Exercise 8, § 1

Let $(E_\ell)$ be a family of fully lattice-ordered spaces, E the product space of the $E_\ell$ (which is fully lattice-ordered). Show that if B is a band in E, then each of its projections $B_\ell = \mathrm{pr}_\ell(B)$ is a band in $E_\ell$, and B is identical to the product of the $B_\ell$. From this, deduce the determination of the bands in the space $\mathbf{R}^A$ of mappings of a set A into $\mathbf{R}$. Show that, in the space $\mathscr{B}(A)$ of bounded real-valued functions on A, every band is the trace on $\mathscr{B}(A)$ of a band in $\mathbf{R}^A$.