E-approximation properties

In this paper, we introduce the E -approximation property and the E u -approximation property which generalize Sinha and Karn’s p -approximation property. Characterizations of these properties are given parallel to Lima and Oja’s results on the strong approximation property and the weak bounded approximation property. Representation theorems for the dual of L ( X , Y ) under the topology of uniform convergence on E -compact sets and E u -compact sets are also provided. As an application, the representations are used to generalize the main theorem of Choi and Kim in [ 1 ]. These results build up fundamental bases for future investigations of these properties.