The relation between the algebraic dimensions of a vector space and its dual

Let $V$ be an (infinite dimensional) vector space over the field $\mathbb F (=\mathbb R$ or$\mathbb C$). If $\alpha$ is the dimension of $V$, for some cardinal number $\alpha$, I want to know, what is the dimension of $V^*$, the algebraic dual of $V$? It is known that $dim V < dim V^*$ when V is an infinite dimensional vector space.

In general, if $\dim_{\mathbb F} V = d$ is infinite, we have $$\dim_{\mathbb F} V^* = |\mathbb F|^d.$$ This theorem is attributed to Paul Erdős and Irving Kaplansky by Jacobson in Lectures in Abstract Algebra II, Linear Algebra (chapter IX.5, theorem 2, p.247).
Because of Cantor's theorem, you have $|\mathbb F|^d > d$, so you indeed have $\dim V^* > \dim V$ as soon as $V$ is infinite-dimensional.