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 $E^*$ ,{{cite book|first1=Gottfried|last1=Köthe|title=Topological Vector Spaces I.|place=Germany|publisher=Springer Berlin Heidelberg|date=1983|page=75}}

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

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:{{cite book|title=Elements of mathematics: Algebra I, Chapters 1 - 3|author=Nicolas Bourbaki|page=400|editor=Hermann|isbn=0201006391|year=1974|language=en}}

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

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.

References

Category:Theorems in functional analysis

Category:Paul Erdős