Gordan's Lemma: Let $A \in \mathbb{R}^{m \times n}$. Then exactly one of the following two statements is true:
- There exists $x \in \mathbb{R}^n$ with $Ax > 0$, or
- There exists nonzero $y \ge 0 \in \mathbb{R}^m$ with $yA = 0$.
Does this statement still hold if $m$ is countably infinite? If so, does this fact follow easily from the finite-dimensional setting or does it require significant new machinery?