Abstract
We study the existence of infima of subsets in Banach spaces ordered by normal cones associated to shrinking Schauder bases. Under these conditions we prove the existence of infima for a class of subsets verifying a weakly compactness property. Moreover we prove that a normal cone associated to a Schauder basis in a reflexive Banach space is strongly minihedral extending the known result for unconditional Schauder bases. Several examples are also discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C.D. Aliprantis, O. Burkinshaw, Locally solid Riesz spaces with applications to economics, Mathematical Surveys and Monographs, vol. 105, American Mathematical Society, Providence (2003).
C.D. Aliprantis, O. Burkinshaw Positive operators, Pure and Applied Math. Series, Vol. 119, Academic Press, New York, Reprinted by Springer in 2006 (1985).
C.D. Aliprantis, R. Tourky, Cones and Duality, Graduate Studies in Mathematics, vol. 84, American Mathematical Society, Providence (2007).
M. Fabian, P. Habala, P. Hajek, V. Montesinos, P. Pelant, V. Zizler, Functional Analysis and Infinite Dimensional Geometry, Springer (2001).
R.E. Fullerton, Geometric properties of a basis in a Banach space, Proc. Int. Cong. Math., Amsterdam (1954).
R.E. Fullerton, Geometric structure of absolute basis systems in a linear topological space, Pacific J. Math., 12 (1962), 137–147.
D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, San Diego (1988).
R.C. James, Weakly compact sets, Trans. Amer. Math. Soc., 113 (1964), 129–140.
V. Katsikis, I. Polyrakis, Positive bases in ordered subspaces with the Riesz decomposition property, Studia Math. 174 (2006), 233–253.
M.A. Krasnoselskii, J. A Lifshits, A.V. Sobolev, Positive linear systems, Heldermann Verlag, Berlin (1989).
C.W. McArthur, Developments in Schauder basis theory, Bull. Amer. Math. Soc., 78 (1972), 877–908.
I. Polyrakis, Lattice-subspaces of C[0;1] and positive bases, J. Math. Anal. Appl., 184 (1994), 1–15.
H.H. Schaefer, Banach Lattices and Positive Operators, Springer, Berlin (1974).
I. Singer, Bases in Banach Spaces I, Springer (1970).
A. W. Wickstead, Compact subsets of partially ordered Banach spaces, Math. Ann., 212 (1975), 271–284.
Author information
Authors and Affiliations
Corresponding author
Additional information
Miguel Sama: The work of this author is partially supported by Ministerio de Educación y Ciencia (Spain), project MTM2006-02629 and Ingenio Mathematica (i-MATH) CSD2006-00032 (ConsoliderIngenio 2010).
Rights and permissions
About this article
Cite this article
Rodríguez-Marín, L., Sama, M. A note on cones associated to Schauder bases. Positivity 13, 575–581 (2009). https://doi.org/10.1007/s11117-008-2215-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11117-008-2215-x