For positive T, show ⟨Tx,y⟩≤⟨Tx,x⟩⟨Ty,y⟩

Question

I saw an answer here for when T is positive AND self adjoint. However, I have to show that the inequality ⟨Tx,y⟩≤⟨Tx,x⟩⟨Ty,y⟩ holds when T is positive.

I assume I'll have to use Cauchy-Schwarz inequality but I don't think I'm doing this right:

⟨Tx,y⟩≤||Tx||⋅||y||=||T||⋅||x||⋅||y||

Any hints towards the right direction would be appreciated.

Answer 1

So let $[x,y]=\left<Tx,y\right>$, we have $[x,x]=\left<Tx,x\right>\geq 0$ and $[y,x]=\left<Ty,x\right>=\left<y,Tx\right>=\overline{\left<Tx,y\right>}=\overline{[x,y]}$, linearity and scalar conjugate are easy to check. Now use Cauchy-Schwarz.

It is $|\left<Tx,y\right>|^{2}\leq\left<Tx,x\right>\left<Ty,y\right>$.