I have a question regarding the definition of a normal fan. The definition is thus:
"Given a non-empty polytope $P \subset \mathbb{R ^d}$, the normal fan $N(P)$ of $P$ is the fan consisting of, for each nonempty face of $P$, the linear functionals in $(\mathbb{R^d})^*$ that are maximized on that face. In other words, for each non-empty face $F$ of $P$, let
$$N_F = \{\textbf{c} \in (\mathbb{R^d})^* \mid \textbf{c} \textbf{x} \geq \textbf{c} \textbf{y} \text{ for all } \textbf{x}\in F, \textbf{y} \in P\}.$$
Then each $N_F$ is a cone generated by the normal vectors to the facets containing $F$, and these cones together form a complete fan $N(P)$".
It is to my understanding that a normal fan is complete, i.e. the union of all of the normal cones over their respective boundaries from said fan generates or gives back the ambient space, which in this case is $(\mathbb{R^d})^*$. However, isn't $(\mathbb{R^d})^* = \mathbb{R^d}$? I know that we can identify a vector space with its dual via the inner product, which in this case is just the standard dot product in $\mathbb{R^d}$. In other words, my question is whether $(\mathbb{R^d})^*$ is some other dual (adjoint) space (not just the whole $\mathbb{R^d}$)?
Any explanation or clarification is most welcomed.
The definitions come from this website (scroll to the bottom section 6.4):
https://riliu.math.ncsu.edu/724/notesse6.html
Edit: The reason for the question is that I am trying to prove a statement of the following form:
If we have a closed convex polyhedron $C$ in $\mathbb{R^n}$, then the union of all normal cones $N_C(x)$, through $x$ taking the values from the boundary of $N_C(x)$, equals $\mathbb{R^n}$.
I think that showing that a normal fan is complete will answer this question directly, but I think there are issues with the ambient space being $(\mathbb{R^n})^*$ in the above definition. Or is there really an issue?