I'd like to prove what I believe is the extremal version of Holder's inequality in $\mathbb{R}^n$. That is, for $x,y\in\mathbb{R^n}$:
$$ \lVert x \rVert_p = \max \frac{\lvert y^T x \rvert}{\lVert y \rVert_q} $$
where
$$\frac{1}{p} + \frac{1}{q} = 1$$
I'm really lost on how to go about doing this! Any advice would be appreciated.
Hint: If you take $y_i=x_i |x_i| ^{p-2}$ and compute $\frac {|y^{T}x|} {\|y\|_q}$ you will get exactly $\|x\|_p$. Hence RHS is greater than or equal to LHS. Of course. RHS $\leq$ LHS is just Holder's inequality.