Definition 2

Let $E$ be a vector space over the field $K$. A hermitian form $f$ on $E$ is said to be positive if $f(x, x) \geq 0$ for all $x \in E$.

It is clear that hermitian forms on a vector space $E$ form a vector space over the field $\mathbf{R}$ (but not over the field $\mathbf{C}$, when $K$ is $\mathbf{C}$) : in this space the positive hermitian forms constitute a pointed convex proper cone (II, p. 10) as a result of def. 2 and prop. 1.