Exercise :
Let $X$ be a Banach space and $A \in \mathcal{L}_c(X)$ (means that $A$ is a compact operator). Suppose that $(\text{id}-A)$ is $"1-1"$. Show that the operator $(\text{id}-A)$ has a continuous inverse over the set $V=(\text{id}-A)(X)$.
Question :
As I fail to comprehend the question fully into functional analysis terms, how would one show that there exists a continuous inverse ? What is sufficient to prove that ?
Since $(\text{id}-A)$ is injective, there would exists an inverse if it was also surjective as well, but I can't seem to find a way around that. Also for the continuity, what's the catch ? Does showing that $(\text{id}-A)$ is an isometry has anything to do with that ?
Any hint on how to approach or elaboration will be the most appreciated since I cannot get a grasp on an intuition.