The problem statement, all variables and given/known data Let V and V' be real normed vector spaces and let f be a linear transformation from V to V'. Prove that f is continuous if V is finite dimensional. The attempt at a solution Let [tex]v_1, v_2, \ldots, v_n[/tex] be a basis for V, let e > 0 and let v in V. I must find a d such that for all u in V such that ||u - v|| < d, ||f(u) - f(v)|| < e. Now [tex]f(u) - f(v) = a_1 f(v_1) + \cdots + a_n f(v_n)[/tex] for some scalars [tex]a_i[/tex]. Thus [tex]\|f(u) - f(v)\| \le \|a_1 f(v_1)\| + \cdots + \|a_n f(v_n)\|[/tex]. If [tex]\|a_i f(v_i)\| < e/n[/tex], then ||f(u) - f(v)|| < e. This is as far as I've gone. Any tips?