Unbounded norm topology beyond normed lattices

In this paper, we generalize the concept of unbounded norm (un) convergence: let X be a normed lattice and Y a vector lattice such that X is an order dense ideal in Y ; we say that a net ( y α ) un-converges to y in Y with respect to X if | | | y α - y | ∧ x | | → 0 for every x ∈ X + . We extend sev...

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Bibliographic Details
Published in:Positivity : an international journal devoted to the theory and applications of positivity in analysis 2018-07, Vol.22 (3), p.745-760
Main Authors: Kandić, M., Li, H., Troitsky, V. G.
Format: Article
Language:English
Subjects:
Calculus of Variations and Optimal Control
Optimization
Continuity (mathematics)
Convergence
Econometrics
Fourier Analysis
Function space
Lattices (mathematics)
Mathematics
Mathematics and Statistics
Operator Theory
Potential Theory
Topology
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Summary:In this paper, we generalize the concept of unbounded norm (un) convergence: let X be a normed lattice and Y a vector lattice such that X is an order dense ideal in Y ; we say that a net ( y α ) un-converges to y in Y with respect to X if | | | y α - y | ∧ x | | → 0 for every x ∈ X + . We extend several known results about un-convergence and un-topology to this new setting. We consider the special case when Y is the universal completion of X . If Y = L 0 ( μ ) , the space of all μ -measurable functions, and X is an order continuous Banach function space in Y , then the un-convergence on Y agrees with the convergence in measure. If X is atomic and order complete and Y = R A then the un-convergence on Y agrees with the coordinate-wise convergence.
ISSN:1385-1292
1572-9281
DOI:10.1007/s11117-017-0541-6