Separation Hahn Banach theorem in vector Banach lattice

Question

Let $X$ be a vector Banach lattice. Let $C$ be a closed cone of positive elements in $X^+$ and let $0\leq x\in X-C$.

Q: Does there exists any bounded positive linear functional $f$ on $X$ by which $x$ and $C$ is separated?

Answer 1

This is not possible. Try \begin{align} X &= \mathbb{R}^2, \\ X^+ &= \{x \in X : x_1, x_2 \ge 0 \},\\ C &= \{(t,t) : t \ge 0 \}, \\ x &= (1,0). \end{align}