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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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