Abstract
We study the relation between almost-symmetries and the geometry of Banach spaces. We show that any almost-linear extension of a transformation that preserves transition probabilities up to an additive error admits an approximation by a linear map, and the quality of the approximation depends on the type and cotype constants of the involved spaces.
Highlights
Introduction and PreliminariesIn the work of N
Kalton [1,2,3], we can find novel ideas and methods for the stability of functional equations that depart from the classical methods of Hyers, Ulam and Rassias [4]
Throughout this note, we will be entirely concerned with finite-dimensional Banach spaces and the twisted sums generated by almost-linear maps
Summary
Theorem 2.2), Kalton provides a sharp bound on the stability of the additive map in Rn for the so-called singular case His proof makes use of probabilistic and geometric methods in Banach space theory. We explore how the quality of the approximation depends on the consideration of various classes of almost-linear extensions These extensions have arbitrary finite-dimensional Banach spaces as domain and codomain. Throughout this note, we will be entirely concerned with finite-dimensional Banach spaces and the twisted sums generated by almost-linear maps. We introduce a special space which will generate the linear approximation to the almost-linear map F : X → Y This space is an extension of X and Y and is called a twisted sum (basically because it “twists” the unit ball of X and Y according to F). For a Hilbert space, the type and cotype constants are always equal to one
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
