Exercise 4, § 4

With the notations of II, p. 72, exerc. 1 suppose also that the space $E$ is locally convex. a) There exists a positive continuous linear form $g$ in $E$ that extends $f$, if and only if $f$ is bounded below in $M \cap (W + P)$ for at least one neighbourhood $W$ of 0 in $E$. b) Given a point $x \in E$, there exists a positive continuous linear form $g$ in $E$ such that $g(x) = 1$, if, and only if, $-x \notin \overline{P}$.