A globalization of the Hahn-Banach theorem

Abstract

The Hahn-Banach theorem is one of many fundamental results in functional analysis whose usefulness is diminished by their proof depending upon an application of the Axiom of Choice. The objection raised in this regard is not philosophical, but purely practical. When the existence of a functional has been established by this non-constructive means, the arbitrariness introduced restricts the information which can be extracted, impeding the discussion of points such as whether the functional can be shown to exist continuously with respect to a parameter [ 1, 15, 201, or equivariantly with respect to a group action [2, 333. This paper is concerned with outlining, starting from ideas which are already established, a context and a technique which allow the effects of this dependence on the Axiom of Choice to be avoided, reformulating the Hahn-Banach theorem in a way which may be applied equally to questions of continuity and of equivariance, to yield the information which is traditionally intended when applying the Hahn-Banach theorem. Before introducing the constructive context in which these ideas will be developed, consider the situations already mentioned, in which the arbitrariness of the existence of a functional established by the Axiom of Choice presents difficulties which need to be resolved. Firstly, take that of continuity in a parameter, which may be stated more precisely in the following way: consider a bundle B of seminormed spaces on a topological space X, together with a continuous mapping