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.

Every infinite-dimensional, separable Fréchet space is homeomorphic to ${\displaystyle \mathbb {R} ^{\mathbb {N} },}${\displaystyle \mathbb {R} ^{\mathbb {N} },} the Cartesian product of countably many copies of the real line ${\displaystyle \mathbb {R} .}${\displaystyle \mathbb {R} .}

Kadec norm: A norm ${\displaystyle \|,円\cdot ,円\|}${\displaystyle \|,円\cdot ,円\|} on a normed linear space ${\displaystyle X}${\displaystyle X} is called a Kadec norm with respect to a total subset ${\displaystyle A\subseteq X^{*}}${\displaystyle A\subseteq X^{*}} of the dual space ${\displaystyle X^{*}}${\displaystyle X^{*}} if for each sequence ${\displaystyle x_{n}\in X}${\displaystyle x_{n}\in X} the following condition is satisfied:

• If ${\displaystyle \lim _{n\to \infty }x^{*}\left(x_{n}\right)=x^{*}(x_{0})}${\displaystyle \lim _{n\to \infty }x^{*}\left(x_{n}\right)=x^{*}(x_{0})} for ${\displaystyle x^{*}\in A}${\displaystyle x^{*}\in A} and ${\displaystyle \lim _{n\to \infty }\left\|x_{n}\right\|=\left\|x_{0}\right\|,}${\displaystyle \lim _{n\to \infty }\left\|x_{n}\right\|=\left\|x_{0}\right\|,} then ${\displaystyle \lim _{n\to \infty }\left\|x_{n}-x_{0}\right\|=0.}${\displaystyle \lim _{n\to \infty }\left\|x_{n}-x_{0}\right\|=0.}

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

Kadec renorming theorem: Every separable Banach space ${\displaystyle X}${\displaystyle X} admits a Kadec norm with respect to a countable total subset ${\displaystyle A\subseteq X^{*}}${\displaystyle A\subseteq X^{*}} of ${\displaystyle X^{*}.}${\displaystyle X^{*}.} The new norm is equivalent to the original norm ${\displaystyle \|,円\cdot ,円\|}${\displaystyle \|,円\cdot ,円\|} of ${\displaystyle X.}${\displaystyle X.} The set ${\displaystyle A}${\displaystyle A} can be taken to be any weak-star dense countable subset of the unit ball of ${\displaystyle X^{*}}${\displaystyle X^{*}}

In the argument below ${\displaystyle E}${\displaystyle E} denotes an infinite-dimensional separable Fréchet space and ${\displaystyle \simeq }${\displaystyle \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 ${\displaystyle \mathbb {R} ^{\mathbb {N} }.}${\displaystyle \mathbb {R} ^{\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 ${\displaystyle \mathbb {R} ^{\mathbb {N} }.}${\displaystyle \mathbb {R} ^{\mathbb {N} }.} A result of Bartle-Graves-Michael proves that then $${\displaystyle E\simeq Y\times \mathbb {R} ^{\mathbb {N} }}$${\displaystyle E\simeq Y\times \mathbb {R} ^{\mathbb {N} }} for some Fréchet space ${\displaystyle Y.}${\displaystyle Y.}

On the other hand, ${\displaystyle E}${\displaystyle E} is a closed subspace of a countable infinite product of separable Banach spaces ${\textstyle X=\prod _{n=1}^{\infty }X_{i}}${\textstyle X=\prod _{n=1}^{\infty }X_{i}} of separable Banach spaces. The same result of Bartle-Graves-Michael applied to ${\displaystyle X}${\displaystyle X} gives a homeomorphism $${\displaystyle X\simeq E\times Z}$${\displaystyle X\simeq E\times Z} for some Fréchet space ${\displaystyle Z.}${\displaystyle Z.} From Kadec's result the countable product of infinite-dimensional separable Banach spaces ${\displaystyle X}${\displaystyle X} is homeomorphic to ${\displaystyle \mathbb {R} ^{\mathbb {N} }.}${\displaystyle \mathbb {R} ^{\mathbb {N} }.}

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

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

• 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.

• Topological vector spaces
• Theorems in functional analysis
• Theorems in topology

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• Short description matches Wikidata