Let $(V,\langle \cdot , \cdot\rangle)$ inner product space on $\mathbb{C}$ and let $T:V \rightarrow V$ prof or disproof. if $\forall v \in V$ $\langle T(v),v\rangle$ $ = 0$ than $T=0$.
I had found very easily counterexample for the case $F= \mathbb{R}$ by defending symmetric matrix that defines the inner product and the same matrix don't work on the complex filed than I thinking that it may be true.
the example for $F= \mathbb {R}$ :
$T(x,y) = (y,y) $ and $A = \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}$
$\langle u,v \rangle = u^T A v$
