Proposition 23

Let E be a topological vector space. If A is an open convex set which contains 0, then $p_A$ is finite and continuous, and $A = W(p_A, 1)$. If A is a closed convex set containing 0, then $p_A$ is lower semi-continuous and $A = V(p_A, 1)$.

If A is open and contains 0, then it is absorbent. For $x \in A$, there exists $\rho < 1$ such that $x / \rho \in A$, and thus $p_A(x) < 1$; this, combined with (9) gives $A = W(p_A, 1)$. Since the convex function $p_A$ is bounded above in the open set A, it is continuous in E (II, p. 18, prop. 21).

Suppose A is closed and contains 0. For every $x \in E$ with $p_A(x) \leq 1$, we have $x \in \rho A$ for all $\rho > 1$, therefore $x \in A$ since A is closed; remembering (9), this shows that $A = V(p_A, 1)$. For all $\mu > 0$, $\mu A$ is therefore the set of $x$ such that $p_A(x) \leq \mu$; as $p_A(x) \geq 0$ in E, this shows that $p_A$ is lower semi-continuous in E (GT, IV, § 6.2).

A positive sublinear function $p$ over E is the gauge of each convex set A where $W(p, 1) \subset A \subset V(p, 1)$.