A surjective bounded operator between Banach spaces sends open sets to open sets.Suggest a slogan
Let \(X\) and \(Y\) be Banach spaces and let \(T\colon X\to Y\) be a continuous linear surjection. Then \(T\) is an open map: \(T(V)\) is open in \(Y\) for every open \(V\subseteq X\).