This is a question from Rudin's Functional Analysis.
Let $T\in \mathcal B(X,Y)$ and $T(X)=Y$. I have to prove that there exists $\delta>0$ such that $\{S\in \mathcal B(X,Y):S(X)=Y\}\supset B_{\delta}(T)$, where the ball is with respect to the metric generated by the operator norm.
I could not proceed. Any help is appreciated.