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Topologies in vector lattices

Published online by Cambridge University Press:  24 October 2008

G. T. Roberts
G. T. Roberts
Affiliation:
King's CollegeCambridge
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Extract

This paper is concerned with topologies for vector lattices (in the sense of (1)) or spaces that satisfy the axioms K 1, 2, 3′ and 4 of (5) that are given by neighbourhoods. The notions of convergence introduced by Kantorovitch(5) and Birkhoff(1) do not in general lead to topologies which can be denned in terms of open sets, as, even for directed systems, the notion of closure derived from them is not such that the closure of every set is closed. In order to ensure this Kantorovitch introduces an axiom (his K6) which excludes even some Banach lattices. These notions of convergence are based more on the lattice aspect than the vector-space aspect of a vector lattice. In this paper the reverse is true, and it deals essentially with the application of the theory of topological vector spaces, as developed by von Neumann(9), Mackey (6, 7) and others, to vector lattices.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 48 , Issue 4 , October 1952 , pp. 533 - 546
Copyright
Copyright © Cambridge Philosophical Society 1952

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References

REFERENCES

(1)Birkhoff, G.Lattice theory, revised ed. (New York, 1948).Google Scholar
(2)Grothendieck, A.Sur la complétion du dual d'un espace vectoriel localement convexe. C.R. Acad. Sci., Paris, 230 (1950), 605–6.Google Scholar
(3)Kakutani, S.Concrete representation, of abstract (L)-spaces and the mean ergodic theorem. Ann. Math., Princeton, (2), 42 (1941), 523–37.Google Scholar
(4)Kakutani, S.Concrete representation of abstract (M)-spaces. Ann. Math., Princeton, (2), 42 (1941), 994–1024.Google Scholar
(5)Kantorovitch, L. V.Lineare halbgeordenete Raume. Mat. Sborn. (N.S.), 2 (1937), 121–65.Google Scholar
(6)Mackey, G. W.On infinite dimensional linear spaces. Trans. Amer. math. Soc. 57 (1945), 155–207.Google Scholar
(7)Mackey, G. W.On convex topological linear spaces. Trans. Amer. math. Soc. 60 (1946), 519–37.Google Scholar
(8)Nakano, H.Modern spectral theory (Tokyo, 1950).Google Scholar
(9)von Neumann, J.On complete topological spaces. Trans. Amer. math. Soc. 37 (1935), 1–20.CrossRefGoogle Scholar
(10)Rennie, B. C.Lattices. Proc. Lond. math. Soc. (2), 52 (1951), 386–400.Google Scholar
(11)Schwartz, L.Théorie des distributions, vol. 1 (Actualités sci. ind. no. 1091, Paris, 1950).Google Scholar