2022/09/20 by Abela, Kevin, Chetcuti, Emmanuel, Weber, Hans
#06B30 #06F30 #46A40 #54H12 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2209.09831
A net (xγ)γ∈Γ in a locally solid Riesz space (X,τ) is said to be unbounded τ-convergent to x if |xγ-x|\wedge u\mathop\oversetτ\longrightarrow 0 for all u∈ X+. We recall that there is a locally solid linear topology \mathfrakuτ on X such that unbounded τ-convergence coincides with \mathfrakuτ-convergence, and moreover, \mathfrakuτ is characterised as the weakest locally solid linear topology which coincides with τ on all order bounded subsets. It is with this motivation that we introduce, for a uniform lattice (L,u), the weakest lattice uniformity u^∗ on L that coincides with u on all the order bounded subsets of L. It is shown that if u is the uniformity induced by the topology of a locally solid Riesz space (X,τ), then the u^*-topology coincides with \mathfrakuτ. This allows comparing the results of this paper with earlier results on unbounded τ-convergence. It will be seen that despite the fact that in the setup of uniform lattices most of the machinery used in the techniques of [M. A. Taylor 2019: Unbounded topologies and uo-convergence in locally solid vector spaces, J. Math. Anal. Appl. \bf472 no.1, 981--1000] is lacking, the concept of `unbounded convergence' well fittingly generalizes to uniform lattices. We shall also answer Questions 2.13, 3.3, 5.10 of [M. A. Taylor 2019: Unbounded topologies and uo-convergence in locally solid vector spaces, J. Math. Anal. Appl. \bf472 no.1, 981--1000] and Question 18.51 of [M. A. Taylor 2018: Unbounded convergence in vector lattices, Thesis University of Alberta].