How to prove Schwarz inequality for Hermitian forms? I'm trying to do something like the proof of the Schwarz inequality for inner product.

If $h(y,y)\neq 0$, then we can take $\alpha=-h(x,y)/h(y,y)$ and calculate $h(x+\alpha y,x+\alpha y)$ which is a nonnegative number. The desired conclusion follows of this calculation.

If $h(y,y)=0=h(x,y)$, then the inequality is trivial.

My question is: how do we know that the case $h(y,y)=0$ and $h(x,y)\neq 0$ is not possible? When we are working with an inner product, this case is not possible because, by definition, $\langle y,y\rangle=0\Rightarrow y=0\Rightarrow\langle x,y\rangle=0$. But we don't have this condition for hermitian forms.

Thanks.
If $h(y,y) = 0$ but $h(x,y) \ne 0$, then taking $\alpha = - t h(x,y)$ with $t &gt; 0$
we would have 
$h(x + \alpha y,  x + \alpha y) = h(x, x) - 2 t |h(x,y)|^2$,
which would be negative for sufficiently large $t$.
Instead of considering $h(y,y)$, you should consider $h(x,y)\neq 0$, then put $\alpha=\frac{-h(x,x)}{h(y,x)}$ – it always works. For $h(x,y)=0$, since $h(x,x)\ge 0$ we get the inequality.