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

作者：de Jeu Marcel; Messerschmidt Miek^*

来源：Advances in Mathematics, 2014, 259: 43-66.

DOI：10.1016/j.aim.2014.03.008

• Cone
• Open Mapping Theorem
• Banach space
• Bounded continuous right inverse
• Ordered Banach space

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 Ando's Theorem for an ordered Banach space is obtained for a Banach space that is, more generally than in Ando'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.

• 出版日期2014-7-10
• 单位西北大学