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Given vector subspaces and of a vector space , we write if is finite-dimensional. We write and say and are commensurable if and .
A Tate vector space is a complete Hausdorff topological vector space that admits a basis of neighborhoods of 0 whose elements are mutually commensurable vector subspaces of .
A vector subspace of a Tate vector space is bounded if for every open vector subspace we have .
The dual of a Tate vector space is equipped with a topology generated by the basis of neighborhoods of 0 whose elements are orthogonal complements to bounded subspaces of .
Tate vector spaces form an pre-abelian category.
John Tate, Residues of differentials on curves, Annales scientifiques de l’École Normale Supérieure, Serie 4, Volume 1 (1968) no. 1, pp. 149-159. DOI.
Alexander Beilinson, Boris Feigin, Barry Mazur, Notes on conformal field theory. PDF
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