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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.
Show Scanned Page ##### MSC:
 46-XX Functional analysis
##### Keywords:
functional analysis
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