Erdős–Kaplansky theorem explained

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 K and let I be some basis of it. Then for the dual space E ^{*} ,^[1] \operatorname{dim}(E ^{*)=\operatorname{card}(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 K ^{I} is given by:^[2] \operatorname{dim}( K ^{I)=\operatorname{card}(K} ^{I).} When I is finite, it's a standard result that \dim( K ^{I)} = \operatorname{card}(I) . This gives us a full characterization of the dimension of this space.

References

1. Book: Gottfried. Köthe. Topological Vector Spaces I.. Germany. Springer Berlin Heidelberg. 1983. 75.
2. Book: Elements of mathematics: Algebra I, Chapters 1 - 3. Nicolas Bourbaki. 400. Hermann. 0201006391. 1974. en.