Proposition 3

Let $A$ be a balanced, closed, convex set in a complex locally convex space $E$ and let $K$ be a non-empty compact convex set in $E$ that does not meet $A$. Then there exists a continuous complex linear form $f$ on $E$ and a number $\alpha > 0$ such that $|f(x)| < \alpha$ in $A$ and $|f(y)| > \alpha$ in $K$.

The proposition follows from II, p. 38, prop. 4 as prop. 2 follows from II, p. 37, prop. 1.