# Question about locality of bounded linear operators between Banach spaces

Suppose $T$ is an invertible bounded linear operator from $X$ to $Y$, which are Banach spaces. Is there a neighborhood $U$ of $T$ in $\mathcal{B}(X,Y)$ (space of bounded linear operators from $X$ to $Y$) in which all the operators are invertible? This is certainly true in the finite dimensional case because the determinant is continuous, but what about the infinite dimensional case?

Yes, this is true (in the operator-norm topology). For $X=Y$ this is a special case of the fact that the invertible elements in any Banach algebra form an open set, see Neumann series.