2024/12/09 by Ball, R. N., Hager, A. W.
#06D22 #06F20 #18F70 #46A40 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2412.06998
In the category \(V\) of unital archimedean vector lattices, four notions of uniform completeness obtain. In all cases completeness requires the convergence of uniformly Cauchy sequences; the completions are distinguished by the manner in which the convergence is regulated. Ordinary uniform convergence is regulated by the canonical unit \(1\). Inner relative uniform convergence, here termed iru-convergence, is regulated by an arbitrary positive element. Outer relative uniform convergence, here termed oru-convergence, is regulated by an arbitrary positive element of a vector lattice containing the given object as a sub-vector lattice. *-convergence is equivalent to ordinary uniform convergence on certain specified quotients of the vector lattice. In each case the complete objects form a full monoreflective subcategory of \(V\), denoted respectively \(ucV\), \(irucV\), \(orucV\), and \(*cV\). In this article we provide a unified development of these completions by means of a novel pointfree variant of the classical Yosida adjunction.