Anderson–Kadec theorem
In mathematics, in the areas of topology and functional analysis, the Anderson–Kadec theorem states[1] 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 [math]\displaystyle{ \R^{\N}, }[/math] the Cartesian product of countably many copies of the real line [math]\displaystyle{ \R. }[/math]
Preliminaries
Kadec norm: A norm [math]\displaystyle{ \|\,\cdot\,\| }[/math] on a normed linear space [math]\displaystyle{ X }[/math] is called a Kadec norm with respect to a total subset [math]\displaystyle{ A \subseteq X^* }[/math] of the dual space [math]\displaystyle{ X^* }[/math] if for each sequence [math]\displaystyle{ x_n\in X }[/math] the following condition is satisfied:
- If [math]\displaystyle{ \lim_{n\to\infty} x^*\left(x_n\right) = x^*(x_0) }[/math] for [math]\displaystyle{ x^* \in A }[/math] and [math]\displaystyle{ \lim_{n\to\infty} \left\|x_n\right\| = \left\|x_0\right\|, }[/math] then [math]\displaystyle{ \lim_{n\to\infty} \left\|x_n - x_0\right\| = 0. }[/math]
Eidelheit theorem: A Fréchet space [math]\displaystyle{ E }[/math] is either isomorphic to a Banach space, or has a quotient space isomorphic to [math]\displaystyle{ \R^{\N}. }[/math]
Kadec renorming theorem: Every separable Banach space [math]\displaystyle{ X }[/math] admits a Kadec norm with respect to a countable total subset [math]\displaystyle{ A \subseteq X^* }[/math] of [math]\displaystyle{ X^*. }[/math] The new norm is equivalent to the original norm [math]\displaystyle{ \|\,\cdot\,\| }[/math] of [math]\displaystyle{ X. }[/math] The set [math]\displaystyle{ A }[/math] can be taken to be any weak-star dense countable subset of the unit ball of [math]\displaystyle{ X^* }[/math]
Sketch of the proof
In the argument below [math]\displaystyle{ E }[/math] denotes an infinite-dimensional separable Fréchet space and [math]\displaystyle{ \simeq }[/math] 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 [math]\displaystyle{ \R^{\N}. }[/math]
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 [math]\displaystyle{ \R^{\N}. }[/math] A result of Bartle-Graves-Michael proves that then [math]\displaystyle{ E \simeq Y \times \R^{\N} }[/math] for some Fréchet space [math]\displaystyle{ Y. }[/math]
On the other hand, [math]\displaystyle{ E }[/math] is a closed subspace of a countable infinite product of separable Banach spaces [math]\displaystyle{ X = \prod_{n=1}^{\infty} X_i }[/math] of separable Banach spaces. The same result of Bartle-Graves-Michael applied to [math]\displaystyle{ X }[/math] gives a homeomorphism [math]\displaystyle{ X \simeq E \times Z }[/math] for some Fréchet space [math]\displaystyle{ Z. }[/math] From Kadec's result the countable product of infinite-dimensional separable Banach spaces [math]\displaystyle{ X }[/math] is homeomorphic to [math]\displaystyle{ \R^{\N}. }[/math]
The proof of Anderson–Kadec theorem consists of the sequence of equivalences [math]\displaystyle{ \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} }[/math]
See also
- Metrizable topological vector space – A topological vector space whose topology can be defined by a metric
Notes
- ↑ Bessaga & Pełczyński 1975, p. 189
References
- Bessaga, C.; Pełczyński, A. (1975), Selected Topics in Infinite-Dimensional Topology, Monografie Matematyczne, Warszawa: Panstwowe wyd. naukowe, https://books.google.com/books?id=7n9sAAAAMAAJ.
- Torunczyk, H. (1981), Characterizing Hilbert Space Topology, Fundamenta Mathematicae, pp. 247–262.
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