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Erdős–Kaplansky theorem

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On the dimension of vector space duals

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 {\displaystyle E} be an infinite-dimensional vector space over a field K {\displaystyle \mathbb {K} } and let I {\displaystyle I} be some basis of it. Then for the dual space E ∗ {\displaystyle E^{*}} ,

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

By Cantor's theorem, this cardinal is strictly larger than the dimension card ⁡ ( I ) {\displaystyle \operatorname {card} (I)} of E {\displaystyle E} . More generally, if I {\displaystyle I} is an arbitrary infinite set, the dimension of the space of all functions K I {\displaystyle \mathbb {K} ^{I}} is given by:

dim ⁡ ( K I ) = card ⁡ ( K I ) . {\displaystyle \operatorname {dim} (\mathbb {K} ^{I})=\operatorname {card} (\mathbb {K} ^{I}).}

When I {\displaystyle I} is finite, it's a standard result that dim ⁡ ( K I ) = card ⁡ ( I ) {\displaystyle \dim(\mathbb {K} ^{I})=\operatorname {card} (I)} . This gives us a full characterization of the dimension of this space.

References

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