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 Kadets and Richard Davis Anderson.

Statement

Every infinite-dimensional, separable Fréchet space is homeomorphic to, the Cartesian product of countably many copies of the real line.

Preliminaries

Kadec norm: A norm on a normed linear space is called a Kadec norm with respect to a total subset of the dual space if for each sequence the following condition is satisfied:
Eidelheit theorem: A Fréchet space is either isomorphic to a Banach space, or has a quotient space isomorphic to.
Kadec renorming theorem: Every separable Banach space admits a Kadec norm with respect to a countable total subset of. The new norm is equivalent to the original norm of. The set can be taken to be any weak-star dense countable subset of the unit ball of

Sketch of the proof

In the argument below denotes an infinite-dimensional separable Fréchet space and the relation of topological equivalence.
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.
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. A result of Bartle-Graves-Michael proves that then
for some Fréchet space.
On the other hand, is a closed subspace of a countable infinite product of separable Banach spaces of separable Banach spaces. The same result of Bartle-Graves-Michael applied to gives a homeomorphism
for some Fréchet space. From Kadec's result the countable product of infinite-dimensional separable Banach spaces is homeomorphic to.
The proof of Anderson–Kadec theorem consists of the sequence of equivalences