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

$q\left(x\right)=q(-x)=0$ then

$x=0$. Each quasi-normed space has a natural translation-invariant quasi-uniformity

${\mathcal{U}}_{q}$ and the functions

${q}^{*}\left(x\right)=q\left(x\right)+q(-x)$,

${q}^{*}M\left(x\right)=q\left(x\right)\vee q(-x)$, and

${q}^{*}E\left(x\right)={(q{\left(x\right)}^{2}+q{(-x)}^{2})}^{1/2}$ define equivalent norms that induce the uniformity

$\mathcal{U}{q}^{*}$. 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

$(E,\parallel \parallel ,\le )$ 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

*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.