Simultaneous convergence in two metrics
E.F. Drake · Montana State University ScholarWorks (Montana State University) · 1974
On a metric space certain concepts of convergence for sequences are defined.Using these concepts, several relations between metrics on a set X are introduced and their interdependence studied.Included are the relations "equivalent" (identical topologies), "comparable" (one topology includes another), And "uniconvergent" (the identity map from X bearing one metric, to X bearing a second metric has closed graph).For X a commutative group, convergence in a translation-invariant metric d is more conveniently studied by introducing the associated metron (the function p: X → R such that ∈x ∈ X[p(x) = d(x,o)]).The same is true of relations.Five classes of metrons on a linear space are considered; metrons, scalar-continuous metrons (the product of scalar and vector is a continuous function of the scalar component), quasinorms (the product of scalar and vector is a jointly continuous function of the scalar and vector components), norms, and inner product norms.A typical question studied is whether, on a given linear space, all metrons of a given class bear a given relation to each other.For norms, only four of the relations studied remain distinct, while for complete norms all coincide. It is proved that non-uniconvergent metrons on a one-dimensional space, and non-uniconvergent inner product norms on a space of countably-infinite Hamel dimension exist.The scalar-continuous metron is considered in detail.On finite dimensional spaces it has simple continuity properties, yet allows surprising convergence behavior.Counterexamples illustrating non-comparable and incomplete scalar-continuous metrons on a one-dimensional space are constructed.Some questions relating to completion remain unsolved.