Determine if the following statement is true or false: There is no inner product on $P_3(\mathbb{R})$ w.r.t. which the differentiation map $T: P_3(\mathbb{R})\to P_3(\mathbb{R})$ defined by $T(f)=f'$ is symmetric.
For this question I try to assume there is an inner product and then prove no such inner products and this is because the differentiation map is not surjective. But I am not sure if I have the correct idea.