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Anderson–Kadec theorem

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In mathematics, in the areas of topology and functional analysis, the Anderson–Kadec theorem states[1] that any two infinite-dimensional, separable Banach spaces, or, more generally, Fréchet spaces, are homeomorphic as topological spaces. The theorem was proved by Mikhail Kadec (1966) and Richard Davis Anderson.

Statement

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Every infinite-dimensional, separable Fréchet space is homeomorphic to the Cartesian product of countably many copies of the real line

Preliminaries

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Kadec norm: A norm on a normed linear space is called a Kadec norm with respect to a total subset of the dual space if for each sequence the following condition is satisfied:

  • If for and then

Eidelheit theorem: A Fréchet space is either isomorphic to a Banach space, or has a quotient space isomorphic to

Kadec renorming theorem: Every separable Banach space admits a Kadec norm with respect to a countable total subset of The new norm is equivalent to the original norm of The set can be taken to be any weak-star dense countable subset of the unit ball of

Sketch of the proof

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In the argument below denotes an infinite-dimensional separable Fréchet space and the relation of topological equivalence (existence of homeomorphism).

A starting point of the proof of the Anderson–Kadec theorem is Kadec's proof that any infinite-dimensional separable Banach space is homeomorphic to

From Eidelheit theorem, it is enough to consider Fréchet space that are not isomorphic to a Banach space. In that case there they have a quotient that is isomorphic to A result of Bartle-Graves-Michael proves that then for some Fréchet space

On the other hand, is a closed subspace of a countable infinite product of separable Banach spaces of separable Banach spaces. The same result of Bartle-Graves-Michael applied to gives a homeomorphism for some Fréchet space From Kadec's result the countable product of infinite-dimensional separable Banach spaces is homeomorphic to

The proof of Anderson–Kadec theorem consists of the sequence of equivalences

See also

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Notes

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References

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  • Bessaga, C.; Pełczyński, A. (1975), Selected Topics in Infinite-Dimensional Topology, Monografie Matematyczne, Warszawa: Panstwowe wyd. naukowe.
  • Torunczyk, H. (1981), Characterizing Hilbert Space Topology, Fundamenta Mathematicae, pp. 247–262.