## Impossible relation

Prove that we cannot find any two linear continuous maps $f,g : X \to X$ such that $(f\circ g -g\circ f)x=x,\ \forall x \in X$.
Solution: Prove by induction that $f \circ g^{n+1}-g^{n+1}\circ f =(n+1)g^n$. Taking norms, we get $(n+1)\|g^n \| \leq 2 \|f\| \|g\| \|g^n\|$. If $\|g^n\| \neq 0$ then $n+1 \leq 2 \|f \| \|g\|,\ \forall n$, which is a contradiction. On the other hand, if $g^n=0$ then $g^{n-1}=0$, which implies $g=0$ and $I=0$. Contradiction. Therefore, we cannot find linear continuous function with given property.