I'm wondering (hoping) if an inequality is true. Please can anyone help me?


Let $V$ be a complex vector space $dim\_{\mathbb{C}}(V)=n$ 
with a hermitian scalar product $h$.
Let $v,a, b \in V$.


Is it true that


$(h(v,v)h(a,a)-{|h(v,a)|}^{2})(h(v,v)h(b,b)-{|h(v,b)|}^{2})\geq |(h(v,v)h(a,b)-h(a,v)\overline{h(b,v)}|^{2}$?


With the overline meaning complex conjugate.


Thank you in advance.



Linear algebra inequality
Yes. The case where $v=0$ is trivial so suppose $v\ne0$. Consider
the projection map from $V$ to the hyperplane orthogonal to $v$
and let $a'$ and $b'$ be the images of $a$ and $b$ under this projection.
Then your inequality reduces to
$$h(a',a')h(b',b')\ge\vert h(a',b') \vert^2,$$the Cauchy-Schwarz inequality.


