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There are presented certain results on extending continuous linear operators defined on spaces of $E$-valued continuous functions (defined on a compact Hausdorff space $X$) to linear operators defined on spaces of $E$-valued measurable functions in a way such that uniformly bounded sequences of functions that converge pointwise in the weak (or norm) topology of $E$ are sent to sequences that converge in the weak, norm or weak* topology of the target space. As an application, a new description of uniform closures of convex subsets of $C(X,E)$ is given. Also new and strong results on integral representations of continuous linear operators defined on $C(X,E)$ are presented. A new classes of vector measures are introduced and various bounded convergence theorems for them are proved.
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참고문헌 (19)
참고문헌 신청
B. J. Pettis / 1938 / On integration in vector spaces / Trans. Amer. Math. Soc 44 (2) : 277 ~ 304
D. H. Tucker / 1970 / Absolute continuity and the Radon-Nikodym theorem / J. Reine Angew. Math 244 : 1 ~ 19
D. R. Lewis / 1970 / Integration with respect to vector measures / Pacific J. Math 33 : 157 ~ 165
H. P. Rosenthal / 1974 / A characterization of Banach spaces containing ℓ1 / Proc. Natl. Acad. Sci. USA 71 : 2411 ~ 2413
J. Diestel / 1973 / Applications of weak compactness and bases to vector measures and vectorial integration / Rev. Roumaine Math. Pures Appl 18 : 211 ~ 224
D. H. Tucker / 1970 / Absolute continuity and the Radon-Nikodym theorem / J. Reine Angew. Math 244 : 1 ~ 19
D. R. Lewis / 1970 / Integration with respect to vector measures / Pacific J. Math 33 : 157 ~ 165
H. P. Rosenthal / 1974 / A characterization of Banach spaces containing ℓ1 / Proc. Natl. Acad. Sci. USA 71 : 2411 ~ 2413
J. Diestel / 1973 / Applications of weak compactness and bases to vector measures and vectorial integration / Rev. Roumaine Math. Pures Appl 18 : 211 ~ 224
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