Let $S,T:X \rightarrow Y$ be bounded linear maps between Banach spaces $X$ and $Y$. Suppose $T$ is a bijection. Prove that there exists $\delta>0$ depending only on $T$ such that $||S-T||<\delta$ implies that $S$ is a bijection.
Edit: I have found the answer to my question and will be posting it here so that the question can be closed.
We let $S=(T(\mathbb{I}−T^{−1}(T−S))$. We let $A=T^{−1}(T−S)$ and then we can use the fact that $(\mathbb{I}−A)$ is invertible for $||A||<1$. So now we only need $||S−T||<1/||T^{−1}||$ so that $||A||=||T^{−1}(T-S)||<1$. The quantity $(1/||T^{−1}||)$ will be the $\delta$ mentioned in the problem. So now we have $S=T(\mathbb{I}−A)$. $T$ is invertible, $(\mathbb{I}−A)$ is invertible, therefore $S$ is invertible.