Let E be a Banach space.
Let $L(E)$ be the set of linear continuous operators from E to E.
Let A be the set of inversible elements of $L(E)$.
Prove that A is open in $L(E)$ and that $T→ T^{−1}$ is smooth in $L(E)$.
What I did:
I proved that if $||T|| < 1$ then $Id − T ∈ A$
I proved also that if $T ∈ A$ and $S ∈ L(E)$ then $T + S ∈ A$.
Now I need to prove that A is open so I need to find a ball of center $T$ such that all operators S belonging to this ball are in A.
For the smoothness of the inverse operator in A, I have no clue on how to prove it.
Thank you for any help or hints