Proposition 13

— Let $\mathscr{H}(E)$ be the set of all continuous hermitian endomorphisms of the hilbertian space $E$. Let $\mathscr{F}$ be a non-empty, directed increasing and bounded subset of $\mathscr{H}(E)$.

(i) The set $\mathscr{F}$ has an upper bound $u_0$ in $\mathscr{H}(E)$; we have

(ii) The filter of sections of $\mathscr{F}$ converges to $u_0$ in the space $\mathscr{L}(E)$ endowed with the topology of simple convergence.

Let $\Sigma$ be the filter of sections of $\mathscr{F}$; for every $u \in \mathscr{H}(E)$, let $\Phi_u$ be the continuous hermitian form on $E$ defined by

Let for $u \in \mathscr{H}(E)$ and $x \in E$. By the polarization formulas (V, p. 2), we have

For every $x \in E$, the mapping $u \mapsto \Psi_u(x)$ from into $\mathbf{R}$ is increasing and bounded, hence has a limit with respect to $\Sigma$. By the preceding formulas, the limit

exists for every pair $(x, y)$ of elements of $E$. It is clear that $\Phi$ is a hermitian form on $E$. If $v_1 \in \mathscr{F}$ and $v_2$ is a bound of $\mathscr{F}$, the hermitian forms $f_1 = \Phi - \Phi_{v_1}$ and $f_2 = \Phi_{v_2} - \Phi$ are positive; there exists a real number $M \geq 0$ such that hence consequently the semi-norms $x \mapsto f_i(x, x)^{1/2}$ are continuous on $E$. Since we conclude that $x \mapsto \Phi(x, x)$ is a continuous function on $E$, and by formulas (19) and (20), that $\Phi$ is continuous on $E \times E$. Therefore there exists (V, p. 16, cor. 2) an element $u_0$ of $\mathscr{H}(E)$ such that $\Phi = \Phi_{u_0}$. Formula (18) is evidently satisfied, hence $u_0$ is the upper bound of $\mathscr{F}$ in $\mathscr{H}(E)$. This proves (i).

We have, by construction

Let $v_1 \in \mathscr{F}$; given a $u \in \mathscr{F}$ such that $u \geq v_1$, let $v = u_0 - u$. If we apply the Cauchy-Schwarz inequality to the positive hermitian form $\Phi_v$ on $E$, we get since $\|v\| \leq \|u_0 - v_1\|$ by V, p. 44, prop. 9. Then by (21) we get $\lim_{u,\Sigma} \|(u_0 - u)(x)\| = 0$ for all $x \in E$; which proves assertion (ii).

In particular, prop. 13 can be applied to the case of an increasing and bounded sequence $(u_n)_{n \in \mathbf{N}}$ of elements of $\mathscr{H}(E)$. Then there exists an element $v$ of $\mathscr{H}(E)$ characterized by

and we have $v(x) = \lim_{n \to \infty} u_n(x)$ for all $x \in E$. Moreover, $v$ is the upper bound of the set of the $u_n$ in $\mathscr{H}(E)$.