I am not familiar with semigroup theory, so please stand with my dummy question.
Say, $A$ is the generator of a semigroup, consider space $X_{n} = D(A^{n})$ with graph norm, $\|f\|_{A^{n}}:=\|f\| + \|A^{n}f\|$.
Now, for $n \in \mathbb{N}$, define $\||x\||:=\|x\|+\|Ax\|+...+\|A^{n}x\|$. I need to prove $\||.\||$ and the standard norm(graph norm given above) are equivalent and furthermore, the space is Banach.
Regarding equivalence: $\||x\||\geq \|x\|_{A^{n}}$ is obvious, but how to prove the other direction?
Any comments are welcome. Cheers.