications 392 (2004), 159-181.

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

- N. Popescu, Abelian categories with applications to rings and modules, London Mathematical Society Monographs 3, Academic Press 1973, xii + 470 pp
- U. Oberst,
*Duality theory for Grothendieck categories and linearly compact rings*, J. Algebra**15**(1970), p. 473 –542, pdf - Mihail Ursul,
*Topological rings satisfying compactness conditions*, Math. and its applications 549, Kluwer 2002, gBooks