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On a tensor product of weakly compact mappings. (English) Zbl 0637.47011
Let X and Y denote quasi-complete locally convex Hausdorff spaces. The author considers operators $$u: C_ 0(S) \to X$$ and $$v: C_ 0(T) \to Y$$ and their tensor product $$u \otimes v$$ from $$C_ 0(S \times T)$$ into the quasi-completion of $$X \otimes Y,$$ endowed with the injective or some stronger topology, as well as their representing vector measures. The main result states that $$u \otimes v$$ is weakly compact in case $$u$$ and $$v$$ are.
Reviewer’s remark: More general results in the Banach space setting were given by J. Diestel and B. Faires [Proc. Am. Math. Soc. 58, 189-196 (1976; Zbl 0343.47015)].

##### MSC:
 47B38 Linear operators on function spaces (general) 46M05 Tensor products in functional analysis 47B06 Riesz operators; eigenvalue distributions; approximation numbers, $$s$$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators 28B05 Vector-valued set functions, measures and integrals
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