Corollary

— Let V be a vector space over K and f a positive hermitian form on V. a) There exists a Hilbert space E and a linear mapping $u : V \to E$ such that $f(x, y) = \langle u(x)|u(y) \rangle$ for x, y in V, and such that u(V) is dense in E. b) If two pairs $(E_i, u_i)$ satisfy the conditions analogous to a), then there exists a unique isomorphism $\phi$ from the Hilbert space $E_1$ onto the Hilbert space $E_2$ such that $u_2 = \phi \circ u_1$.

Let N be the set of all $x \in V$ such that $f(x, x) = 0$. We define a positive and separating hermitian form on the space V/N by $\langle \dot{x}|\dot{y} \rangle = f(x, y)$ for $x \in \dot{x}$ and $y \in \dot{y}$. Let E be the hilbertian space completion of V/N and $u$ the mapping $x \mapsto x + N$ from V into E. Then the conditions of a) are satisfied.

Under the hypotheses of b), N is equal to the kernel of $u_1$ and to that of $u_2$. Hence there exists a bijective linear mapping $\phi_0$ from $u_1(V)$ onto $u_2(V)$ such that $u_2(x) = \phi_0(u_1(x))$ for all $x \in V$. We verify immediately that $\phi_0$ is an isomorphism of prehilbertian spaces, hence an isometry. Since $u_i(V)$ is dense in $E_i$ for $i = 1, 2$, $\phi_0$ extends uniquely to an isometry $\phi$ from $E_1$ onto $E_2$, and b) follows.

We say that the hilbertian space E is the separated completion of V (for the form f).