A lexicographical order induced by Schauder bases

Abstract

In this paper, we show that every Banach space with a Schauder basis can be seen as a totally ordered vector space. Indeed, this order can be considered as a lexicographical order since it is a generalization of lexicographical order in $\mathbb{R}^{n}.$ We also provide order structural properties of the order by approaching geometrical (cone) sense.

Keywords

• total order
• cone
• Schauder basis

References

1. [1] C.D. Aliprantis, C. Bernard and R. Tourky Economic equilibrium: Optimality and price decentralization, Positivity, 6 (3), 205–241, 2002.
2. [2] C.D. Aliprantis and O. Burkinshaw, Locally solid Riesz spaces with applications to economics, American Mathematical Society, Providence, RI, 2003.
3. [3] C.D. Aliprantis and O. Burkinshaw, Positive Operators, Springer, Dordrecht, 2006.
4. [4] C.D. Aliprantis and R. Tourky, Cones and duality, 84, American Mathematical So- ciety, Providence, RI, 2007.
5. [5] C.D. Aliprantis, R. Tourky and C.Y. Nicholas, Cone conditions in general equilibrium theory, J. Econom. Theory 1, 96–121, 2000.
6. [6] C. Bessaga and A. Pelczynski, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17, 151–164, 1958.
7. [7] M. Ehrgott, Multicriteria optimization, Springer Science and Business Media, 2005.
8. [8] E. Jonge and A.C.M Rooij, Introduction to Riesz spaces, Mathematisch Centrum, 1977.

9. [9] M. Kucuk, M. Soyertem and Y. Kucuk,On constructing total orders and solving vector optimization problems with total orders, J. Global Optim. 50, (2), 235–247, 2011.
10. [10] M. Kucuk, M. Soyertem and Y. Kucuk,The generalization of total ordering cones and vectorization to separable Hilbert spaces, J. Math. Anal. Appl. 389, (2), 1344–1351, 2012.
11. [11] M. Kucuk, M. Soyertem, Y. Kucuk and I. Atasever, Vectorization of set-valued maps with respect to total ordering cones and its applications to set-valued optimization problems, J. Math. Anal. Appl. 385 (1), 285–292, 2012.
12. [12] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I-II Function Spaces, Springer, 1996.
13. [13] W.A. Luxemburg and A.C. Zaanen, Riesz Spaces I, North-Holland, Amsterdam, 1971.
14. [14] N.P. Meyer, Banach Lattices, Springer, Berlin, 1991.
15. [15] A. Pelczynski and I. Singer, On non-equivalent basis and conditional basis in Banach spaces, Studia Math. 1 (25), 5–25, 1964.
16. [16] A.L. Quoc and D.T. Quoc, Tykhonov well-posedness for lexicographic equilibrium problems, Optimization, 65 (11), 1929–1948, 2016.
17. [17] H.H. Schaefer, Banach lattices and positive operators, Springer-Verlag, 1974.
18. [18] J. Schauder, Zur Theoriestetiger Abbildungenin Funktionalraumen, Math. Z. 26, 47– 65, 1927.
19. [19] A.C. Zaanen, Introduction to Operator Theory in Riesz Spaces, Springer-Verlag, 1997.