Английская Википедия:Anderson–Kadec theorem
Шаблон:Short description 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>\R^{\N},</math> the Cartesian product of countably many copies of the real line <math>\R.</math>
Preliminaries
Kadec norm: A norm <math>\|\,\cdot\,\|</math> on a normed linear space <math>X</math> is called a Шаблон:Visible anchor with respect to a total subset <math>A \subseteq X^*</math> of the dual space <math>X^*</math> if for each sequence <math>x_n\in X</math> the following condition is satisfied:
- If <math>\lim_{n\to\infty} x^*\left(x_n\right) = x^*(x_0)</math> for <math>x^* \in A</math> and <math>\lim_{n\to\infty} \left\|x_n\right\| = \left\|x_0\right\|,</math> then <math>\lim_{n\to\infty} \left\|x_n - x_0\right\| = 0.</math>
Eidelheit theorem: A Fréchet space <math>E</math> is either isomorphic to a Banach space, or has a quotient space isomorphic to <math>\R^{\N}.</math>
Kadec renorming theorem: Every separable Banach space <math>X</math> admits a Kadec norm with respect to a countable total subset <math>A \subseteq X^*</math> of <math>X^*.</math> The new norm is equivalent to the original norm <math>\|\,\cdot\,\|</math> of <math>X.</math> The set <math>A</math> can be taken to be any weak-star dense countable subset of the unit ball of <math>X^*</math>
Sketch of the proof
In the argument below <math>E</math> denotes an infinite-dimensional separable Fréchet space and <math>\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>\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>\R^{\N}.</math> A result of Bartle-Graves-Michael proves that then <math display=block>E \simeq Y \times \R^{\N}</math> for some Fréchet space <math>Y.</math>
On the other hand, <math>E</math> is a closed subspace of a countable infinite product of separable Banach spaces <math display="inline">X = \prod_{n=1}^{\infty} X_i</math> of separable Banach spaces. The same result of Bartle-Graves-Michael applied to <math>X</math> gives a homeomorphism <math display=block>X \simeq E \times Z</math> for some Fréchet space <math>Z.</math> From Kadec's result the countable product of infinite-dimensional separable Banach spaces <math>X</math> is homeomorphic to <math>\R^{\N}.</math>
The proof of Anderson–Kadec theorem consists of the sequence of equivalences <math display=block>\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
Notes
References
Шаблон:Functional analysis Шаблон:Topological vector spaces
