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arXiv · 1309.2612

Bounded convergence theorems

Piotr Niemiec
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Abstract

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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Piotr Niemiec. 2013-09-10. Bounded convergence theorems. https://doi.org/10.4134/jkms.j150749

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