We consider the optimization of the vector of grasping forces that support a known generalized force acting on the grasped object---a rigid body or a mechanism. Working in the framework of finite-dimensional normed vector spaces and their dual spaces, the cost function to be minimized is assumed to be a norm on the space of grasping forces. We present an expression for the optimum which depends on the external force and the kinematics of the grasping system. Next, assuming that optimal grasping forces are applied using force control, and assuming that there is a bound on the norm of the admissible grasping forces, we characterize the largest norm of an external force that the grasping system may support, that is, the norm of the worst-case loading that may be applied and still be supported. A few simple examples are given for the sake of illustration.