Dear Colleagues,

Could you please help me in solving the following problem:

Let $\displaystyle T:X\longrightarrow X$ be a bounded linear operator on a complex inner product space $\displaystyle X$. If $\displaystyle \langle Tx,x \rangle =0$ for all $\displaystyle x\in X$, show that $\displaystyle T=0$.

Moreover, this is does not hold in the case of real inner product space.

Regards,

Raed.