# Article

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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
Summary:
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.
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