An order topology in ordered topological vector spaces

Carter, Lyne H.

doi:10.1090/S0002-9947-1976-0390704-2

An order topology in ordered topological vector spaces
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by Lyne H. Carter PDF

Trans. Amer. Math. Soc. 216 (1976), 131-144 Request permission

Abstract:

An order topology $\Omega$ that can be defined on any partially-ordered space has as its closed sets those that contain the (o)-limits of all their (o)-convergent nets. In this paper we study the situation in which a topological vector space with a Schauder basis is ordered by the basis cone. In a Fréchet space $(E,\tau )$, we obtain necessary and sufficient conditions both for $\tau \subset \Omega$ and for $\tau = \Omega$. Characterizations of (o)- and $\Omega$-convergence and of $\Omega$-closed sets are obtained. The equality of the order topology with the strong topology in certain dual Banach spaces is related to weak sequential completeness through the concept of a shrinking basis.

References

Unconditional bases and semiorderedness

90

33

41

Mahlon M. Day, Normed linear spaces, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21, Springer-Verlag, New York-Heidelberg, 1973. MR 0344849
J. Grosberg and M. Krein, Sur la décomposition des fonctionnelles en composantes positives, C. R. (Doklady) Acad. Sci. URSS (N.S.) 25 (1939), 723–726 (French). MR 0002019
John T. Hofler, Continuous lattice ordering by Schauder basis cones, Proc. Amer. Math. Soc. 30 (1971), 527–532. MR 415264, DOI 10.1090/S0002-9939-1971-0415264-X
Robert C. James, Bases and reflexivity of Banach spaces, Ann. of Math. (2) 52 (1950), 518–527. MR 39915, DOI 10.2307/1969430
Graham Jameson, Ordered linear spaces, Lecture Notes in Mathematics, Vol. 141, Springer-Verlag, Berlin-New York, 1970. MR 0438077, DOI 10.1007/BFb0059130
L. V. Kantorovič, B. Z. Vulih, and A. G. Pinsker, Partially ordered groups and partially ordered linear spaces, Amer. Math. Soc. Transl. (2) 27 (1963), 51–124. MR 0151532, DOI 10.1090/trans2/027/08
S. Karlin, Bases in Banach spaces, Duke Math. J. 15 (1948), 971–985. MR 29103, DOI 10.1215/S0012-7094-48-01587-7

Riesz spaces

Charles W. McArthur, Convergence of monotone nets in ordered topological vector spaces, Studia Math. 34 (1970), 1–16. MR 259559, DOI 10.4064/sm-34-1-1-16
Isaac Namioka, Partially ordered linear topological spaces, Mem. Amer. Math. Soc. 24 (1957), 50. MR 94681
Anthony L. Peressini, Ordered topological vector spaces, Harper & Row, Publishers, New York-London, 1967. MR 0227731
A. V. Potepun, Generalized regularity of $K$-lineals, and properties of the order topology, Dokl. Akad. Nauk SSSR 207 (1972), 541–543 (Russian). MR 0315394
Helmut H. Schaefer, Topological vector spaces, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1966. MR 0193469
Ivan Singer, Bases in Banach spaces. I, Die Grundlehren der mathematischen Wissenschaften, Band 154, Springer-Verlag, New York-Berlin, 1970. MR 0298399, DOI 10.1007/978-3-642-51633-7
B. Z. Vulikh, Introduction to the theory of partially ordered spaces, Wolters-Noordhoff Scientific Publications, Ltd., Groningen, 1967. Translated from the Russian by Leo F. Boron, with the editorial collaboration of Adriaan C. Zaanen and Kiyoshi Iséki. MR 0224522, DOI 10.1177/108056997003300203