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On the representation of linear transformations in integral form. (Russian. English summary) Zbl 0126.31802
English summary: Let \(X\) be a Banach space and let \(P\) be an arbitrary set. Suppose \(E\) be a linear lattice of real-valued functions on \(P\) with the following properties: \(\min\{f, 1\}\in E\) for every \(f\in E\). For any non-negative linear functional \(I\) on \(E\) (i. e., \(I(f)\geq 0\) for \(f\in E\), \(f\geq 0\), and \(I(\alpha f+\beta g) =\alpha I(f)+\beta I(g)\) for \(f, g\in E\) and \(\alpha, \beta\) real), and for any sequence \(\{f_n\}\) such that \(f_n\in E\), \(f_n \geq f_{n-1}\), \(n=1,2,\ldots\), and \(\lim f_n=0\), holds the relation \(\lim I(f_n)= 0\). It is proved, that a linear transformation \(T: E\to X\) (no matter if continuous in any topology or not) may by written in the form \(T(f)=\int f\,d\mu\), \(f\in E\), where \(\mu\) is some \(X\)-valued measure defined on a \(\delta\)-ring of subsets of \(P\), if and only if, for every \(g\in E\), \(g\geq 0\), the set \(\{T(f): | f|\leq g,\;f\in E\}\) is relatively weakly compact in \(X\). From this theorem some corollaries are derived for the case that \(P\) is a topological space and \(E\) consists of continuous functions. E.g. \(E\) may be the linear lattice of all continuous functions with compact support.
46-XX Functional analysis
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