Continuous lattice ordering by Schauder basis cones

Hofler, John T.

doi:10.1090/S0002-9939-1971-0415264-X

Continuous lattice ordering by Schauder basis cones
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by John T. Hofler

Proc. Amer. Math. Soc. 30 (1971), 527-532

DOI: https://doi.org/10.1090/S0002-9939-1971-0415264-X

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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

Ja. M. Ceĭtlin, Unconditionality of a basis and partial order, Izv. Vysš. Učebn. Zaved. Matematika 1966 (1966), no. 2 (51), 98–104 (Russian). MR 0198203
Thurlow A. Cook, Weakly equicontinuous Schauder bases, Proc. Amer. Math. Soc. 23 (1969), 536–537. MR 248496, DOI 10.1090/S0002-9939-1969-0248496-5
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
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 239391, DOI 10.1090/S0002-9947-1969-0239391-0
Anthony L. Peressini, Ordered topological vector spaces, Harper & Row, Publishers, New York-London, 1967. MR 0227731

Basis cone base theory