Order-Preserving Variants of the Basic Principles of Functional Analysis

Year 2019, Volume: 2 Issue: 1, 10 - 17, 17.06.2019

Adilbek Zaitov

https://doi.org/10.33401/fujma.503688

Cited By: 2

Abstract

We will establish order-preserving versions of the basic principles of functional analysis such as Hahn-Banach, Banach-Steinhaus, open mapping, and Banach-Alaoglu theorems.

Keywords

Functional analysis, Order-preserving functional, Vector space with an order unit

References

• [1] M. Grabisch, M. Roubens, Application of the Choquet Integral in Multicriteria Decision Making, In the book ‘Fuzzy Measures and Integrals – Theory and Applications’ (eds M. Grabisch, T. Murofushi, M. Sugeno), Physica Verlag, (2000), 348-374.
• [2] T. Radul, Idempotent measures: Absolute retracts and soft maps, (2018), arXiv:1810.09140v1 [math.GN].
• [3] W. Rudin, Functional Analysis (2nd edition), International Editions (McGraw-Hill Book Co – Singapore for manufacture and export), 1991.
• [4] A. Peperko, Uniform boundedness principle for nonlinear operators on cones of functions, Hindawi J. Func. Spaces, 2018, Article ID 6783748, 5 pages, available at https://doi.org/10.1155/2018/6783748
• [5] A. Zaitov, On extension of order-preserving functionals, Doklady Akademii Nauk Uzbekistan, 5 (2005), 3-7.
• [6] A. Zaitov, Banach-Alaoglu theorem for order-preserving functionals, Theses of reports of the international scientific conference ”Operator Algebras and Quantum Probability Theory”, Tashkent, (2005), 81-83.
• [7] A. Zaitov, Open mapping theorem for order-preserving operators, The collection of theses of the International conference of young scientists devoted 1000 to the anniversary of Mamun Academy of Khwarezm, Tashkent, (2006), 4-5.
• [8] V. Paulsen, M. Tomforde, Vector spaces with an order unit, Indiana Univ. Math. J., 58(3) (2009), 1319-1359.