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- Title
Unbounded norm topology beyond normed lattices.
- Authors
Kandić, M.; Li, H.; Troitsky, V. G.
- Abstract
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 (yα)<inline-graphic></inline-graphic> un-converges to y in Y with respect to X if |||yα-y|∧x||→0<inline-graphic></inline-graphic> for every x∈X+<inline-graphic></inline-graphic>. 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=L0(μ)<inline-graphic></inline-graphic>, the space of all μ<inline-graphic></inline-graphic>-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=RA<inline-graphic></inline-graphic> then the un-convergence on Y agrees with the coordinate-wise convergence.
- Subjects
TOPOLOGICAL groups; LATTICE theory; SYMMETRIC spaces; STOCHASTIC convergence; VECTORS (Calculus)
- Publication
Positivity, 2018, Vol 22, Issue 3, p745
- ISSN
1385-1292
- Publication type
Article
- DOI
10.1007/s11117-017-0541-6