Strong topologies on vector-valued function spaces

Let (X, parallel to . parallel to(X)) be a real Banach space and let E be an ideal of L-0 over a sigma-finite measure space (Omega, Sigma, mu). Let (X) be the space of all strongly Sigma-measurable functions f: Omega --> X such that the scalar function (f) over tilde, defined by (f) over tilde(omega) = parallel to f(omega)parallel to(X) for omega is an element of Omega, belongs to E. The paper deals with strong topologies on E(X). In particular, the strong topology beta(E(X), E(X)((n) over tilde)) (E(X)((n) over tilde) = the order continuous dual of E(X)) is examined. We generalize earlier results of [PC] and [FPS] concerning the strong topologies.