MAT 544 Real

whenever ‖~v‖V < δ. In this case the bounded linear operator dF~v0 is called the (Frechet) derivative of F at ~v0. Prove that if ‖ • ‖V is a norm on V which is equivalent to ‖ • ‖V , then F is differentiable at ~v9 with respect to ‖ • ‖V if and only if it is differentiable at ~v0 with respect to ‖ • ‖V , and in this case the derivatives are equal. Similarly, if ‖ • ‖W is a norm on W which is equivalent to ‖ • ‖W , prove that F is differentiable at ~v0 with respect to ‖ • ‖W if and only if it is so with respect to ‖ • ‖W , and in this case the derivatives are equal. In particular, if V and W are finite dimensional, then all norms on V , respectively on W , are equivalent, hence differentiability is an intrinsic property.