Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-12-02T23:44:56.410Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability and closed graph theorems in classes of bornological spaces

Published online by Cambridge University Press:  18 May 2009

T. K. Mukherjee  and
W. H. Summers
T. K. Mukherjee
Affiliation:
Department of Mathematics, Jadavpur University, Calcutta-700009, India
W. H. Summers
Affiliation:
Department of Mathematics, University of Arkansas, Fayetteville, AR 72701, USA
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the general theory of locally convex spaces, the idea of inductive limit is pervasive, with quotient spaces and the less obvious notion of direct sum being among the instances. Bornological spaces provide another important example. As is well known (cf. [7]), a Hausdorff locally convex space E is bornological if, and only if, E is an inductive limit of normed vector spaces. Going even further in this direction, a complete Hausdorff bornological space is an inductive limit of Banach spaces.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 23 , Issue 2 , July 1982 , pp. 151 - 162
Copyright
Copyright © Glasgow Mathematical Journal Trust 1982

References

REFERENCES

1.Corson, H. H., Normality in subsets of product spaces, Amer. J. Math. 81 (1959), 785–796.CrossRefGoogle Scholar
2.De Wilde, M., Ultrabornological spaces and the closed graph theorem, Bull. Soc. Roy. Sci. Liège 40 (1971), 116–118.Google Scholar
3.Dierolf, S. and Lurje, P., Deux exemples concernant des espaces (ultra) bornologiques, C. R. Acad. Sci. 282 (1976), 1347–1350.Google Scholar
4.Hogbe-Nlend, H., Théorie des bornologies et applications, Lecture Notes in Mathematics 213 (Springer, 1971).Google Scholar
5.Köthe, G., Topological vector spaces I (Springer, 1969).Google Scholar
6.Mahowald, M., Barrelled spaces and the closed graph theorem, J. London Math. Soc. 36 (1961), 108–110.CrossRefGoogle Scholar
7.Robertson, A. P. and Robertson, W. J., Topological vector spaces (Cambridge, 1964).Google Scholar
8.Robertson, W. J. and Yeomans, F. E., On the stability of barrelled topologies, I, Bull. Austral. Math. Soc. 20 (1979), 385–395.CrossRefGoogle Scholar
9.Saxon, S. and Levin, M., Every countable-codimensional subspace of a barrelled space is barrelled, Proc. Amer. Math. Soc. 29 (1971), 91–96.CrossRefGoogle Scholar
10.Schwartz, L., Sur le théorème du graphe fermé, C. R. Acad. Sci. 263 (1966), 602–605.Google Scholar
11.Valdivia, M., Absolutely convex sets in barrelled spaces, Ann. Inst. Fourier (Grenoble) 21 (1971), 3–13.Google Scholar
12.Valdivia, M., A class of bornological barrelled spaces which are not ultrabornological, Math. Ann. 194 (1971), 43–51.CrossRefGoogle Scholar
13.Valdivia, M., On final topologies, J. Reine Angew. Math. 251 (1971), 193–199.Google Scholar
14.Valdivia, M., On nonbornological barrelled spaces, Ann. Inst. Fourier (Grenoble) 22 (1972), 27–30.CrossRefGoogle Scholar
15.Valdivia, M., Some properties of the bornological spaces, Publ. Dip. Math. (Lyon) 10 (1973), 51–56.Google Scholar
16.Valdivia, M., A class of precompact sets in Banach spaces, J. Reine Angew. Math. 276 (1975), 130–136.Google Scholar
17.Valdivia, M., Sur certains hyperplans qui ne sont pas ultra-bornologiques dans les espaces ultra-bornologiques, C. R. Acad. Sci. Sér. A 284 (1977), 935–937.Google Scholar