Anderson–Kadec theorem

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 $$\R^{\N},$$ the Cartesian product of countably many copies of the real line $$\R.$$

Preliminaries
Kadec norm: A norm $$\|\,\cdot\,\|$$ on a normed linear space $$X$$ is called a  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 $$\R^{\N}.$$

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 $$\R^{\N}.$$

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

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