Proposition 3

Tag 07X7, permanent. Cite it as [Bourbaki, Tag 07X7].

— Let f be a positive hermitian form on E. Put

p(x) = f(x, x)^{1/2}

for all x \in E. Then p is a semi-norm on E, and is a norm if and only if f is separating.

It is enough to prove the inequality $p(x + y) \leq p(x) + p(y)$ . But we have

f(x + y, x + y) = f(x, x) + f(y, y) + f(x, y) + \overline{f(x, y)}

and, by Cauchy-Schwarz inequality

f(x + y, x + y) \leq f(x, x) + f(y, y) + 2(f(x, x) f(y, y))^{1/2}

= (f(x, x)^{1/2} + f(y, y)^{1/2})^2 .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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