Generalized Orlicz spaces of Banach-valued functions: Basic theory and duality

For a measure space Ω, we extend the theory of Orlicz spaces generated by an even convex integrand φ:Ω×X→[0,∞] to the case when the range Banach space X is arbitrary. We settle fundamental structural properties such as completeness, characterize separability, reflexivity and represent the dual space. This representation includes the case when X′ has no Radon-Nikodym property or φ is unbounded. We apply our theory to represent convex conjugates and Fenchel-Moreau subdifferentials of integral functionals, leading to the first general such result on function spaces with non-separable range space. For this, we prove a new interchange criterion between infimum and integral for non-separable range spaces, which we consider to be of independent interest.