Question

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.

Answer 1

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.

Answer 2

Cauchy-Schwarz in the orthogonal complement to v?