= Positive semidefinite Hermitian form
{title2=$h(v,v)\geq0$}
A <Hermitian form> is positive semidefinite when $h(v,v)\geq0$ for every <vector>. It need not define an <inner product>, because a nonzero <vector> may have zero squared <norm>. The <Cauchy-Schwarz inequality> still holds: applying nonnegativity to $v+zw$ and minimizing the quadratic expression in $z$ gives $|h(v,w)|^2\leq h(v,v)h(w,w)$ when $h(w,w)>0$; when $h(w,w)=0$, varying $z$ forces $h(v,w)=0$. Thus its zero-norm <vectors> are exactly the <radical of a Hermitian form>. Quotienting this <radical of a Hermitian form> gives a positive <inner product>; taking its <Hilbert space completion> then gives a <Hilbert space>. This is the final positivity step in the <Gupta-Bleuler null-state quotient>.
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