This is Problem 26, Chapter 4 in Rudin's Functional Analysis
$X, Y$ are Banach Spaces and $\mathcal{B}(X, Y)$ denotes the collection of all continuous linear maps from $X$ to $Y$.
Assume $T \in \mathcal{B}(X, Y)$ and $T(X) = Y$. Show that there is a $\delta > 0 $ such that $S(X) = Y$ for all $S \in \mathcal{B}(X, Y)$ with $|| S - T || < \delta$
By Open Mapping theorem this is equivalent to $T(B_X) \supset rB_Y$ for some $r > 0$ where $B$s denote the open unit balls.
This result is of course true in the finite dimensional case.
I have proved a similar result: Collection of Compact operators(these are never surjective) form a closed set but I have not made any progress on this one.
I am looking for hints if possible.