A strong open mapping theorem for surjections from cones onto Banach 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 Andô’s Theorem for an ordered
Banach space is obtained for a Banach space that is, more generally than in
Andô’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.