Non-separably valued Orlicz spaces, part I

TL;DR

This paper extends the theory of Orlicz spaces to arbitrary Banach range spaces, characterizing their structure and duals, and applies these results to convex analysis of integral functionals with non-separable spaces.

Contribution

It introduces a comprehensive framework for Orlicz spaces with non-separable range Banach spaces, including dual representation and an interchange criterion for infimum and integral.

Findings

Characterization of completeness, separability, and reflexivity of the spaces.

Representation of dual spaces including non-Radon-Nikodym cases.

First general representation of convex conjugates and subdifferentials for such spaces.

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.