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

设X为Banach空间，E为有限度量空间\（（\ Omega，\ Sigma，\ lambda）\）上的完美Banach函数空间，使得\（L ^ \ infty \ subset E \ subset L ^ 1 \ ）。让\（E '\）表示的Köthe双é和\（\ tau蛋白（E，E'）\）代表的自然麦基拓扑ê。结果表明，在局部凸空间\（（E，\ tau（E，E'））\）和Banach空间X之间的每个核算子\（T：E \ rightarrow X \）都是Bochner可表示的。特别地，我们获得线性运算符\（T：L ^ \ infty \ rightarrow X \）当且仅当它的表示度量\（m_T：\ Sigma \ rightarrow ）时，局部凸空间\（（L ^ \ infty，\ tau（L ^ \ infty，L ^ 1））\）和Banach空间X之间是核的X \）具有Radon-Nikodym属性，并且\（| m_T |（\ Omega）= \ Vert T \ Vert _ {nuc} \）（= T的核范数）。作为一个应用，证明了\（L ^ \ infty \）上的某些自然内核运算符是核的。此外，表明每个核算子\（T：L ^ \ infty \ rightarrow X \）都通过某个Orlicz空间\（L ^ \ varphi \）进行因式分解，即\（T = S \ circ i_ \ infty \），其中\（S：L ^ \ varphi \ rightarrow X \）是Bochner可表示且紧凑的运算符，\（i_ \ infty：L ^ \ infty \ rightarrow L ^ \ varphi \）是包含关系图。