Suppose $X=Y=\mathbb R^n$. Usually, to apply a separating hyperplane theorem in $X$ (the primal space), we associate both $X$ and $Y$ with the standard Euclidean norm.
My question is that could I instead define the $l_1$-norm
$$\|x\|_1=\sum_{i=1}^n |x_i|, \forall x\in X$$
and the $l_{\infty}$-norm
$$\|y\|_{\infty}=\max_{1\leq i\leq n}\{| y_i|\}, \forall y\in Y$$
and a dual pair $\langle.,.\rangle: Y\times X\to \mathbb R$ by
$$\langle y, x \rangle= \sum_{i=1}^n a_i x_i y_i $$
where $a_i>0$ for all $1\leq i\leq n$?
If so, could I apply the separation theorem with this duality pair for a $l_1$-norm compact set $S\subset X$ and $z\notin S$, $z\in X$: there exists $y^*\in Y$ and $\|y^*\|_{\infty}\leq 1$ such that
$$\langle y^*, z \rangle>\sup_{x\in S}\langle y^*, x \rangle$$
?
Yes, thats perfectly possible.
But in your situation, you could also invoke the separation theorem in the Euclidean space, replace the vector $(y_i)_i$ by $(y_i/a_i)_i$ [to get your dual pair] and rescale $y$ [to get $\|y\|_\infty \le 1$].