Anderson–Kadec theorem explained

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

\| ⋅ \| 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\toinfty} x ^{*\left(x} _{n\right)} = x ^{*(x} _{0 )} for x ^{*} \in A and \lim _{n\toinfty} \left\|x _{n\right\|} = \left\|x _{0\right\|,} then \lim _{n\toinfty} \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 \| ⋅ \| 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 $$E\; \backslash simeq\; Y\; \backslash times\; \backslash R^$$for some Fréchet space Y.

On the other hand,

E is a closed subspace of a countable infinite product of separable Banach spaces $X\; =\; \backslash prod\_^\; X\_i$ of separable Banach spaces. The same result of Bartle-Graves-Michael applied to X gives a homeomorphism$$X\; \backslash simeq\; E\; \backslash times\; Z$$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

References

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