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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 𝒰 q and the functions q * (x)=q(x)+q(-x), q * M(x)=q(x)q(-x), and q * E(x)=(q(x) 2 +q(-x) 2 ) 1/2 define equivalent norms that induce the uniformity 𝒰q * . 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,,) 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:
46A40Ordered topological linear spaces, vector lattices
54E15Uniform structures and generalizations
46B40Ordered normed spaces