### Impacto

Ferrera Cuesta, Juan and Gómez Gil, Javier and Llavona, José G.
(1983)
*On completion of spaces of weakly continuous functions.*
The Bulletin of the London Mathematical Society, 15
(3).
260-264 .
ISSN 0024-6093

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## Abstract

Let E and F be two Banach spaces and let A be a nonempty subset of E . A mapping f:A→F is said to be weakly continuous if it is continuous when A has the relative weak topology and F has the topology of its norm. Let A={E} , B= {A⊂E:A is bounded} and C= {A⊂E:A is weakly compact}. Then C w (E;F) , C wb (E;F) and C wk (E;F) are the spaces of all mappings f:E→F whose restrictions to subsets A⊂E belonging to A , B and C , respectively, are weakly continuous. Clearly, C w (E;F)⊂C wb (E;F)⊂C wk (E;F) , and they are all endowed with the topology of uniform convergence on weakly compact subsets of E . The authors show that C wk (E;F) is the completion of C w (E;F) . They also show that, when E has no subspace isomorphic to l 1 , then C wb (E;F)=C wk (E;F) . When E has the Dunford-Pettis property and contains a subspace isomorphic to l 1 , the authors prove that C wb (E;F) is a proper subspace of C wk (E;F) . The same conclusion holds when E is a Banach space that contains a subspace isomorphic to l ∞ .

Item Type: | Article |
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Uncontrolled Keywords: | Topology of uniform convergence on weakly compact subsets |

Subjects: | Sciences > Mathematics > Mathematical analysis |

ID Code: | 15376 |

References: | R. M. ARON, C. HERVES and M. VALDIVIA, 'Weakly continuous mappings on Banach spaces', J. Fund. Anal, (to appear). R. M. ARON and J. B. PROLLA, 'Polynomial approximation of differentiable functions on Banach spaces', J. reine angew. Math., 313 (1980), 195-216. C. H. DOWKER, 'On a theorem of Hanner', Ark. Mat., 2 (1954), 307-313. R. E. EDWARDS, Functional analysis (Holt-Rinehart-Winston, 1965). K. FLORET, Weakly compact sets, Lecture Notes in Mathematics 801 (Springer-Verlag, Berlin, 1980). J.GOMEZ GIL, Espacios de funciones debilmente diferenciables, doctoral thesis (Univ. Complutense Madrid, 1981). A. GROTHENDIECK, 'Sur les applications faiblement compactes d'espaces du type C{K)\ Canadian J. Math., 5 (1953), 129-173. J. LINDENSTRAUSS and L. TZAFRIRI, Classical Banach spaces I (Springer-Verlag, Berlin, 1977). A. PELCZYNSKI, 'On Banach spaces containing L,(/i)\ Studia Math., 30 (1968), 231-246. R. S. PHILLIPS, 'On linear transformations', Trans. Amer. Math. Soc, 48 (1940), 516-541. H. P. ROSENTHAL, 'Some recent discoveries in the isomorphic theory of Banach spaces', Bull. Amer. Math. Soc. 84, 5 (1978), 803-831. Z. SEMADENI, Banach spaces of continuous functions, Monografia Matematyczne 55 (PWN Warszawa 1971). |

Deposited On: | 25 May 2012 08:20 |

Last Modified: | 06 Feb 2014 10:23 |

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