Remark

It follows from the existence of orthonormal bases that two non-degenerate positive hermitian forms on two vector spaces of the same finite dimension are equivalent (§ 1, no 6). Let then L be a finite-dimensional hermitian space over A, whose metric form is positive non-degenerate, and let $V_1$ and $V_2$ be two linear varieties of the same dimension in L (§ 6, no 6); since the restrictions of the metric form to the directions $T_1$ and $T_2$ of $V_1$ and $V_2$ on the one hand, and to the orthogonal subspaces $T_1^0$ and $T_2^0$ on the other hand, are equivalent, there exists a unitary automorphism $u$ of the space $T$ of the translations of $L$ such that $u(T_1) = T_2$ and $u(T_1^0) = T_2^0$; there therefore exists a displacement $v$ of $L$ such that $v(V_1) = V_2$. Let $(a, b), (a', b')$ be two pairs of points of $L$; for there to exist a displacement $v$ of $L$ such that $v(a) = a'$ and $v(b) = b'$, it is therefore necessary and sufficient (with the notation of § 6, no 6) that one has $e(a, b) = e(a', b')$; the element $\sqrt{e(a, b)}$ of $A$ (chap. VI, § 2, no 4) is called the distance between $a$ and $b$ in the hermitian space $L$.