Remark

In the space $\mathbf{R}^A$, the subspace $\mathscr{B}(A)$ of bounded real-valued functions on $A$ satisfies condition 1) of Def. 4; moreover, for every subset $X$ of $\mathscr{B}(A)$ that is bounded above in $\mathscr{B}(A)$, the upper envelope of $X$ belongs to $\mathscr{B}(A)$. However, if $A$ is infinite, a subset of $\mathscr{B}(A)$ may be bounded above in $\mathbf{R}^A$ without being bounded above in $\mathscr{B}(A)$, in which case $\mathscr{B}(A)$ is not a band in $\mathbf{R}^A$.

It follows at once from Def. 4 that if $B$ is a band in $E$ then, for every nonempty subset $X$ of $B$ that is bounded below in $E$, $\inf X$ belongs to $B$. Every band $B$ in $E$, equipped with the ordered vector space structure induced by that of $E$, is a fully lattice-ordered space and, for every subset $X \subset B$ that is bounded above in $B$, the supremum of $X$ in $B$ is identical with its supremum in $E$.

The intersection of any family of bands in a fully lattice-ordered space $E$ is also a band. For every subset $M \subset E$, there exists a smallest band containing $M$ (since $E$ itself is a band); this band will be called the band generated by $M$.

The properties of bands in a fully lattice-ordered space rest on the following proposition: