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.