Erdős-Kaplansky theorem
# 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 spacees, in particular it shows that the algebraic dual space is not isomorph to the vector space itself.

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 $\{b_{i}\}_{i\in I}$ be some basis of it. Then for the dual space $E^{*}$

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

## Literature

* Nathan Jacobson: Structure of rings. American Mathematical Society, Colloquium Publications, Vol. 37, 1956.
* Gottfried Köthe: Topological linear spaces, Springer-Verlag, Basic Teachings of Mathematical Sciences, Volume 107, 1960.
