Topologies/Norms on Real Closed Fields Let $\mathbf{R}$ be a real closed field containing $\Bbb{R}$, and let $\mathbf{C} := \mathbf{R}[i]$. Is there a natural norm or topology on $\mathbf{R}$ or $\mathbf{C}$?
My interest in this question lies in the fact that I am considering vector spaces over $\mathbf{C}$ with $\mathbf{C}$-valued inner products, and positive operators on such vector spaces. I want to know whether positive operators on such vector spaces are also positive on their completion. I suspect that it is true.
 A: Both ${\bf R}$ and ${\bf C}$ are vector spaces over $\mathbb{R}$ and as every vector space has a basis they are each isomorphic to $\mathbb{R}^I$ for some set $I$.
Let $\{e_i\}_{i\in I}$ be a basis of $\mathbb{R}^I$. There is an injective map: $$f\colon \mathbb{R}^I\to \prod_{i\in I}\mathbb{R}$$
where for $v\in \mathbb{R}^I$ we let $\{f(v)\}_i$ be the coefficient on $e_i$ of $v$.  Note this map is never surjective if $I$ infinite, as for any $v\in \mathbb{R}^I$ only finitely many of the $\{f(v)\}_i$ are non-zero.
The cartesian product $\prod_{i\in I}\mathbb{R}$ may be regarded as a topological space: formally it is the product $\prod_{i\in I}\mathbb{R}$ in the category of topological spaces.  From this universal property one can easily verify that the open sets in $\prod_{i\in I}\mathbb{R}$ are generated by subsets of the form:$$\prod_{i\in I}U_i$$
where the $U_i\subseteq \mathbb{R}$ are open and for all but finitely many $i\in I$ we have $U_i=\mathbb{R}$.
In summary ${\bf R}$ and ${\bf C}$ may each be identified with a vector space $\mathbb{R}^I$, which inherits a subspace topology from being identified with a subset of the product over $i\in I$ of $\mathbb{R}$ in the category of topological spaces.
