A strong open mapping theorem for surjections from cones onto Banach spaces

TL;DR

This paper extends the Open Mapping Theorem to surjective maps from cones onto Banach spaces, establishing conditions for openness and existence of bounded right inverses, with applications to ordered Banach spaces.

Contribution

It generalizes the Open Mapping Theorem for cone mappings, introduces a continuous bounded right inverse, and improves results related to ordered Banach spaces and function spaces.

Findings

Surjective cone maps are open if and only if they are surjective.

Existence of continuous bounded positively homogeneous right inverses.

Applications to ordered Banach spaces and function spaces.

Abstract

We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael's Selection Theorem to yield the existence of a continuous bounded positively homogeneous right inverse of such a surjective map; a strong version of the usual Open Mapping Theorem is then a special case. As another consequence, an improved version of the analogue of And\^o's Theorem for an ordered Banach space is obtained for a Banach space that is, more generally than in And\^o's Theorem, a sum of possibly uncountably many closed not necessarily proper cones. Applications are given for a (pre)-ordered Banach space and for various spaces of continuous functions taking values in such a Banach space or, more generally, taking values in an arbitrary Banach space that is a finite sum of closed not necessarily proper cones.