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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 $q\left(x\right)=q\left(-x\right)=0$ then $x=0$. Each quasi-normed space has a natural translation-invariant quasi-uniformity ${𝒰}_{q}$ and the functions ${q}^{*}\left(x\right)=q\left(x\right)+q\left(-x\right)$, ${q}^{*}M\left(x\right)=q\left(x\right)\vee q\left(-x\right)$, and ${q}^{*}E\left(x\right)={\left(q{\left(x\right)}^{2}+q{\left(-x\right)}^{2}\right)}^{1/2}$ define equivalent norms that induce the uniformity $𝒰{q}^{*}$. For the quasi-norm $q\left(x\right)=\parallel {x}^{+}\parallel$ defined on a normed lattice, the norms defined by ${q}_{L}^{*}$, ${q}_{M}$, and ${q}_{E}^{*}$ are equivalent to the original norm $\parallel \parallel$ 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 $q\left(x\right)=\parallel {x}^{+}\parallel$. The authors characterize the positive continuous linear functionals of a normed lattice $\left(E,\parallel \parallel ,\le \right)$ as those linear functionals that are continuous when considered as maps from $\left(E,q\right)$ 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