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 $\R^,$ 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 $\lim_ x^*\left(x_n\right) = x^*(x_0)$ for $x^* \in A$ and $\lim_ \left\, x_n\right\, = \left\, x_0\right\, ,$ then $\lim_ \left\, x_n - x_0\right\, = 0.$

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

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 $\R^.$

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

On the other hand, $E$ is a closed subspace of a countable infinite product of separable Banach spaces $X = \prod_^ X_i$ 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 $\R^.$

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

See also

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Notes

References

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{{DEFAULTSORT:Anderson-Kadec theorem

Topological vector spaces
Theorems in functional analysis
Theorems in topology