2016/06/30 by Miek Messerschmidt
Mathematics · #Advanced Banach Space Theory #Algorithm #Banach space #Bounded function #Combinatorics #Computer science #Cone (formal languages) #Corollary #Discrete mathematics #Eberlein–Šmulian theorem #Functional Equations Stability Results #Geometry #Holomorphic and Operator Theory #Lipschitz continuity #Lp space #Mathematical analysis #Mathematical proof #Mathematics #Open mapping theorem (functional analysis) #Pure mathematics #Space (punctuation) #math.FA #msc:32A12 #msc:46A30 #msc:46B20 #msc:46B40
paper · pdf · doi:10.1016/j.jfa.2018.02.008
Major rewrite. Large parts were removed which a referee pointed out can be proven through much easier methods
openalex publication_date 2018/02/21 · arxiv created 2018/02/22 · arxiv updated 2018/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
A version of the classical Klee-Andô Theorem states the following: For every Banach space X, ordered by a closed generating cone C⊆ X, there exists some α>0 so that, for every x∈ X, there exist x±∈ C so that x=x+-x- and ‖x+‖+‖x-‖≤α‖x‖. The conclusion of the Klee-Andô Theorem is what is known as a conormality property. We prove stronger and somewhat more general versions of the Klee-Andô Theorem for both conormality and coadditivity (a property that is intimately related to conormality). A corollary to our result shows that the functions x↦ x±, as above, may be chosen to be bounded, continuous, and positively homogeneous, with a similar conclusion yielded for coadditivity. Furthermore, we show that the Klee-Andô Theorem generalizes beyond ordered Banach spaces to Banach spaces endowed with arbitrary collections of cones. Proofs of our Klee-Andô Theorems are achieved through an Open Mapping Theorem for cone-valued multi-functions/correspondences. We very briefly discuss a potential further strengthening of The Klee-Andô Theorem beyond what is proven in this paper, and motivate a conjecture that there exists a Banach space X, ordered by a closed generating cone C⊆ X, for which there exist no Lipschitz functions (⋅)±:X→ C satisfying x=x+-x- for all x∈ X.