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 Kadets (1966) and Richard Davis Anderson.

Contents

Statement of the theoremEdit

Every infinite-dimensional, separable Fréchet space is homeomorphic to  , the Cartesian product of countably many copies of the real line  .

PreliminariesEdit

Kadec norm: A norm   on a normed linear space   is called a Kadec norm with respect to a total subset   of the dual space   if for each sequence   the following condition is satisfied:

  • If   for   and  , then  .

Eidelheit theorem: A Fréchet space   is either isomorphic to a Banach space, or has a quotient space isomorphic to  .

Kadec Renorming Theorem: Every separable Banach space   admist a Kadec norm with respect to a countable total subset   of  . The new norm is equivalent to the original norm   of  . The set   can be taken to be any weak-star dense countable subset of the unit ball of  

Sketch of the proofEdit

In the argument below   denotes an infinite dimensional separable Fréchet space and   the relation of topological equivalence (existence of homeomorphism).

A starting point of the proof of Anderson-Kadec theorem is Kadec's proof that any infinite-dimensional separable Banach space is homeomorphic to  .

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  . A result of Bartle-Graves-Michael proves that then

 

for some Fréchet space  .

On the other hand,   is a closed subspace of a countable infinite product of separable Banach spaces   of separable Banach spaces. The same result of Bartle-Graves-Michael applied to   gives a homeomorphism

 

for some Fréchet space  . From Kadec's result the countable product of infinite-dimensional separable Banach spaces   is homeomorphic to  .

The proof of Anderson-Kadec theorem consists of the sequence of equivalences

 

NotesEdit

  1. ^ Bessaga, C.; Pełczyński, A. (1975). Selected Topics in Infinite-Dimensional Topology. Panstwowe wyd. naukowe. p. 189.

ReferencesEdit

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