Theorem 1

1° In order that a linear form $L$ on a Riesz space $E$ be relatively bounded, it is necessary and sufficient that it be the difference of two positive linear forms.

2° The ordered vector space $\Omega$ of relatively bounded linear forms on $E$ is a Riesz space that is fully lattice-ordered.

If $L = U - V$, where $U$ and $V$ are positive linear forms on $E$, the relation $-x \leq y \leq x$ implies

whence it is immediate that $|L(y)| \leq U(x) + V(x)$; thus, $L$ is relatively bounded. Suppose, conversely, that $L$ is relatively bounded; it all comes down to proving that there exists a positive linear form $N$ such that $N(x) \geq L(x)$ for all $x \geq 0$, because $N - L$ will then be a positive linear form.

Now, if a positive linear form $N$ has this property then, for every $x \geq 0$ and for $0 \leq y \leq x$, we have $N(x) \geq N(y) \geq L(y)$, therefore $N(x) \geq \sup_{0 \leq y \leq x} L(y)$; if we prove that the real-valued function

defined on the set $P$ of elements $\geq 0$ of $E$, may be extended to a positive linear form on $E$ (to be denoted $M$ as well), we will have demonstrated the first part of the theorem and will have proved, moreover, that $M$ is the supremum of $0$ and $L$ in $\Omega$. Since $M(x) \geq 0$ on $P$, it all comes down to proving that

for every pair of elements $x \geq 0,\ x' \geq 0$ of $E$ (No. 1, Prop. 3). By definition,

On the other hand, for every $z$ such that $0 \leq z \leq x + x'$, we have $x + x' = z + u$ with $u \geq 0$; by the decomposition lemma (§ 1, No. 1), there exist two elements $y,\ y'$ such that $0 \leq y \leq x,\ 0 \leq y' \leq x'$ and such that $z = y + y',\ u = (x - y) + (x' - y')$; then

therefore $M(x + x') = \sup_{0 \leq z \leq x + x'} L(z) \leq M(x) + M(x')$, which completes the proof of the first part of the theorem. Moreover, we have thus shown that $\Omega$ is a Riesz space and that, for every relatively bounded linear form $L$ on $E$ and for every $x \geq 0$,

(1)

It remains to see that $\Omega$ is fully lattice-ordered; for this, it suffices to show that any set $H$ of positive linear forms, bounded above and directed for the relation $\leqslant$, has a supremum in $\Omega$.

More generally, we have the following lemma: