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Korovkin-type approximation theory in Riesz spaces. (English) Zbl 1491.46005

Summary: In this paper, we prove the following Riesz spaces’ version of the Korovkin theorem. Let \(E\) and \(F\) be two Archimedean Riesz spaces with \(F\) uniformly complete, let \(W\) be a nonempty subset of \(E^{+}\), and let \((T_{n})\) be a given sequence of (r-u)-continuous elements of \(\mathcal {L}(E,F)\), such that \(| T_{n}-T_{m}| x=| (T_{n}-T_{m})x|\) for all \(x\in E^{+},m,n\geq n_{0}\) (for a given \(n_{0}\in \mathbb {N} )\). If the sequence \((T_{n}x)_{n}(r-u)\)-converges for every \(x\in W\), then \((T_{n})(r-u)\)-converges also pointwise on the ideal \(E_{W}\), generated by \(W\), to a linear operator \(S_{0}:E_{W}\to F\). We also prove a similar Korovkin-type theorem for nets of operators. Some applications for \(f\)-algebras and orthomorphisms are presented.

MSC:

46A40 Ordered topological linear spaces, vector lattices
06F25 Ordered rings, algebras, modules
41A35 Approximation by operators (in particular, by integral operators)
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