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Quasi-uniform structures in linear lattices. (English) Zbl 0803.46007
This paper concerns normed linear spaces defined over the field of real numbers. The authors call a normed lattice in which the parallelogram law holds for positive elements an \(E\) space, and they call a nonnegative sublinear function \(q\) a quasi-norm provided that if \(q(x)= q(-x)= 0\) then \(x= 0\). Each quasi-normed space has a natural translation-invariant quasi-uniformity \({\mathcal U}_ q\) and the functions \(q^*(x)= q(x)+ q(- x)\), \(q^* M(x)= q(x)\lor q(-x)\), and \(q^* E(x)= (q(x)^ 2+ q(-x)^ 2)^{1/2}\) define equivalent norms that induce the uniformity \({\mathcal U}{q^*}\). For the quasi-norm \(q(x)= \| x^ +\|\) defined on a normed lattice, the norms defined by \(q^*_ L\), \(q_ M\), and \(q^*_ E\) are equivalent to the original norm \(\| \|\) and coincide with this norm respectively when the normed lattice is an \(L\) space (\(M\) space, or \(E\) space). Consequently, every normed lattice is determined (in the sense of L. Nachbin) by the quasi-uniformity derived from the quasi-norm \(q(x)= \| x^ +\|\). The authors characterize the positive continuous linear functionals of a normed lattice \((E,\| \|,\leq)\) as those linear functionals that are continuous when considered as maps from \((E,q)\) to the real line with the right-ray topology. They use this characterization and the algebraic version of the Hahn-Banach theorem to obtain an alternative proof of Proposition 33.15 from G. J. O. Jameson’s “Topology and normed spaces” (1974; Zbl 0285.46002). This alternative proof illustrates a theme of the paper: It is helpful in determining some global aspects of a normed lattice from the study of its positive cone to decompose the normed lattice into its two quasi-pseudometric structures.

MSC:
46A40 Ordered topological linear spaces, vector lattices
54E15 Uniform structures and generalizations
46B40 Ordered normed spaces
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[1] P. Fletcher and W.F. Lindgren, Quasi-uniform spaces , Lect. Notes Pure Appl. Math. 77 (1982). · Zbl 0501.54018
[2] G.J.O. Jameson, Topology and normed spaces , Chapman and Hall, Ltd., London (1974). · Zbl 0285.46002
[3] L. Nachbin, Topology and order , Robert E. Kriegler Publishing Co., Huntington, New York (1976).
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