On nestedly complete topological vector lattices

On nestedly complete topological vector lattices A topological vector lattice E is called (σ-)nestedly complete if every downward directed net (resp., decreasing sequence) of order intervals in E whose ‘diameters’ tend to zero has a nonempty intersection. Some characterizations of the (σ-)nested completeness are given, and it is shown that if E is metrizable and nestedly complete, so is each of its quotients E/I, where I is a closed ideal in E. Conversely, if a closed ideal I in E is (sequentially) complete and E/I is (σ-)nestedly complete, so is E. However, the nested completeness is not a three-space property: an example is given where both I and E/I are nestedly complete while E is not. It is also shown that the nested completeness and the related notion of nested density come up quite naturally when extending some positive linear operators. Finally, the nested and other completeness type properties of vector lattices C(S) are investigated. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

On nestedly complete topological vector lattices

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SP Birkhäuser Verlag Basel
Copyright © 2010 by The Author(s)
Mathematics; Econometrics; Calculus of Variations and Optimal Control; Optimization; Potential Theory; Operator Theory; Fourier Analysis
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