Let X be a Banach space and E be a perfect Banach function space over a finite measure space (Ω,Σ,λ)\documentclass[12pt]{minimal}
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\begin{document}$$(\Omega ,\Sigma ,\lambda )$$\end{document} such that L∞⊂E⊂L1\documentclass[12pt]{minimal}
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\begin{document}$$L^\infty \subset E\subset L^1$$\end{document}. Let E′\documentclass[12pt]{minimal}
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\begin{document}$$E'$$\end{document} denote the Köthe dual of E and τ(E,E′)\documentclass[12pt]{minimal}
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\begin{document}$$\tau (E,E')$$\end{document} stand for the natural Mackey topology on E. It is shown that every nuclear operator T:E→X\documentclass[12pt]{minimal}
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\begin{document}$$T:E\rightarrow X$$\end{document} between the locally convex space (E,τ(E,E′))\documentclass[12pt]{minimal}
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\begin{document}$$(E,\tau (E,E'))$$\end{document} and a Banach space X is Bochner representable. In particular, we obtain that a linear operator T:L∞→X\documentclass[12pt]{minimal}
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\begin{document}$$T:L^\infty \rightarrow X$$\end{document} between the locally convex space (L∞,τ(L∞,L1))\documentclass[12pt]{minimal}
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\begin{document}$$(L^\infty ,\tau (L^\infty ,L^1))$$\end{document} and a Banach space X is nuclear if and only if its representing measure mT:Σ→X\documentclass[12pt]{minimal}
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\begin{document}$$m_T:\Sigma \rightarrow X$$\end{document} has the Radon-Nikodym property and |mT|(Ω)=‖T‖nuc\documentclass[12pt]{minimal}
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\begin{document}$$|m_T|(\Omega )=\Vert T\Vert _{nuc}$$\end{document} (= the nuclear norm of T). As an application, it is shown that some natural kernel operators on L∞\documentclass[12pt]{minimal}
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\begin{document}$$L^\infty $$\end{document} are nuclear. Moreover, it is shown that every nuclear operator T:L∞→X\documentclass[12pt]{minimal}
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\begin{document}$$T:L^\infty \rightarrow X$$\end{document} admits a factorization through some Orlicz space Lφ\documentclass[12pt]{minimal}
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\begin{document}$$L^\varphi $$\end{document}, that is, T=S∘i∞\documentclass[12pt]{minimal}
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\begin{document}$$T=S\circ i_\infty $$\end{document}, where S:Lφ→X\documentclass[12pt]{minimal}
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\begin{document}$$S:L^\varphi \rightarrow X$$\end{document} is a Bochner representable and compact operator and i∞:L∞→Lφ\documentclass[12pt]{minimal}
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\begin{document}$$i_\infty :L^\infty \rightarrow L^\varphi $$\end{document} is the inclusion map.
