Definition 1

Let E be a vector space over the field K. A hermitian form (on the left) on E is a map f from E × E into K satisfying the following conditions (for $x_1, x_2, x, y_1, y_2, y$ in E and $\lambda, \mu$ in K):

(1)

(2)

(3)

When the field K is R, the notion of hermitian form on E reduces to that of symmetric bilinear form on E × E (A, III, § 6, No. 3).

We note that the second condition (1) and the second condition (2) follow from the other three.

¹ For the reader specially interested in hilbertian spaces, we point out that only No. 7 of § 1 and No. 8 of § 4 depend on results of chapters III and IV. For this the reader can consult « Summary of some important properties of Banach spaces » which appears at the end of this volume. The only references to chapters I and II concern the definition of a convex set and of a semi-norm (II, p. 1 and p. 7), that of a topological direct sum (I, p. 4), of a total family and a topologically independent family (I, p. 12).

From (1) and (2) we deduce immediately that

In particular, if E is finite dimensional, and if $(e_j)_{1 \leq j \leq n}$ is a basis of E, then for $x = \sum_{j=1}^n \xi_j e_j$ and $y = \sum_{j=1}^n \eta_j e_j$, we have,

with the notation $\alpha_{jk} = f(e_j, e_k)$; moreover, relation (3) amounts to $\alpha_{jk} = \overline{\alpha}_{kj}$ for every pair of indices $j, k$; this implies in particular that the numbers $\alpha_{jj}$ are real.

From (3), the number $Q(x) = f(x, x)$ is real for all $x \in E$. Moreover, we immediately establish the following formulas, known as polarization formulas

(5)

(6)