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.
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:
$$|x^\top y|\leq \|x\|_1\|y\|_\infty$$
Among other things, this question is easily construed as a reasonable one, about the continuity of pairings $\langle, \rangle \colon V\times V^*\to \mathbb C$. Among other true things that can be said, it is a general fact that separately continuous pairings on Frechet spaces turn out to be jointly continuous for general reasons. ... thus... what is the real question?