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Zbl 0803.46007
Ferrer, J.; Gregori, V.; Alegre, C.
Quasi-uniform structures in linear lattices.
(English)
[J] Rocky Mt. J. Math. 23, No.3, 877-884 (1993). ISSN 0035-7596

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 ${\cal U}\sb q$ and the functions $q\sp*(x)= q(x)+ q(- x)$, $q\sp* M(x)= q(x)\lor q(-x)$, and $q\sp* E(x)= (q(x)\sp 2+ q(-x)\sp 2)\sp{1/2}$ define equivalent norms that induce the uniformity ${\cal U}{q\sp*}$. For the quasi-norm $q(x)= \Vert x\sp +\Vert$ defined on a normed lattice, the norms defined by $q\sp*\sb L$, $q\sb M$, and $q\sp*\sb E$ are equivalent to the original norm $\Vert \Vert$ 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 $q(x)= \Vert x\sp +\Vert$. The authors characterize the positive continuous linear functionals of a normed lattice $(E,\Vert \Vert,\le)$ 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 {\it 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.
[P.Fletcher (Blacksburg)]
MSC 2000:
*46A40 Ordered topological linear spaces
54E15 Uniform structures and generalizations
46B40 Ordered normed spaces

Keywords: normed lattice; parallelogram law; nonnegative sublinear function; translation-invariant quasi-uniformity; positive continuous linear functionals; Hahn-Banach theorem

Citations: Zbl 0285.46002

Cited in: Zbl 0930.46004

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