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On Order Properties of Order Bounded Transformations

  • Charalambos D. Aliprantis (a1)

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W. A. J. Luxemburg and A. C. Zaanen in [7] and W. A. J. Luxemburg in [5] have studied the order properties of the order bounded linear functionals of a given Riesz space L. In this paper we consider the vector space (L, M) of the order bounded linear transformations from a given Riesz space L into a Dedekind complete Riesz space M.

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References

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1. Day, M. M., The spaces L with 0 < p < 1, Bull. Amer. Math. Soc. 46 (1940), 816823.
2. Fremlin, D. H., Topological Riesz spaces and measure theory (Cambridge University Press, London, 1974).
3. Jameson, G., Ordered linear spaces (Springer-Verlag, Berlin, New York, 1970).
4. Kantorovich, L. V., Concerning the general theory of operations in particular ordered spaces, Dan SSSR, (1936), 271274 (Russian).
5. Luxemburg, W. A. J., Notes on Banach function spaces, Proc. Acad. Sc. Amsterdam, Note XV, A68, (1965), 415446.
6. Luxemburg, W. A.J.and Zaanen, A. C., The linear modulus of an integral transformation, Proc. Acad. Sc. Amsterdam, A75 (1971), 442447.
7. Luxemburg, W. A.J.and Zaanen, A. C., Notes on Banach function spaces, Proc. Acad. Sc. Amsterdam, Note VI, A66, (1963), 669-681, Note IX, A67 (1964), 507-518; Note X, A67 (1964), 493506.
8. Luxemburg, W. A.J.and Zaanen, A. C., Riesz spaces. I (North Holland, Amsterdam, 1971).
9. Nakano, H., Modulared semi-ordered linear spaces (Maruzen Co., Tokyo, 1950).
10. Ogasawara, T., Vector lattices, I and II, Tokyo, 1948 (In Japanese).
11. Peressini, A. L., Ordered topological vector spaces (Harper and Row, New York, 1967).
12. Riesz, F., Sur quelques notions fondamentales dans la théorie générale des opérations linéaires, Ann. of Math. 41 (1940), 174206. (This work was first published in 1937 in Hungarian.)
13. Vulikh, B. Z., Introduction to the theory of partially ordered spaces, translation from the Russian (Wolters-Noordhoff, Groningen, 1967).
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On Order Properties of Order Bounded Transformations

  • Charalambos D. Aliprantis (a1)

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