$ \| * \|_a ,\| * \|_b $ - norms in vector space $ X $. Any linear functional $f: X \to K$ is continuous towards $ \| * \|_a \iff$ is continuous towards $ \| * \|_b $. Prove that these norms are equivalent.
I need to prove that $ C_1 \| x \|_a \leq \| x \|_b \leq C_2 \| x \|_a $. I know that $ | f(x) | \leq M_1\| x \|_a$ and $ | f(x) |\leq M_2\| x \|_b$. But what should be next?