2017/03/30 by Marko Kandić, Kandić, M., Huijie Li +3 · 2 citations
Computer Science · Mathematics · #46A40 (Secondary) #46B42 (Primary) #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.1703.10654
openalex publication_date 2017/03/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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) un-converges to y in Y with respect to X if ‖| yα-y| \wedge 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=L0(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=\mathbb RA then the un-convergence on Y agrees with the coordinate-wise convergence.