Let $\| \cdot \|$ be a norm on $\mathbb{R}^n$. The associated dual norm, denoted $\| \cdot \|_*$ is defined as $\| z \|_* = \sup\{ z^{t} x : \| x \| < 1 \}$.
Does someone know how prove that the dual norm of the $\mathcal l_{p}$ norm is the $\mathcal l_{q}$ norm? It's not homework. I've been reading about norms and it was stated without proof in a book. Thanks.