### Topological dual of non-locally convex Orlicz-Bochner spaces

Marian Nowak (1999)

Commentationes Mathematicae Universitatis Carolinae

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Let ${L}^{\varphi}\left(X\right)$ 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 ${L}^{\varphi}{\left(X\right)}^{*}$ of ${L}^{\varphi}\left(X\right)$ can be represented in the form: ${L}^{\varphi}{\left(X\right)}^{*}={L}^{\varphi}{\left(X\right)}_{n}^{\sim}\oplus {L}^{\varphi}{\left(X\right)}_{s}^{\sim}$, where ${L}^{\varphi}{\left(X\right)}_{n}^{\sim}$ and ${L}^{\varphi}{\left(X\right)}_{s}^{\sim}$ denote the order continuous dual and the singular dual of ${L}^{\varphi}\left(X\right)$ respectively. The spaces ${L}^{\varphi}{\left(X\right)}^{*}$, ${L}^{\varphi}{\left(X\right)}_{n}^{\sim}$ and ${L}^{\varphi}{\left(X\right)}_{s}^{\sim}$ 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...