Geodesic
Countable enlargements of norm topologies and the quasidistinguished property
Glasgow mathematical journal, Tome 35 (1993) no. 2, pp. 235-238

Voir la notice de l'article provenant de la source Cambridge University Press

Let E be a Hausdorff locally convex space with continuous dual E1 and let M be a subspace of the algebraic dual E* such that M ∩ E1 = {0} and dim M = א0. In the terminology of [4] the Mackey topology τ(E, E1 + M) is called a countable enlargement of τ(E, E1). There has been some interest in the question of when barrelledness is preserved under countable enlargements (see [4], [5], [6], [8], [9]). In this note we are concerned with the preservation of the quasidistinguished property for normed spaces under countable enlargements; this was posed as on open question by B. Tsirulnikov in [7]. According to [7] a Hausdorff locally convex space E is quasidistinguished if every bounded subset of its completion Ê is contained in the completion of a bounded subset of Ê (equivalently, in the closure in Ê of a bounded subset of E). Any normed space is clearly quasidistinguished and remains so under a finite enlargement (dim M < χ0) since the enlarged topology is normable. (See the Main Theorem of [7] for a general result on the preservation of the quasidistinguished property under finite enlargements.) We shall write QDCE for a countable enlargement which preserves the quasidistinguished property. 
Catalan, F. X.; Tweddle, I. Countable enlargements of norm topologies and the quasidistinguished property. Glasgow mathematical journal, Tome 35 (1993) no. 2, pp. 235-238. doi: 10.1017/S0017089500009794
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