Exercise 6, § 3

a) Let H be a hyperplane in E a vector space over $\mathbf{R}$ and let p be a sublinear function on E. Let f be a linear form on H, such that $f(y) \leq p(y)$ in H. Let a be a point of $\mathbf{C} H$, and let h be the linear form on E which extends f and is such that $h(a) = \inf_{y \in H} (f(y + p(a - y)))$. Then $h(x) \leq p(x)$ in E. Show that for every linear form g on E extending f and such that $g(x) \leq p(x)$ in E, we also have $g(a) \leq h(a)$. b) Let V be a vector subspace of E and f a linear form on V such that $f(y) \leq p(y)$ in V. Let S be a non-empty set of E. Show that there exists a linear form h on E that extends f, such that $h(x) \leq p(x)$ in E and that there is no other linear form g on E extending f such that $g(x) \leq p(x)$ in E that is distinct from h and such that $g(x) \geq h(x)$ in S. (Consider the set $\mathscr{F}$ of pairs (V', f') where V' is a vector subspace containing V and f' a linear form on V' extending f and such that $f'(z) \leq p(z)$ in V' and further such that there is no other linear form $f''$ on V' with the same properties and such that $f''(z) \geq f'(z)$ in S $\cap$ V'. Order $\mathscr{F}$ and use a) and th. 2 of S, III, § 2.4.)