Definition 3

Let $E$ be a vector space over the field $K$. A semi-norm $p$ on $E$ is said to be prehilbertian if there exists a positive hermitian form $f$ on $E$ such that $p(x) = f(x, x)^{1/2}$ for all $x \in E$.

Observe that for a semi-norm $p$ on $E$, there exists at most one positive hermitian form $f$ such that $p(x) = f(x, x)^{1/2}$ for all $x \in E$; this follows from the polarization formulas (V, p. 2).