tailieunhanh - Báo cáo hóa học: " Demi-linear duality"

Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Demi-linear duality | Li et al. Journal of Inequalities and Applications 2011 2011 128 http content 2011 1 128 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access Demi-linear duality Ronglu Li 1 Aihong Chen1 and Shuhui Zhong2 Correspondence rongluli@yahoo. department of Mathematics Harbin Institute of Technology Harbin 150001 . China Full list of author information is available at the end of the article Springer Abstract As is well known there exist non-locally convex spaces with trivial dual and therefore the usual duality theory is invalid for this kind of spaces. In this article for a topological vector space X we study the family of continuous demi-linear functionals on X which is called the demi-linear dual space of X. To be more precise the spaces with non-trivial demi-linear dual for which the usual dual may be trivial are discussed and then many results on the usual duality theory are extended for the demi-linear duality. Especially a version of Alaoglu-Bourbaki theorem for the demilinear dual is established. Keywords demi-linear duality equicontinuous Alaoglu-Bourbaki theorem 1 Introduction Let K e R c and X be a locally convex space over K with the dual X . There is a beautiful duality theory for the pair X X9 see 1 Chapter 8 . However it is possible that X 0 even for some Fréchet spaces such as Lp 0 1 for 0 p 1. Then the usual duality theory would be useless and hence every reasonable extension of X will be interesting. Recently Ly U X Y the family of demi-linear mappings between topological vector spaces X and Y is firstly introduced in 2 . Ly U X Y is a meaningful extension of the family of linear operators. The authors have established the equicontinuity theorem the uniform boundedness principle and the Banach-Steinhaus closure theorem for the extension Ly U X Y . Especially for demi-linear functionals on the spaces of test functions Ronglu Li et al have established a theory which is a