Proposition 2

Let $A$ be a non-empty balanced open convex set of a complex topological vector space $E$, and $B$ be a non-empty convex set 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 $|f(y)| \geq \alpha$ in $B$.

For, there exists a continuous real linear form $g$ on $E$ and a real number $\alpha$ such that $g(x) < \alpha$ in $A$ and $g(y) \geq \alpha$ in $B$ (II, p. 37, prop. 1). As $0 \in A$, we have $\alpha > 0$. We show that the continuous complex linear form $f(x) = g(x) - ig(ix)$ and the number $\alpha$ have the properties required. For, since $\Re f = g$, we have $|f(y)| \geq \alpha$ in $B$. On the other hand, for all $x \in A$ and all real $\vartheta$, the point $e^{i\vartheta}x$ belongs to $A$, since $A$ is balanced, and we have $f(x) = e^{-i\vartheta}f(e^{i\vartheta}x)$; then there exists a number $\vartheta$ such that $|f(x)| = \Re(e^{i\vartheta}f(x)) = g(e^{i\vartheta}x) < \alpha$, and the proposition follows.