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 $E^{*}$,

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

By Cantor's theorem, this cardinal is strictly larger than the dimension $\operatorname {card} (I)$ 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:

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

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