LAUSR

Let $$(E,\Vert \cdot \Vert _E)$$(E,‖·‖E) be a Banach function space, $$E'$$E′ the Köthe dual of E and $$(X,\Vert \cdot \Vert _X)$$(X,‖·‖X) be a Banach space. It is shown that every Bochner representable operator $$T:E\rightarrow X$$T:E→X maps relatively $$\sigma (E,E')$$σ(E,E′)-compact sets in E onto relatively norm compact sets in X. If, in particular, the associated norm $$\Vert \cdot \Vert _{E'}$$‖·‖E′ on $$E'$$E′ is order continuous, then every Bochner representable operator $$T:E\rightarrow X$$T:E→X is $$(\gamma _E,\Vert \cdot \Vert _X)$$(γE,‖·‖X)-compact, where $$\gamma _E$$γE stands for the natural mixed topology on E. Applications to Bochner representable operators on Orlicz spaces are given.