Exercise 13, § 1

a) Let E be a fully lattice-ordered space. If $u \neq 0$ is an arbitrary element of E, show that the set of $x \in E$ such that there exists an integer $n$ for which $|x| \leq n|u|$, is a linear subspace $C_u$ that is a fully lattice-ordered and co-lattice subspace (Exer. 3) of E. For every $x \in C_u$, one denotes by $\|x\|$ the infimum of the scalars $\lambda > 0$ such that $|x| \leq \lambda|u|$; show that $\|x\|$ is a norm on $C_u$ (make use of the fact that E is archimedean (Exer. 5)), that $C_u$, equipped with this norm, is complete, and that $\|x\| = \sup(\|x^+\|, \|x^-\|)$.

b) Let $\mathscr{J}$ be the set of components of $u$ in the bands of the space $C_u$; show that $\mathscr{J}$ is the set of elements $c$ of $C_u$ such that $0 \leq c \leq u$ and $\inf(c, u - c) = 0$ (use Riesz’s theorem). Show that $\mathscr{J}$ is closed in the normed space $C_u$ and that, for the order relation induced by that of $C_u$, $\mathscr{J}$ is a complete Boolean algebra (S, III, §1, Exers. 11 and 17;¹ it suffices to show that if $(c_\iota)$ is a family of elements of $\mathscr{J}$ then $\sup_\iota c_\iota$ belongs to $\mathscr{J}$.

c) Let $x$ be any element of $C_u$; for every $\lambda \in \mathbf{R}$ let $c(\lambda)$ be the component of $u$ in the band of $C_u$ generated by $(\lambda u - x)^+$. Show that if $\lambda \leq \mu$ then $c(\lambda) \leq c(\mu)$; $c(\lambda) = 0$ for $\lambda < -\|x\|$, and $c(\lambda) = u$ for $\lambda > \|x\|$. Show that if $c \in \mathscr{J}$ is such that $c \leq c(\lambda)$, then the component of $x$ in the band generated by $c$ is $\leq \lambda c$ (observe that the bands generated by $c(\lambda)$ and by $(\lambda u - x)^+$ are identical, and that the component of $\lambda u - x$ in this band is equal to that of $(\lambda u - x)^+$; from this, deduce that the component of $x$ in this same band is $\leq \lambda c(\lambda)$). Similarly, show that if $c \in \mathscr{J}$ is such that $c \leq u - c(\lambda)$, then the component of $x$ in the band generated by $c$ is $\geq \lambda c$.

d) It is known (GT, II, §4, Exer. 12) that there exists an order structure isomorphism $c \mapsto \theta_c$ of the Boolean algebra $\mathscr{J}$ onto the Boolean algebra formed by the characteristic functions of the sets in a compact, totally disconnected space S that are both open and closed. Show that the relation $\sum_i \lambda_i c_i = 0$ implies that the function $\sum_i \lambda_i \theta_{c_i}$ is 0 on S (make use of the decomposition lemma); deduce from this remark and c) that the mapping $c \mapsto \theta_c$ may be extended to an isomorphism $x \mapsto \theta_x$ of the normed space $C_u$ onto the normed space $\mathscr{C}(S; \mathbf{R})$ of continuous real-valued functions on S, so that $(\theta_x)^+ = \theta_{x^+}$. (With the help of c), show that for every $x \in C_u$ and every $\varepsilon > 0$, there exists an increasing sequence $(\lambda_i)_{0 \leq i \leq n}$ of real numbers such that $\lambda_i - \lambda_{i-1} \leq \varepsilon$ and $0 \leq x - \sum_{i=1}^n \lambda_{i-1} (c(\lambda_{i-1}) - c(\lambda_i)) \leq \varepsilon u .$

e) In order that S be finite, it is necessary and sufficient that $C_u$ be finite-dimensional; from this, deduce (with the help of Exer. 12) that every finite-dimensional fully lattice-ordered space is isomorphic to a product space $\mathbf{R}^n$ (cf. §2, Exer. 7).

f) Show that S is an extremally disconnected space (GT, I, §11, Exer. 21) (make use of the fact that $\mathscr{J}$ is a complete Boolean algebra). An extremally disconnected compact space is called a Stone space. Show that if X is a Stone space then, for every lower semi-continuous numerical function $f$ on X, the upper semi-continuous regularization $g$ of $f$ (GT, IV, §6, No. 2) is continuous (for every $a < g(x)$ show that $x$ cannot belong to the closure of the (open) set of $y$ such that $g(y) < a$, by observing that $f(z) \leq a$ at a point $z$ in the closure of this set). From this, deduce that $\mathscr{C}(X; \mathbf{R})$ is then a fully lattice-ordered space. Show that when the Stone space X is infinite, the supremum in $\mathscr{C}(X; \mathbf{R})$ of a set that is bounded above is not necessarily equal to its upper envelope (consider a non-closed open set in X); however, these two functions are equal on the complement of a meager set (consider the set of points where their difference is $\geq 1/n$).

g) Let X be a Stone space; show that if $f$ is a bounded real-valued function, defined on the complement of a nowhere dense subset M of X and continuous on $X - M$, then

¹ The terms 'Boolean algebra' (GT, I, §6, Exer. 20) and 'Boolean lattice' (S, III, §1, Exer. 17) mean the same thing (réseau booléien); completeness means that every subset has a supremum and an infimum (S, III, §1, Exer. 11).

f may be extended by continuity to all of X (make use of the fact that $\mathscr{C}(X; \mathbf{R})$ is fully lattice-ordered). From this, deduce that the set $\mathscr{Q}(X; \overline{\mathbf{R}})$, of continuous numerical functions $f$ on X (finite or not) such that $f^{+1}(\infty)$ and $f^{-1}(-\infty)$ are nowhere dense in X, is a fully lattice-ordered vector space.

h) Show that the band $B_u$ generated by $u$ in E is isomorphic (as an ordered space) to an isolated subspace (Exer. 4) of $\mathscr{Q}(X; \overline{\mathbf{R}})$ (regard an element $f \geq 0$ of $\mathscr{Q}(X; \overline{\mathbf{R}})$ as the supremum of the elements $\inf(f, n)$).