Simple Energy Estimate problem

Question

I am doing a computation and I need the following inequality (if true) $$ \int_{-\infty}^\infty{fv^2 {\mbox{d}}x}\leq \|f\|_{L^2}\|v\|_{L^2}^2, $$ for any function $v$ and $f$ in $L^2(\mathbb{R})$. It seems it should hold but I cannot quite use Cauchy-Schwarz or Holder to make it work. If it is true, please provide the argument so I can use it in my following computations. Thank you.

Answer 1

It is false. By Basic Hilbert space theory this inequality would imply that $\|v^{2}\| \leq \|v\|^{2}$ and it is easy to see that this is false. [Example $v=I_{(0,\frac 1 2)}$].