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.

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.