It is shown that exact controllability in finite time for linear control systems given on an infinite dimensional separable Banach space in integral form (mild solution) can never arise using locally L 1-controls, if the operator through which the control acts on the system is compact. This improves a previous result of the author, by removing the assumption that the state space have a basis. It is suggested by the recent discovery that a separable Banach space need not have a basis.