Geodesic
On the closed graph theorem
Glasgow mathematical journal, Tome 17 (1976) no. 2, pp. 89-97

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

Our main purpose is to describe those separated locally convex spaces which can serve as domain spaces for a closed graph theorem in which the range space is an arbitrary Banach space of (linear) dimension at most c, the cardinal number of the real line R. These are the δ-barrelled spaces which are considered in §4. Many of the standard elementary Banach spaces, including in particular all separable ones, have dimension at most c. Also it is known that an infinite dimensional Banach space has dimension at least c (see e.g. [8]). Thus if we classify Banach spaces by dimension we are dealing, in a natural sense, with the first class which contains infinite dimensional spaces. 
Popoola, J. O.; Tweddle, I. On the closed graph theorem. Glasgow mathematical journal, Tome 17 (1976) no. 2, pp. 89-97. doi: 10.1017/S0017089500002780
@article{10_1017_S0017089500002780,
     author = {Popoola, J. O. and Tweddle, I.},
     title = {On the closed graph theorem},
     journal = {Glasgow mathematical journal},
     pages = {89--97},
     year = {1976},
     volume = {17},
     number = {2},
     doi = {10.1017/S0017089500002780},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089500002780/}
}
TY  - JOUR
AU  - Popoola, J. O.
AU  - Tweddle, I.
TI  - On the closed graph theorem
JO  - Glasgow mathematical journal
PY  - 1976
SP  - 89
EP  - 97
VL  - 17
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089500002780/
DO  - 10.1017/S0017089500002780
ID  - 10_1017_S0017089500002780
ER  - 
%0 Journal Article
%A Popoola, J. O.
%A Tweddle, I.
%T On the closed graph theorem
%J Glasgow mathematical journal
%D 1976
%P 89-97
%V 17
%N 2
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089500002780/
%R 10.1017/S0017089500002780
%F 10_1017_S0017089500002780

[1] 1.Adasch, N., Tonnelierte Räume und zwei Sätze von Banach, Math. Ann. 186 (1970), 209–214. Google Scholar | DOI

[2] 2.Bourbaki, N., Espaces vectoriels topologiques, Chap. III–V (Paris, 1967), 121. Google Scholar

[3] 3.De Wilde, M. and Houet, C., On increasing sequences of absolutely convex sets in locally convex spaces, Math. Ann. 192 (1971), 257–261. Google Scholar | DOI

[4] 4.Dugundji, J., Topology (Boston, 1966), 175–176. Google Scholar

[5] 5.Husain, T., Two new classes of locally convex spaces, Math. Ann. 166 (1966), 289–299. Google Scholar | DOI

[6] 6.Iyahen, S. O., On the closed graph theorem, Israel J. Math. 10 (1971), 96–105. Google Scholar | DOI

[7] 7.Kalton, N. J., Some forms of the closed graph theorem, Proc. Cambridge Philos. Soc. 70 (1971), 401–408. Google Scholar | DOI

[8] 8.Lacey, H. E., The Hamel dimension of any infinite dimensional separable Banach space is c. Amer. Math. Monthly 80 (1973), 298. Google Scholar

[9] 9.Levin, M. and Saxon, S., A note on the inheritance of properties of locally convex spaces by subspaces of countable codimension, Proc. Amer. Math. Soc. 29 (1971), 97–102. Google Scholar | DOI

[10] 10.Persson, A., A remark on the closed graph theorem in locally convex vector spaces, Math. Scand. 19 (1966), 54–58. Google Scholar | DOI

[11] 11.Pták, V., Completeness and the open mapping theorem, Bull. Soc. Math. France 86 (1958), 41–74. Google Scholar | DOI

[12] 12.Robertson, A. P. and Robertson, W. J., Topological vector spaces, 2nd edition (Cambridge, 1973). Google Scholar

[13] 13.Saxon, S. and Levin, M., Every countable-codimensional subspace of a barrelled space is barrelled, Proc. Amer. Math. Soc. 29 (1971), 91–96. Google Scholar | DOI

[14] 14.Sulley, L. J., On B(I) and B,(I) locally convex spaces, Proc. Cambridge Philos. Soc. 68 (1970), 95–97. Google Scholar | DOI

[15] 15.Valdivia, M., Absolutely convex sets in barrelled spaces, Ann. Inst. Fourier, Grenoble 21 (1971), 3–13. Google Scholar | DOI

[16] 16.Webb, J. H., Countable-codimensional subspaces of locally convex spaces, Proc. Edinburgh Math. Soc. 18 (Series II) (1973), 167–172. Google Scholar | DOI

Cité par Sources :