Ferrera Cuesta, Juan
(1982)
*Spaces of weakly continuous functions.*
Pacific Journal of Mathematics , 102
(2).
pp. 285-291.
ISSN 0030-8730

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## Abstract

This paper is very much in the spirit of a paper by H. Corson [Trans. Amer. Math. Soc. 101 (1961),

1–15; MR0132375 (24 2220)]. Let E be a real Banach space. The bw-topology on E is the finest

topology which agrees with the weak topology on all bounded subsets of E. Cwb(E) [Cwbu(E)]

is the set of real functions which are weakly continuous [weakly uniformly continuous] on all

bounded sets in E. Cwb(E) is always barrelled; a sufficient condition is given for Cwb(E) to be

bornological (under the compact-open topology). As a main result, the following are shown to be

equivalent: (1) E is reflexive; (ii) Cwbu(E) is a Fr´echet space; (iii) Cwbu(E) is a Pt´ak space; (iv)

Cwbu(E) is complete; (v) Cwbu(E) is barrelled; (vi) Cwbu(E) = Cwb(E).

Item Type: | Article |
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Subjects: | Sciences > Mathematics > Mathematical analysis |

ID Code: | 15380 |

References: | H. H. Corson, The weak topology of a Banach space, Trans. Amer. Math. Soc , 101 (1961), 1-15. J. Diestel, Geometry of Banach Spaces, Selected Topics. Lect. Notes in Math., 485 Springer-Verlag, New York, 1975. J. Ferrera J. Gomez and J. L. G. Llavona, On completion of spaces of weakly continuous, functions, to appear in J. London Math. Soc. L. Gilman and M. Jerison, Rings of Continuous Functions, Van Norstrand, 1960. J. Kelley, General Topology, Springer. G. Kothe, Topological Vector Spaces I, Springer-Verlag, New York, 1969. L. Nachbin, Topological vector spaces of continuous functions, Proc. Nat. Acad. Sci. U.S.A., 40 (1954), 471-474. T. Shirota, On locally convex vector spaces of continuous functions, Proc. Japan Acad., 30 (1954), 294-298. M. Talagrand, Sur une conjecture de B. H. Corson, Bull. Sci. Math., (2), 99 (1975), 211-212. M. Valdivίa, Some new results on weak compactness, J. Functional Anal., 24 (1977), 1-10. S. Warner, The topology of compact convergence on continuous function spaces, Duke Math. J., 25, 265-282. |

Deposited On: | 28 May 2012 08:33 |

Last Modified: | 06 Feb 2014 10:23 |

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