On completion of spaces of weakly continuous functions.

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 ∞ .