Exercise 11, § 2

a) Show that, in the space $\mathbf{R}^n$, every convex set A of dimension $n$ contains at least one interior point (consider an affinely independent system of $n + 1$ points of A). Deduce that if A is everywhere dense in $\mathbf{R}^n$ then $A = \mathbf{R}^n$. b) Let E be the normed space $l^1(\mathbf{N})$ of absolutely convergent series of real numbers $x = (\xi_n)$ (I, p. 4); show that the set P of $x$, such that $\xi_n \geq 0$ for every index $n$, is a proper convex cone, which generates E but does not contain any interior point. c) Let E be a Hausdorff topological vector space on which there exists a non-continuous linear form $f$ (cf. II, p. 86, exerc. 17, a)). Show that the sets A and B defined by the relations f(x) \geqslant 0,\ f(x) < 0\text{ are convex, non-empty, complementary, everywhere dense and that each of them generates E (algebraically).}