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 ${\displaystyle E}$ be an infinite-dimensional vector space over a field ${\displaystyle \mathbb {K} }$ and let ${\displaystyle I}$ be some basis of it. Then for the dual space ${\displaystyle E^{*}}$,^[1]

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

By Cantor's theorem, this cardinal is strictly larger than the dimension ${\displaystyle \operatorname {card} (I)}$ of ${\displaystyle E}$. More generally, if ${\displaystyle I}$ is an arbitrary infinite set, the dimension of the space of all functions ${\displaystyle \mathbb {K} ^{I}}$ is given by:^[2]

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

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