I'm trying show that if $p,q$ are Holder Conjugates then: $\forall\, a\in\mathbb{R}^{n}:\,\Vert a\Vert_{q}=\max_{x\in\mathbb{R}^{n},\,\Vert x\Vert_{p}=1}\left$ Where $\left$ is the Euclidian Inner-Product on $\mathbb{R}^{n}$ .
Immediately from Holder's Inequality I get that: $\max_{x\in\mathbb{R}^{n},\,\Vert x\Vert_{p}=1}\left\le\Vert a\Vert_{q}$ To show the other direction of the inequality I wanted to pick a $v\in\mathbb{R}^{n}$ such that $\Vert v\Vert_{p}=1$ and $\left=\Vert a\Vert_{p}$ but I can't seem to manage to do that. Since it's also easy to show that: $\max_{x\in\mathbb{R}^{n},\,\Vert x\Vert_{p}=1}\left=\max_{x\in\mathbb{R}^{n},\,\Vert x\Vert_{p}\leq1}\left$ It would also suffice to find a $v$ with $\Vert v\Vert_{p}\leq1$.
Help would be appreciated!