Verify that the function space $\operatorname{Map}(T, \mathbf{R})$ described in Chapter 1 is partially ordered by the relation $\geq$ if we define $g \geq f$ iff $g(t) \geq f(t)$ for all $t \in T$. Verify that with this partial ordering $L:=\operatorname{Map}(T, \mathbf{R})$ is an ordered vector space. Describe the positive and negative cones $L_{+}$and $L_{. .}$for this case.