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Spaces of continuous functions taking their values in the \(\varepsilon\)-product. (English) Zbl 1116.46002

The author deals with linear spaces endowed with a bornology consisting of Banach disks in the sense of H.Hogbe–Nlend and L.Waelbroeck, and calls them \(b\)-spaces. He defines nuclearity as well as the \(\varepsilon\)-product in the class of \(b\)-spaces. Then he proves his main result: Theorem. Let \(X\) be a compact or locally compact \(\sigma\)-compact topological space, let \(N\) be a nuclear \(b\)-space, and let \(E\) be a \(b\)-space. Then the spaces of continuous vector-valued functions \(C(X,N\varepsilon E)\) and \(N\varepsilon C(X,E)\) are naturally isomorphic as \(b\)-spaces.

MSC:

46A32 Spaces of linear operators; topological tensor products; approximation properties
46M05 Tensor products in functional analysis
46A17 Bornologies and related structures; Mackey convergence, etc.

Citations:

Zbl 0139.06902
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