Duality and some topological properties of vector-valued function spaces

Let \(E\) be an ideal of \(L^0\) over \(\sigma\)-finite measure space \((\Omega, \Sigma, \mu)\) and let \((X, \| \cdot \|_X)\) be a real Banach space. Let \(E(X)\) be a subspace of the space \(L^0(X)\) of \(\mu\)-equivalence classes of all strongly \(\Sigma\)-measurable functions \(f\colon \Omega \to X\) and consisting of all those \(f\in L^0(X)\), for which the scalar function \(\tilde{f} = \|f (\cdot)\|_X\) belongs to \(E\). Let \(E\) be equipped with a Hausdorff locally convex-solid topology \(\xi\) and let \(\xi\) stand for the topology on \(E(X)\) associated with \(\xi\). We examine the relationship between the properties of the space \((E(X), \xi)\) and the properties of both the spaces \((E, \xi)\) and \((X, \|· \|_X)\). In particular, it is proved that \(E(X)\) (embedded in a natural way) is an order closed ideal of its bidual iff \(E\) is an order closed ideal of its bidual and \(X\) is reflexive. As an application, we obtain that \(E(X)\) is perfect iff \(E\) is perfect and \(X\) is reflexive.