Geodesic
Quasidistinguished countable enlargements of normed spaces
Glasgow mathematical journal, Tome 38 (1996) no. 1, pp. 65-68

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

If E is a Hausdorff locally convex space and M is an -dimensional subspace of the algebraic dual E* that is transverse to the continuous dual E′, then, according to [7], the Mackey topology τ(E, E′ + M) is a countable enlargement (CE) of τ(E, E′) [or of E]. Much is still unknown as to when CEs preserve barrelledness (cf. [14]). E is quasidistinguished (QD) if each bounded subset of the completion Ê is contained in the completion of a bounded subset of E [12]. Clearly, each normed space is QD, and Tsirulnikov [12] asked if each CE of a normed space must be a QDCE, i.e., must preserve the QD property. Since CEs preserve metrizability (but not normability), her question was whether metrizable spaces so obtained must be QD, and was moderated by Amemiya's negative answer (cf. [5, p. 404]) to Grothendieck's query, who had asked if all metrizable spaces are QD, having proved the separable ones are [4]. 
Saxon, S. A.; Ruiz, L. M. Sànchez. Quasidistinguished countable enlargements of normed spaces. Glasgow mathematical journal, Tome 38 (1996) no. 1, pp. 65-68. doi: 10.1017/S001708950003127X
@article{10_1017_S001708950003127X,
     author = {Saxon, S. A. and Ruiz, L. M. S\`anchez},
     title = {Quasidistinguished countable enlargements of normed spaces},
     journal = {Glasgow mathematical journal},
     pages = {65--68},
     year = {1996},
     volume = {38},
     number = {1},
     doi = {10.1017/S001708950003127X},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S001708950003127X/}
}
TY  - JOUR
AU  - Saxon, S. A.
AU  - Ruiz, L. M. Sànchez
TI  - Quasidistinguished countable enlargements of normed spaces
JO  - Glasgow mathematical journal
PY  - 1996
SP  - 65
EP  - 68
VL  - 38
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S001708950003127X/
DO  - 10.1017/S001708950003127X
ID  - 10_1017_S001708950003127X
ER  - 
%0 Journal Article
%A Saxon, S. A.
%A Ruiz, L. M. Sànchez
%T Quasidistinguished countable enlargements of normed spaces
%J Glasgow mathematical journal
%D 1996
%P 65-68
%V 38
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1017/S001708950003127X/
%R 10.1017/S001708950003127X
%F 10_1017_S001708950003127X

[1] 1.Catálan, F. X. and Tweddle, I., Countable enlargements of norm topologies and the quasidistinguished property. Glasgow Math. J. 35 (1993), 235–238. Google Scholar | DOI

[2] 2.van Douwen, E. K., The integers and topology, pp. 111–168 in Handbook of set-theoretic topology, (eds.: Kunen, K. and Vaughan, J. E.). (North Holland 1984). Google Scholar | DOI

[3] 3.Engelking, R., General Topology. PWN 1977. Google Scholar

[4] 4.Grothendieck, A., Sur les espaces (F) et (DF). Summa Brasil. Math. 3 (1954) 57–123. Google Scholar

[5] 5.Köthe, G., Topological Vector Spaces (Springer Verlag, 1969). Google Scholar

[6] 6.Robertson, A. P. and Robertson, W. J., Topological Vector Spaces (Cambridge 1973). Google Scholar

[7] 7.Robertson, W. J, Tweddle, I. and Yeomans, F. E., On the stability of barrelled topologies III. Bull. Austr. Math. Soc. 22 (1980), 99–112. Google Scholar | DOI

[8] 8.Saxon, S. A., The codensity character of topological vector spaces, in Topological Vector Spaces, Algebras & Related Areas (eds. Lau, A. and Tweddle, I.), (Longman, 1994), 24–36. Google Scholar

[9] 9.Saxon, S. A. and Ruiz, L. M. Sánchez, Optimal cardinals for metrizable barrelled spaces. J. London Math. Soc. (2) 51 (1995), 137–147. Google Scholar | DOI

[10] 10.Saxon, S. A. and Ruiz, L. M Sánchez, Barrelled countable enlargements and the bounding cardinal. J. London Math. Soc., to appear. Google Scholar

[11] 11.Saxon, S. A. and Ruiz, L. M. Sánchez, Barralled countable enlargements and the dominating cardinal. Preprint. Google Scholar

[12] 12.Tsirulnikov, B., On the locally convex noncomplete quasi-distinguished spaces. Bull. Soc1. Roy. Sci. Liège 47 (1978), 147–152. Google Scholar

[13] 13.Tweddle, I., Barrelled spaces whose bounded sets have at most countable dimension. J. London Math. Soc. (2) 29 (1984), 276–282. Google Scholar | DOI

[14] 14.Tweddle, I., Saxon, S. A. and Ruiz, L. M. Sánchez, Barrelled countable enlargements, in Topological Vector Spaces, Algebras & Related Areas (eds. Lau, A. and Tweddle, I.) (Longman, 1994), 3–15. Google Scholar

Cité par Sources :