Banach–Mazur theorem

The Banach - Mazur in 1933, named after Stefan Banach and Stanislaw Mazur, is a classic set from the branch of functional analysis. Among the separable Banach spaces there are those that contain a copy of every separable Banach space other. The Banach space of continuous functions with the supremum norm is such a universal Banach space.

Wording of the sentence

Is a compact space, it should be the Banach space of continuous functions with the supremum norm. Assuming for the Cantor discontinuum, we obtain a separable Banach space already containing an isometric copy of every separable Banach space.

  • Banach - Mazur: there is an isometric linear operator for every separable Banach space.

The following proof sketch shows how you can find such isometries. It is the unit ball in the dual space of. This is according to the theorem of Banach - Alaoglu compact in the weak -* topology and because of the separability even metrizable. Then there exists a continuous, surjective map, because after a result from the topology is compact metrizable space every one continuous image of Cantor'schen discontinuum. Is now defined by, then obviously linear and because also isometric, where the last equality follows from the Hahn- Banach and the second to last from the surjectivity of.

From this it follows easily the following corollary, which is also called the Banach - Mazur.

  • Corollary: there is an isometric linear operator for every separable Banach space.

For each defined as a continuous function that, and so that the intervals of linear. The figure then defines an isometric embedding and the claim follows from the above set of Banach - Mazur.

Comments

  • Together with the fact that possesses a Schauder basis, there are applications in the theory of base sequences in separable Banach spaces; Examples can be found in the below book by Terry J. Morrison.
  • The property of having a Schauder basis, not inherited by subspaces, as is well known, a Schauder basis and there are separable Banach spaces without Schauder basis, and those can be obtained by the theorem of Banach - Mazur as subspaces of. For the same reason, the approximation property can not be inherited by subspaces.
  • Is a universal separable Banach space with respect to the subspace formation in the class of all separable Banach spaces, that is just the contents of the set of Banach - Mazur. There are also universal separable Banach spaces with respect to the quotient: One can show that every separable Banach space is isometrically isomorphic to a quotient of the sequence space.
  • Aleksander Pełczyński 1962 has shown that the following statements on a separable Banach space are equivalent:

Swell

  • S. Banach, S. Mazur: On the theory of linear dimension, Studia Mathematica (1933 ), Volume 4, pages 100-112
  • A. Pełczyński: On the universality of some Banach spaces (Russian), Vestnik Leningrad. Univ. Ser. Meh Mat. Astr. 13 (1962 ), pp. 22-29 ( German translation, PDF, 761 kB)
  • P. Wojtaszczyk: Banach spaces for analysts, Cambridge Studies in Advanced Mathematics 25 (1991 )
  • Terry J. Morrison: Functional Analysis, Theory An Introduction to Banach Space, Wiley (2001) ISBN 0471372145
  • Functional Analysis
  • Set ( mathematics)
Uniform norm Cantor set Isometry Quotient space (linear algebra)
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