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

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

Preliminaries

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

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

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

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

Sketch of the proof

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

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

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

for some Fréchet space ${\displaystyle Y}$.

On the other hand, ${\displaystyle E}$ 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 ${\displaystyle X}$ gives a homeomorphism

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

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

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

1. 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.