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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 Kadets (1966) and Richard Davis Anderson.

Statement

Every infinite-dimensional, separable Fréchet space is homeomorphic to $$\mathbb {R} ^{\mathbb {N} }$$, the Cartesian product of countably many copies of the real line $$\mathbb {R}$$ .

Preliminaries

Kadec norm: A norm $$\|\cdot \|$$ on a normed linear space X is called a Kadec norm with respect to a total subset $${\displaystyle A\subset X^{*}}$$ of the dual space $$X^{*}$$ if for each sequence $$x_{n}\in X$$ the following condition is satisfied:

If lim $${\displaystyle \lim _{n\to \infty }x^{*}(x_{n})=x^{*}(x_{0})}$$ for $${\displaystyle x^{*}\in A}$$ and $${\displaystyle \lim _{n\to \infty }\|x_{n}\|=\|x_{0}\|}$$ , then lim $${\displaystyle \lim _{n\to \infty }\|x_{n}-x_{0}\|=0}.$$

Eidelheit theorem: A Fréchet space E is either isomorphic to a Banach space, or has a quotient space isomorphic to $$\mathbb {R} ^{\mathbb {N} }.$$

Kadec renorming theorem: Every separable Banach space X admits a Kadec norm with respect to a countable total subset $${\displaystyle A\subset X^{*}}$$ of $$X^{*}$$. The new norm is equivalent to the original norm $$\|\cdot \|$$ of X. The set A can be taken to be any weak-star dense countable subset of the unit ball of $$X^{*}$$

Sketch of the proof

In the argument below E denotes an infinite-dimensional separable Fréchet space and $$\simeq$$ 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 $$\mathbb {R} ^{\mathbb {N} }.$$

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 $$\mathbb {R} ^{\mathbb {N} }$$. A result of Bartle-Graves-Michael proves that then

$${\displaystyle E\simeq Y\times \mathbb {R} ^{\mathbb {N} }}$$

for some Fréchet space Y.

On the other hand, is a closed subspace of a countable infinite product of separable Banach spaces $${\displaystyle X=\prod _{n=1}^{\infty }X_{i}}$$ of separable Banach spaces. The same result of Bartle-Graves-Michael applied to X gives a homeomorphism

$${\displaystyle X\simeq E\times Z}$$

for some Fréchet space Z. From Kadec's result the countable product of infinite-dimensional separable Banach spaces X is homeomorphic to$$\mathbb {R} ^{\mathbb {N} }.$$

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

{\displaystyle {\begin{aligned}\mathbb {R} ^{\mathbb {N} }&\simeq (E\times Z)^{\mathbb {N} }\\&\simeq E^{\mathbb {N} }\times Z^{\mathbb {N} }\\&\simeq E\times E^{\mathbb {N} }\times Z^{\mathbb {N} }\\&\simeq E\times \mathbb {R} ^{\mathbb {N} }\\&\simeq Y\times \mathbb {R} ^{\mathbb {N} }\times \mathbb {R} ^{\mathbb {N} }\\&\simeq Y\times \mathbb {R} ^{\mathbb {N} }\\&\simeq E\end{aligned}}}

Notes

Bessaga, C.; Pełczyński, A. (1975). Selected Topics in Infinite-Dimensional Topology. Panstwowe wyd. naukowe. p. 189.

References

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

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