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

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

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 $A\subseteq X^{*}$ 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 $E\simeq Y\times \mathbb {R} ^{\mathbb {N} }$ for some Fréchet space $Y.$

On the other hand, $E$ is a closed subspace of a countable infinite product of separable Banach spaces ${\textstyle X=\prod _{n=1}^{\infty }X_{i}}$ 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 & Pełczyński 1975, p. 189

References

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.

[1] Bessaga & Pełczyński 1975, p. 189

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