Q:

Does the Cauchy Schwarz inequality hold on the L1 and L infinity norm?

So i am wondering if the Cauchy Schwarz inequality holds for all p-norms, not just when p=2, which is the euclidean space.
Thank you.

A:

Cauchy Schwarz inequality can be generalized as follows:
\begin{equation}\label{d}
|x^\top y|\leq \|x\|\|y\|_{\star}, \forall x,y \in \mathbb{R^{n}}
\end{equation}
where $\|\|_\star$ is called the dual norm. 
Since dual of $l_2$ norm is itself, using $l_2$ in above we reach to Cauchy Schwarz inequality. Since $l_1$ and $l_\infty$ are dual, we obtain:
$$|x^\top y|\leq \|x\|_1\|y\|_\infty$$

