On a type of mixed norm spaces

In this paper the mixed norm spaces L((B,p,q)) and their duals are investigated. In the case p, 2 < infinity it is proved that the dual of L((B,p,q)) is L((B,p',q')), where p(-1) + p'(-1) = 1 and q(-1) + q'(-1) = 1. For p = 2 and p = infinity an isometric iso- morphism is discussed between the mixed norm space L((B,2,infinity)) and L(infinity)(B,l(2)), the L(infinity)-space of l(2)-valued functions. Here a measurability theorem is proved for l(2)-valued functions. The dual of an important subspace of L((B,2,infinity)) is characterized as a space of vector measures. Finally, as an application we show that if B is finitely generated then the dual of L((B,2,infinity)) is L((B,2,1)).