Decide if the given endomorphism $F:V\rightarrow V$ is normal

Question

Decide if the given endomorphism $F:V\rightarrow V$ is normal

Let $V := \mathbb{C}[x]_{\leq 2}$ with $ \langle g,h\rangle := \int_{-1}^{1}g(x)\overline{h(x)}dx$ and $F$ given by the derivative $F(g):=g'$

I would love to get some tipps on how to solve this one. Any help is appreciated

Answer 1

Tip: Your operator satisfies $F^3 = 0$ so $F$ is nilpotent. When can a nilpotent operator be normal?