Remark

— 1) Suppose f is positive and separating, and let x, y be two vectors $\neq 0$. The proof of Cauchy-Schwarz inequality shows that, if the two members of (7) are equal, then there exists a scalar $\xi$ such that $f(x + \xi y, x + \xi y) = 0$, hence $x + \xi y = 0$, in other words, $x$ and $y$ are linearly dependent; the converse is immediate. The proof of inequality (8) shows that the equality $p(x + y) = p(x) + p(y)$ is possible only if $x$ and $y$ are linearly dependent; if $y = \lambda x$, the preceding equality can be written as $|1 + \lambda| = 1 + |\lambda|$, and implies that $\lambda$ is real and positive.

2) Let $f$ be a positive hermitian form on $E$, and let $E$ be assigned the semi-norm $x \mapsto f(x, x)^{1/2}$; if $f$ is the positive, separating hermitian form defined on $E/N$ associated with $f$, then the normed space obtained by assigning the norm $x \mapsto f(\dot{x}, \dot{x})^{1/2}$ to $E/N$ is the normed space associated with $E$ (II, p. 5).