A nonstandard density theorem for weak topologies on Banach and Bochner spaces

We prove a nonstandard density result. It asserts that if a particular formula is true for functions in a set K of linear continuous functions between Banach spaces E and D, then it remains valid for functions that are limits, in the uniform convergence topology on a given class M of subsets of E, of nets of vectors in K. We then apply this result to various class M and sets K in the context of E-valued Bochner integrable functions defined on a finite measure space