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 N , {\displaystyle \mathbb {R} ^{\mathbb {N} },} the Cartesian product of countably many copies of the real line R . {\displaystyle \mathbb {R} .}
Preliminaries Kadec norm: A norm ‖ ⋅ ‖ {\displaystyle \|\,\cdot \,\|} on a normed linear space X {\displaystyle X} is called a Kadec norm with respect to a total subset A ⊆ X ∗ {\displaystyle A\subseteq X^{*}} of the dual space X ∗ {\displaystyle X^{*}} if for each sequence x n ∈ X {\displaystyle x_{n}\in X} the following condition is satisfied:
If lim n → ∞ x ∗ ( x n ) = x ∗ ( x 0 ) {\displaystyle \lim _{n\to \infty }x^{*}\left(x_{n}\right)=x^{*}(x_{0})} for x ∗ ∈ A {\displaystyle x^{*}\in A} and lim n → ∞ ‖ x n ‖ = ‖ x 0 ‖ , {\displaystyle \lim _{n\to \infty }\left\|x_{n}\right\|=\left\|x_{0}\right\|,} then lim n → ∞ ‖ x n − x 0 ‖ = 0. {\displaystyle \lim _{n\to \infty }\left\|x_{n}-x_{0}\right\|=0.}
Eidelheit theorem: A Fréchet space E {\displaystyle E} is either isomorphic to a Banach space, or has a quotient space isomorphic to R N . {\displaystyle \mathbb {R} ^{\mathbb {N} }.}
Kadec renorming theorem: Every separable Banach space X {\displaystyle X} admits a Kadec norm with respect to a countable total subset A ⊆ X ∗ {\displaystyle A\subseteq X^{*}} of X ∗ . {\displaystyle X^{*}.} The new norm is equivalent to the original norm ‖ ⋅ ‖ {\displaystyle \|\,\cdot \,\|} of X . {\displaystyle X.} The set A {\displaystyle A} can be taken to be any weak-star dense countable subset of the unit ball of X ∗ {\displaystyle X^{*}}
Sketch of the proof In the argument below E {\displaystyle E} denotes an infinite-dimensional separable Fréchet space and ≃ {\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 R 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 R N . {\displaystyle \mathbb {R} ^{\mathbb {N} }.} A result of Bartle-Graves-Michael proves that then
E ≃ Y × R N {\displaystyle E\simeq Y\times \mathbb {R} ^{\mathbb {N} }}
for some Fréchet space Y . {\displaystyle Y.}
On the other hand, E {\displaystyle E} is a closed subspace of a countable infinite product of separable Banach spaces X = ∏ n = 1 ∞ X i {\textstyle X=\prod _{n=1}^{\infty }X_{i}} of separable Banach spaces. The same result of Bartle-Graves-Michael applied to X {\displaystyle X} gives a homeomorphism
X ≃ E × Z {\displaystyle X\simeq E\times Z}
for some Fréchet space Z . {\displaystyle Z.} From Kadec's result the countable product of infinite-dimensional separable Banach spaces X {\displaystyle X} is homeomorphic to R N . {\displaystyle \mathbb {R} ^{\mathbb {N} }.}
The proof of Anderson–Kadec theorem consists of the sequence of equivalences
R N ≃ ( E × Z ) N ≃ E N × Z N ≃ E × E N × Z N ≃ E × R N ≃ Y × R N × R N ≃ Y × R N ≃ E {\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}}}
See also Metrizable topological vector space – Topological vector space whose topology can be defined by a metric
Notes
References Bessaga, C.; Pełczyński, A. (1975), Selected Topics in Infinite-Dimensional Topology, Monografie Matematyczne, Warszawa: Panstwowe wyd. naukowe. Toruńczyk, H. (1981), Characterizing Hilbert space topology, Fundamenta Mathematicae, vol. 111, pp. 247–262.