Can I prove Hölder's inequality given duality of $p$- and $q$-norms?

Question

The only fact I know is $$||x||_p=\max_{||z||_q\le1}|z^Tx|$$ Of course, $\frac1p+\frac1q=1$.

Based on the definition of $||x||_p$, I want to prove Hölder's inequality that for all $x,y$: $$|x^Ty|\le||x||_p||y||_q$$

I know how to prove Hölder's inequality based on Young's inequality and the definition of $p$-norms, but do I need those low-level proofs here?

Answer 1

Take any nonzero $x$ and let $y=x/\|x\|_p$. Take any $z\in \ell^q$ and put $w=z/\|z\|_q$. Then $\|y\|_p=\|w\|_q=1$. Applying $$\tag{1}||x||_p = \max_{||z||_q \leq 1} z^T x$$ to $y$ and $w$, we get $$ |w^Ty|\leq\|y\|_p=1. $$ That is, $$ \frac{|z^Tx|}{\|x\|_p\,\|z\|_q}\leq1, $$ which is Hölder.

Regarding the "low level" proof, as far as I can tell to show $(1)$ you need Hölder.