Abstract:Let $L^\varphi (X)$ be an Orlicz-Bochner space defined by an Orlicz function $\varphi $ taking only finite values (not necessarily convex) over a $\sigma $-finite atomless measure space. It is proved that the topological dual $L^\varphi (X)^*$ of $L^\varphi (X)$ can be represented in the form: $L^\varphi (X)^*=L^\varphi (X)^\sim _n\oplus L^\varphi (X)^\sim _s$, where $L^\varphi (X)^\sim _n$ and $L^\varphi (X)^\sim _s$ denote the order continuous dual and the singular dual of $L^\varphi (X)$ respectively. The spaces $L^\varphi (X)^*$, $L^\varphi (X)^\sim _n$ and $L^\varphi (X)^\sim _s$ are examined by means of the H. Nakano's theory of conjugate modulars. (Studia Mathematica 31 (1968), 439--449). The well known results of the duality theory of Orlicz spaces are extended to the vector-valued setting.
Keywords: vector-valued function spaces, Orlicz functions, Orlicz spaces, Orlicz-Bochner spaces, topological dual, order dual, order continuous linear functionals, singular linear functionals, modulars, conjugate modulars
AMS Subject Classification: 46E30, 46E40, 46A20