Compute norm of vector using optimization over inner product

Question

Let $x$ be a vector, and let $p,q$ numbers such that $\frac{1}{p} + \frac{1}{q} = 1$. Why $||x||_p = \max_{y : ||y||_q = 1}y^T x$? I tried to prove it with Hölder inequality, but I did not succeeded to finish the proof.

Answer 1

Let $x \neq 0$. $y^{T}x \leq \|x\|_p$ is just Holder's inequality. Put $y_i=\frac 1 A|x_i|^{p-1} sign(x_i) $ where $A=\|x\|_p^{p/q}$ to see that the value $\|x\|_p$ is actually attained. The result is trivial when $x=0$.