A strong open mapping theorem for surjections from cones onto Banach spaces (original) (raw)
Related papers
Right inverses of surjections from cones onto Banach spaces
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; an improved version of the usual Open Mapping Theorem is then a special case. As another consequence, a stronger 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.
Geometric properties and continuity of the pre-duality mapping in Banach space
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2014
We use the preduality mapping in proving characterizations of some geometric properties of Banach spaces. In particular, those include nearly strongly convexity, nearly uniform convexity-a property introduced by K. Goebel and T. Sekowski-, and nearly very convexity.
On intermediate value theorem in ordered Banach spaces for noncompact and discontinuous mappings
International Journal of Nonlinear Analysis and Applications, 2016
In this paper, a vector version of the intermediate value theorem is established. The main theorem of this article can be considered as an improvement of the main results have been appeared in [textit{On fixed point theorems for monotone increasing vector valued mappings via scalarizing}, Positivity, 19 (2) (2015) 333-340] with containing the uniqueness, convergent of each iteration to the fixed point, relaxation of the relatively compactness and the continuity on the map with replacing topological interior of the cone by the algebraic interior. Moreover, by applying Ascoli-Arzela's theorem an example in order to show that the main theorem of the paper [textit{An intermediate value theorem for monotone operators in ordered Banach spaces}, Fixed point theory and applications, 2012 (1) (2012) 1-4] may fail, is established.
Common Fixed Point Theorems in Cone Banach Spaces ABSTRACT| FULL TEXT
Recently, E. Karapınar (Fixed Point Theorems in Cone Banach Spaces, Fixed Point Theory Applications, Article ID 609281, 9 pages, 2009) presented some fixed point theorems for self-mappings satisfying certain contraction principles on a cone Banach space. Here we will give some generalizations of this theorem.
arXiv Functional Analysis, 2019
For classes of topological vector spaces, we analyze under which conditions open-mapping, bounded-inverse, and closed-graph properties are equivalent. We show that closure under quotients with closed subspaces and closure under closed graphs are sufficient. We show that the class of barreled Pták spaces is exactly the largest class of locally-convex topological vector spaces, which contains all Banach spaces, is closed under quotients with closed subspaces, is closed under closed graphs, is closed under continuous images, and for which an open-mapping theorem, a bounded-inverse theorem, and a closed-graph theorem holds.