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Erdős–Kaplansky theorem

From Wikipedia, the free encyclopedia

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

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Let be an infinite-dimensional vector space over a field and let be some basis of it. Then for the dual space ,[1]

By Cantor's theorem, this cardinal is strictly larger than the dimension of . More generally, if is an arbitrary infinite set, the dimension of the space of all functions is given by:[2]

When is finite, it's a standard result that . This gives us a full characterization of the dimension of this space.

References

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  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.