Metrics with good Corona properties

TL;DR

This paper explores various metric corona properties in topological vector spaces, showing that some well-behaved spaces lack these properties, and discusses open questions regarding classical function spaces.

Contribution

It introduces several definitions of metric corona properties and demonstrates that certain complete metrizable spaces lack these properties, highlighting open problems in the field.

Findings

Classical function spaces have good corona properties.

Some complete metrizable topological vector spaces lack good corona properties.

Open question: do locally convex topological vector spaces have good corona properties?

Abstract

In this note, we give several definitions of metric corona properties which could be of interest in Set Topology, Functional Analysis and Approximation Theory, and prove that there are complete metrizable t.v.s. which are nice in the sense that they have a metric which is invariant by translations, but they do not have good corona properties. All classical function spaces satisfy good corona properties but it is an open question to know if this also holds for the more general setting of locally convex t.v.s.