vector norms and matrix norms

Let x be a vector. How do you show that $\displaystyle \left\| x^{*} \right\| _{p} = \left\| x \right\| _{q}$

where $\displaystyle \frac{1}{p} + \frac{1}{q} = 1$ ?

By using this definition of $\displaystyle \left\| x^{*} \right\| _{p} = max_{ \left\| y \right\| _{p} =1} \left\| x^{*} y \right\| _{p} $

and Holder's inequality, I am able to prove that

$\displaystyle \left\| x^{*} \right\| _{p} \leq \left\| x \right\| _{q}$

But how do you show the other side of the inequality?