Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Continuous lattice ordering by Schauder basis cones


Author: John T. Hofler
Journal: Proc. Amer. Math. Soc. 30 (1971), 527-532
MSC: Primary 46A40
DOI: https://doi.org/10.1090/S0002-9939-1971-0415264-X
MathSciNet review: 0415264
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let (E, $ \tau $) be a barrelled Hausdorff space lattice ordered by the cone of an unconditional Schauder basis $ ({x_n},{f_n})$. It is shown that under such an ordering (E, T) is a locally convex lattice. Necessary and sufficient conditions are given for the lattice operations to be continuous with respect to the weak topologies on E and its topological dual $ E'$: the lattice operations are $ \sigma (E,E')$-continuous on E if and only if $ \{ {f_n}:n \in \omega \} $ is a Hamel basis for $ E'$.


References [Enhancements On Off] (What's this?)

  • [1] Ja. M. Ceitlin, Unconditional basis and semiorderedness, Izv. Vysš. Učebn. Zaved. Matematika 1966, no. 2 (51), 98-104; English transl., Amer. Math. Soc. Transl. (2) 90 (1969), 17-25. MR 33 #6362. MR 0198203 (33:6362)
  • [2] T. A. Cook, Weakly equicontinuous Schauder bases, Proc. Amer. Math. Soc. 23 (1969), 536-537. MR 40 #1748. MR 0248496 (40:1748)
  • [3] C. W. McArthur, Convergence of monotone nets in ordered topological vector spaces, Studia Math. 34 (1970), 1-16. MR 0259559 (41:4197)
  • [4] C. W. McArthur and J. R. Retherford, Some applications of an inequality in locally convex spaces, Trans. Amer. Math. Soc. 137 (1969), 115-123. MR 39 #748. MR 0239391 (39:748)
  • [5] A. L. Peressini, Ordered topological vector spaces, Harper and Row, New York, 1967. MR 37 #3315. MR 0227731 (37:3315)
  • [6] S. A. Saxon, Basis cone base theory, Dissertation, Florida State University, Tallahassee, Fla., 1969.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46A40

Retrieve articles in all journals with MSC: 46A40


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0415264-X
Keywords: Continuous lattice operations, Schauder basis cones, unconditional Schauder basis
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society