In optimization theory, we often find ourselves wanting to prove bounds (upper or lower) on functions. The Cauchy-Schwarz Inequality is a powerful and commonly used inequality to help instantiate these bounds on functions.
$$ | \langle x,y \rangle| \leq \|x\|_2 \space \|y\|_2 $$
While Cauchy-Schwarz is powerful, this is a specific instance of a more general form — the Holder’s Inequality.
Holder’s Inequality utilizes the dual norm and is defined as the following:
$$ | \langle x,y \rangle| \leq \|x\| \space \|y\|_* $$
For $\|x\|$ and $\|y\|*$, we satisfy the above inequality when the dual norm $\| \cdot \|*$ is set such that it is a Holder Conjugate of our norm defined on $\| x \|$.
So $| \langle x,y \rangle| \leq \|x\|_p \space \|y\|_q$ when $\frac{1}{p} + \frac{1}{q} = 1$, which gives us: