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

Preliminaries

Kadec norm: A norm $\Vert \cdot \Vert$ on a normed linear space $X$ is called a Kadec norm 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 $\underset{n\to \mathrm{\infty }}{\lim }{x}^{*}\left(x_n\right)=x^{*}(x_0)$ for $x^{*}\in A$ 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 ${\mathbb{ℝ}}^{\mathbb{\mathbb{N}}}.$ 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 $\Vert \cdot \Vert$ of $X.$ The set $A$ can be taken to be any weak-star dense countable subset of the unit ball of $X^{*}$

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 ${\mathbb{ℝ}}^{\mathbb{\mathbb{N}}}.$ 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 ${\mathbb{ℝ}}^{\mathbb{\mathbb{N}}}.$ A result of Bartle-Graves-Michael proves that then $$E\simeq Y\times {\mathbb{ℝ}}^{\mathbb{\mathbb{N}}}$$ 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 ${\mathbb{ℝ}}^{\mathbb{\mathbb{N}}}.$ The proof of Anderson–Kadec theorem consists of the sequence of equivalences $$\begin{array}{rl}{\mathbb{ℝ}}^{\mathbb{\mathbb{N}}}& \simeq (E\times Z{)}^{\mathbb{\mathbb{N}}}\\ & \simeq E^{\mathbb{\mathbb{N}}}\times Z^{\mathbb{\mathbb{N}}}\\ & \simeq E\times E^{\mathbb{\mathbb{N}}}\times Z^{\mathbb{\mathbb{N}}}\\ & \simeq E\times {\mathbb{ℝ}}^{\mathbb{\mathbb{N}}}\\ & \simeq Y\times {\mathbb{ℝ}}^{\mathbb{\mathbb{N}}}\times {\mathbb{ℝ}}^{\mathbb{\mathbb{N}}}\\ & \simeq Y\times {\mathbb{ℝ}}^{\mathbb{\mathbb{N}}}\\ & \simeq E\end{array}$$

See also

• Metrizable topological vector space – A topological vector space whose topology can be defined by a metric

References

• Bessaga, C.; Pełczyński, A. (1975), Selected Topics in Infinite-Dimensional Topology, Monografie Matematyczne, Warszawa: Panstwowe wyd. naukowe.
• Torunczyk, H. (1981), Characterizing Hilbert Space Topology, Fundamenta Mathematicae, pp. 247–262.