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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?

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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.

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