Definition 2

Let A be a field, L an affine space over A and T the space of translations of L (Chapter II, 2nd ed., App. II). If T is endowed with a non-degenerate hermitian form $\Phi$, L is said to be a hermitian space over A, and $\Phi$ the metric form of L.

If J is the identity (which implies that A is commutative), L is rather called a Euclidean space.

If $a$ and $b$ are two points of L, put $e(a, b) = \Phi(b - a, b - a)$. Let c be a third point of L. For $b - a$ and $c - a$ to be orthogonal, it is necessary, by formula (17) of $§ 1$, No. 5, that we have $e(b, c) = e(a, b) + e(a, c)$, and this condition is sufficient when $J = 1$ and A is not of characteristic 2 (“Pythagorean theorem”).

Two linear varieties of L are said to be orthogonal if their directions (Chapter II, 2nd ed., App. II, No. 3) are orthogonal. A linear variety of L is said to be isotropic (resp. totally isotropic) if its direction is isotropic (resp. totally isotropic). A vector of T is said to be orthogonal to a linear variety of L if it is orthogonal to the direction of that variety.

Let V be a linear variety in L, and $x$ a point of L. The set of points $y$ of L such that $y - x$ is orthogonal to V is a linear variety W passing through $x$; W is said to be the totally orthogonal variety (or, more simply, the orthogonal variety) to V passing through x. If L is of finite dimension, the dimension of W is equal to the codimension of V. Moreover, if V is nonisotropic, the directions of V and W are supplementary (§ 4, No. 1, cor. of prop. 1); then W meets V in a single point $x_1$; taking an origin in V, one sees at once that, for fixed V, the mapping $x \to x_1$ is an affine idempotent linear mapping; it is called the orthogonal projection of L onto V; the linear mapping associated with it (Chapter II, 2nd ed., App. II, No. 4) is the orthogonal projector of T onto the direction of V (No. 3).