Unbounded Norm Topology beyond Normed Lattices
نویسنده
چکیده
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|∧x∥∥→ 0 for every x ∈ X+. We extend several known results about unconvergence 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 = R then the un-convergence on Y agrees with the coordinate-wise convergence.
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