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- W2018073033 abstract "The subject of this paper is a further study of linear bounded transformations T from the space C(S, E) into F, where E and F are Banach spaces and C(S, E) is the Banach space of all continuous functions defined on a compact Hausdortf space S with values in E, endowed with the usual uniform norm. The paper is closely related to a paper of one of us [3] in which, under the assumption that S = I is a compact interval of the reals, essentially three subjects were treated: (1) Characterization of those transformations which can be represented by finitely additive, “E*-regular” set functions U of bounded semi-variation defined on the Bore1 field ~8 of S with values in L(E, F); (2) h c aracterization of the compact and weakly compact transformations by properties of their representing measures U; and (3) a detailed study of the cases in which F is weakly complete, and, F being weakly complete, E is reflexive. In the latter case, it was found that every linear bounded transformation from C(I, E) into F is weakly compact. We have extended these results to the case where S is a compact HausdorfI space. Our considerations are based on recent results of N. Dinculeanu [6-S]. Our first theorem points out a direct relation between T and its representing measure U : 9’ + L(E, F**) from which necessary and sufficient conditions are obtained that U takes its values in L(E, F). Dinculeanu has posed the question of characterizing those transformations T for which the corresponding measure U (a) is regular (in the norm of L(E, F**) and (b) is regular and has values in L(E, F). This question is answered in Theorem 3 which at the same time shows that both cases coincide." @default.
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- W2018073033 date "1969-10-01" @default.
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- W2018073033 title "Linear bounded transformations on the space of continuous functions" @default.
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- W2018073033 doi "https://doi.org/10.1016/0022-1236(69)90012-3" @default.
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