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We study properties of Banach spaces C(L) of all continuous scalar (real or complex) functions on compact lines L. First we show that if L is a separable compact line, then for every closed linear subspace X of C(L) with separable dual the quotient space C(L)/X possesses a sequence of continuous linear functionals separating its points. Next we show that for any compact line L the space C(L) contains...
We say that a function f from [0,1] to a Banach space X is increasing with respect to E⊂X* if x∗∘f is increasing for every x*∈E. A function f:[0,1]m→X is separately increasing if it is increasing in each variable separately. We show that if X is a Banach space that does not contain any isomorphic copy of c0 or such that X* is separable, then for every separately increasing function f:[0,1]m→X with...
We show that if X is a WGG Banach space and it does not contain any isomorphic copy of l1, then for every bounded Pettis integrable function f : [0, 1]^2 --> X* there exists a scalarly equivalent function for which the Fubini theorem for the Pettis integral holds. On the other hand, we show that for every bounded Pettis integrable function f : [0, 1]^2 --> l^2 (R) there exists a scalarly equivalent...
Let K be a compact Hausdorff space, mi a positive Radon measure on K, and let G be a compact group with the Haar measure lambda. We consider properties of the following generalization of translations on groups: we associate with every bounded to mi x R[lambda]- measurable function f : K x G --> C the function T[sub f] : G --> L^[infinity] (mi x R[lambda]) given by T[sub f](t) = f[sub t] where...
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