Anderson–Kadec theorem

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 of the theorem

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 $A\subset X^{*}$ of the dual space $X^{*}$ if for each sequence $x_{n}\in X$ the following condition is satisfied:

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

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 $X$ admist a Kadec norm with respect to a countable total subset $A\subset X^{*}$ 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 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

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

for some Fréchet space $Y$ .

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

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

{\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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[1] Bessaga, C.; Pełczyński, A. (1975). Selected Topics in Infinite-Dimensional Topology. Panstwowe wyd. naukowe. p. 189.