Ordered vector spaces, wedges and cones

Prove Proposition 9.3:

Prove Proposition 9.4: if $(\mathcal{X}, \leq )$ is an ordered vector space and $P$ is the wedge of positive elements, $\leq$ is antisymmetric if and only if $P \cap (-P) = (0)$.

Show that $e$ is an order unit for $(\mathcal{X}, \leq )$ if and only if for every $x$ in $\mathcal{X}$ there is a $\delta > 0$ such that $e \pm tx \geq 0$ for $0 \leq t \leq \delta$.

Show that $C(\mathbb {R})$, the space of all continuous real-valued functions on $\mathbb {R}$, has no order unit.

Prove (9.11): with $(\mathcal{X}, \leq )$ an ordered vector space, $\mathcal{Y}$ a linear manifold in $\mathcal{X}$ that is cofinal, $f : \mathcal{Y} \to \mathbb {R}$ a positive linear functional, and, for $x$ in $\mathcal{X}$,

the function $q$ is a sublinear functional on $\mathcal{X}$.

Characterize the order units of $C_b(X)$. Does $C_b(X)$ always have an order unit?

Characterize the order units of $C_0(X)$ if $X$ is locally compact. Does $C_0(X)$ always have an order unit?

Let $\mathcal{X} = M_2(\mathbb {R})$, the $2 \times 2$ matrices over $\mathbb {R}$. Define $A$ in $M_2(\mathbb {R})$ to be positive if $A = A^*$ and $\left\langle Ax, x \right\rangle \geq 0$ for all $x$ in $\mathbb {R}^2$. Characterize the order units of $M_2(\mathbb {R})$.

If $1 \leq p < \infty$ and $\mathcal{X} = L^p(0,1)$, define $f \leq g$ to mean that $f(x) \leq g(x)$ a.e. Show that $\mathcal{X}$ is an ordered vector space that has no order unit.

Suppose that $\mathbb {R}^{n}$ is totally ordered in such a way that

1. $x_1 \geq y_1$ and $x_2 \geq y_2 \Rightarrow x_1+x_2 \geq y_1+y_2$;

2. $x \geq y$ and $\lambda \geq 0$ implies that $\lambda x \geq \lambda y$ for $\lambda \in \mathbb {R}$;

3. $x \geq y$ and $y \geq x$ implies that $x = y$. Prove that $\mathbb {R}^{n}$ is isomorphic (as a totally ordered space) to the product of $n$ lines (with the usual order), endowed with the lexicographical order (see Problem 8).