Let X be a complex Banach space and T in L(X, X) a linear operator. Let (T* f)(x)=f (Tx), (where x in X, f in X*) define a linear operator T* in L(X*, X*).

Assuming that T is continuous, show that T* is continuous with ||T*||<=||T||.

Would anybody be kind enough to offer a proof of this result or point me in the direction of a book where the result is proved?

Thanks