We cast the problem of optimal decentralized control as one of minimizing a closed-loop norm subject to a subspace constraint on the controller. In this note, we consider continuous linear operators on Banach spaces, and show that a simple property called quadratic invariance is necessary and sufficient for the constraint set to be preserved under feedback, and thus allows optimal synthesis to be recast as a convex optimization problem. These results hold for any norm and any Banach space.