Erdős–Kaplansky theorem

The Erdős–Kaplansky theorem is a theorem from functional analysis. The theorem makes a fundamental statement about the dimension of the dual spaces of infinite-dimensional vector spaces; in particular, it shows that the algebraic dual space is not isomorphic to the vector space itself. A more general formulation allows to compute the exact dimension of any function space. The theorem is named after Paul Erdős and Irving Kaplansky.

Statement

Let $E$ be an infinite-dimensional vector space over a field $\mathbb{𝕂}$ and let $I$ be some basis of it. Then for the dual space $E^{*}$,

$\dim (E^{*})=\mathrm{card}({\mathbb{𝕂}}^I).$

By Cantor's theorem, this cardinal is strictly larger than the dimension $\mathrm{card}(I)$ of $E$. More generally, if $I$ is an arbitrary infinite set, the dimension of the space of all functions ${\mathbb{𝕂}}^I$ is given by:

$\dim ({\mathbb{𝕂}}^I)=\mathrm{card}({\mathbb{𝕂}}^I).$

When $I$ is finite, it's a standard result that $\dim ({\mathbb{𝕂}}^I)=\mathrm{card}(I)$. This gives us a full characterization of the dimension of this space.