Non-separably valued Orlicz spaces, part I

Abstract

For a measure space $\Omega$ we extend the theory of Orlicz spaces generated by an even convex integrand $\varphi \colon \Omega \times X \to \left[ 0, \infty \right]$ to the case when the range Banach space $X$ is arbitrary. Besides settling fundamental structural properties such as completeness, we characterize separability, reflexivity and represent the dual space. This representation includes the cases when $X'$ has no Radon-Nikodym property or $\varphi$ 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 of independent interest.