Suppose F = C,V is finite-dimensional, T in L(V), all the eigenvalues of T have absolute value less than 1, and epsilon > 0. Prove that there exists a positive integer m such that ||T^mv|| leq e ||v|| for every v in V. Solution Given that F =C and V is finite dimensional As all the eigen values are <1 Hence For Epsilon >0 T(v) /||V|| <1 Hence ||T^m(V) || = |T(V)|m But as |x+y|<=|x|+|y| |T(V)|m<= |T(v)|+T(V)+.... m times As each eigen value is <1 ||T^m(V) ||.