Anderson–Kadec theorem

Template:Short description In mathematics, in the areas of topology and functional analysis, the Anderson–Kadec theorem states 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 ${ℝ}^{\mathbb{N}},$ the Cartesian product of countably many copies of the real line $ℝ.$

Preliminaries

Kadec norm: A norm $\Vert \cdot \Vert$ on a normed linear space $X$ is called a Template:Visible anchor with respect to a total subset $A\subseteq X^{\ast }$ of the dual space $X^{\ast }$ if for each sequence $x_n\in X$ the following condition is satisfied:

• If $\underset{n\to \mathrm{\infty }}{lim}{x}^{\ast }\left(x_n\right)=x^{\ast }(x_0)$ for $x^{\ast }\in A$ and $\underset{n\to \mathrm{\infty }}{lim}\Vert x_n\Vert =\Vert x_0\Vert ,$ then $\underset{n\to \mathrm{\infty }}{lim}\Vert x_n-x_0\Vert =0.$

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

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

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 ${ℝ}^{\mathbb{N}}.$

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 ${ℝ}^{\mathbb{N}}.$ A result of Bartle-Graves-Michael proves that then $$E\simeq Y\times {ℝ}^{\mathbb{N}}$$ for some Fréchet space $Y.$

On the other hand, $E$ is a closed subspace of a countable infinite product of separable Banach spaces $X={\prod }_{n=1}^{\mathrm{\infty }}{X}_i$ 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 ${ℝ}^{\mathbb{N}}.$

The proof of Anderson–Kadec theorem consists of the sequence of equivalences $$\begin{array}{cc}{ℝ}^{\mathbb{N}}& \simeq (E\times Z{)}^{\mathbb{N}}\\ & \simeq E^{\mathbb{N}}\times Z^{\mathbb{N}}\\ & \simeq E\times E^{\mathbb{N}}\times Z^{\mathbb{N}}\\ & \simeq E\times {ℝ}^{\mathbb{N}}\\ & \simeq Y\times {ℝ}^{\mathbb{N}}\times {ℝ}^{\mathbb{N}}\\ & \simeq Y\times {ℝ}^{\mathbb{N}}\\ & \simeq E\end{array}$$

See also

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References

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