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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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##### References:
 [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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