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On completion of spaces of weakly continuous functions.

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
Uncontrolled Keywords:Topology of uniform convergence on weakly compact subsets
Subjects:Sciences > Mathematics > Mathematical analysis
ID Code:15376
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Deposited On:25 May 2012 08:20
Last Modified:06 Feb 2014 10:23

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