Characterization of very smooth points for nonseparable duals

Characterize the very smooth points of the Bochner space Lp(μ, X) in terms of the very smooth points of X for 1 < p < ∞ when the Banach space X has a nonseparable dual.

Background

The paper establishes, when X∗ is separable, that a point f in the unit sphere of Lp(μ, X) is very smooth if and only if the normalized values f(t)/∥f(t)∥ are very smooth points of X almost everywhere on the set where f(t) is nonzero, for 1 < p < ∞. The proof uses the relationship between very smoothness and weak-star–weak points of continuity, together with the paper’s pointwise characterization for Bochner function spaces.

Question 3.7 leaves open whether the same equivalence can be proved for Banach spaces whose duals are not separable.