Dense barrelled subspaces of uncountable codimension
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- by Stephen A. Saxon and Wendy J. Robertson PDF
- Proc. Amer. Math. Soc. 107 (1989), 1021-1029 Request permission
Abstract:
Let $E$ be a Hausdorff barrelled space. If there exists a dense barrelled subspace $M$ such that $(\operatorname {codim} (M) \geq c)[\operatorname {codim} (M) = \operatorname {dim} (E)]$, we say that ($M$ is a satisfactory subspace [11]) [$E$ is barrelledly fit], respectively. Robertson, Tweddle and Yeomans [11] proved that $E$ has a barrelled countable enlargement (BCE) if it has a satisfactory subspace. (Trivially) $E$ has a satisfactory subspace if $\dim (E) \geq c$ and $E$ is barrelledly fit. We show that $E$ is barrelledly fit (and $\dim (E) \geq c$) if $E \ncong \varphi$ and either (i) $E$ is an (LF)-space, or (ii) $E$ is an infinite-dimensional separable space and the continuum hypothesis holds. Conclusion: barrelledly fit spaces and their permanence properties arise from and advance the study of BCE’s.References
- José Bonet and Pedro Pérez Carreras, Remarks on the stability of barreled-type topologies, Bull. Soc. Roy. Sci. Liège 52 (1983), no. 5, 313–318. MR 731286
- José Bonet and Pedro Pérez Carreras, On the three-space problem for certain classes of Baire-like spaces, Bull. Soc. Roy. Sci. Liège 51 (1982), no. 9-12, 381–385. MR 705014
- Marc De Wilde and Bella Tsirulnikov, Barrelledness and the supremum of two locally convex topologies, Math. Ann. 246 (1979/80), no. 3, 241–248. MR 563402, DOI 10.1007/BF01371045 —, Barrelled spaces with a $B$-complete completion, Manuscripta Math. 33 (1981), 411-427.
- Lech Drewnowski, A solution to a problem of De Wilde and Tsirulnikov, Manuscripta Math. 37 (1982), no. 1, 61–64. MR 649564, DOI 10.1007/BF01239945
- John Horváth, Topological vector spaces and distributions. Vol. I, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966. MR 0205028 G. Köthe, Topological vector spaces I, Springer-Verlag, Berlin, 1969.
- Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR 597342
- A. P. Robertson and W. J. Robertson, Topological vector spaces, Cambridge Tracts in Mathematics and Mathematical Physics, No. 53, Cambridge University Press, New York, 1964. MR 0162118
- W. J. Robertson, S. A. Saxon, and A. P. Robertson, Barrelled spaces and dense vector subspaces, Bull. Austral. Math. Soc. 37 (1988), no. 3, 383–388. MR 943441, DOI 10.1017/S0004972700027003
- I. Tweddle and F. E. Yeomans, On the stability of barrelled topologies. II, Glasgow Math. J. 21 (1980), no. 1, 91–95. MR 558281, DOI 10.1017/S0017089500004043
- W. Roelcke and S. Dierolf, On the three-space-problem for topological vector spaces, Collect. Math. 32 (1981), no. 1, 13–35. MR 643398
- Stephen Saxon and Mark Levin, Every countable-codimensional subspace of a barrelled space is barrelled, Proc. Amer. Math. Soc. 29 (1971), 91–96. MR 280972, DOI 10.1090/S0002-9939-1971-0280972-0
- Stephen A. Saxon and P. P. Narayanaswami, Metrizable (LF)-spaces, (db)-spaces, and the separable quotient problem, Bull. Austral. Math. Soc. 23 (1981), no. 1, 65–80. MR 615133, DOI 10.1017/S0004972700006900
- P. P. Narayanaswami and Stephen A. Saxon, (LF)-spaces, quasi-Baire spaces and the strongest locally convex topology, Math. Ann. 274 (1986), no. 4, 627–641. MR 848508, DOI 10.1007/BF01458598
- Stephen A. Saxon and P. P. Narayanaswami, Metrizable [normable] $(\textrm {LF})$-spaces and two classical problems in Fréchet [Banach] spaces, Studia Math. 93 (1989), no. 1, 1–16. MR 989565, DOI 10.4064/sm-93-1-1-16
- Stephen A. Saxon and Albert Wilansky, The equivalence of some Banach space problems, Colloq. Math. 37 (1977), no. 2, 217–226. MR 511780, DOI 10.4064/cm-37-2-217-226 S. Saxon, The codensity character of topological vector spaces (in preparation). —, Barrelled spaces and two axiomatic conditions (in preparation). —, The fit and flat components of barrelled spaces (in preparation).
- Bella Tsirulnikov, On conservation of barrelledness properties in locally convex spaces, Bull. Soc. Roy. Sci. Liège 49 (1980), no. 1-2, 5–25 (English, with French summary). MR 586948
- I. Tweddle, Barrelled spaces whose bounded sets have at most countable dimension, J. London Math. Soc. (2) 29 (1984), no. 2, 276–282. MR 744098, DOI 10.1112/jlms/s2-29.2.276
- Manuel Valdivia, On suprabarrelled spaces, Functional analysis, holomorphy, and approximation theory (Proc. Sem., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1978) Lecture Notes in Math., vol. 843, Springer, Berlin, 1981, pp. 572–580. MR 610847
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 107 (1989), 1021-1029
- MSC: Primary 46A07
- DOI: https://doi.org/10.1090/S0002-9939-1989-0990433-2
- MathSciNet review: 990433