So it's a standard advanced-linear algebra fact that if $\dim V=\kappa\ge\infty$ then $\dim V^*=\kappa'>\kappa$, and this proof makes sense enough to me. What I'd be interested in showing or learning, is actually calculating $\kappa'$ as a cardinality, in terms of $\kappa$. Clearly it is just a function of $\kappa$ (doesn't depend on base field), and knowing just this restricted statement above says certain cardinalities can't arise as the dimensions of dual spaces.
For example, is $\dim k^{\mathbb{N}}$ equal to the cardinality of the continuum? Yes, and for $k=\mathbb{R}$ I think you can argue based on decimal expansions.
In general, I see that you can argue as follows: $\kappa'$ is the cardinality of a minimal set $S\subset\mathcal{P}(\kappa)$ of subsets of $\kappa$ with the following property: every subset of $\kappa$ is a union of finitely many elements of $S$. I'd assume I could prove that minimality exists by Zorn, and that this finite union is unique by minimality (maybe this part isn't true?), but the goal again, is just expressing this cardinality $\kappa'$ in terms of $\kappa$. Any ideas?
