Banach-Mazur theorem

The Banach-Mazur theorem from 1933, named after Stefan Banach and Stanisław Mazur , is a classic theorem from the branch of functional analysis . Among the separable Banach spaces there are some that contain a copy of every other separable Banach space. The Banach space of continuous functions with the supremum norm is such a universal Banach space. ${\displaystyle C([0,1])}$ ${\displaystyle [0,1]\rightarrow {\mathbb {R} }}$

Formulation of the sentence

If a space is compact , then one denotes the Banach space of the continuous functions from to with the supremum norm . ${\displaystyle K}$ ${\displaystyle C(K)}$ ${\displaystyle K}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \|\cdot \|_{\infty }}$

First version

In the first version of the set of Banach Mazur is the Cantor'sche discontinuum : ${\displaystyle K}$ ${\displaystyle \Delta }$

For every separable Banach space there is an isometric linear operator from to . ${\displaystyle E}$ ${\displaystyle E}$ ${\displaystyle C(\Delta )}$

The following evidence sketch shows how to find such isometrics. Let it be the unit sphere in the dual space of . According to the Banach-Alaoglu theorem, this is compact with regard to the weak - * - topology and even metrizable because of the separability . Then there is a continuous, surjective mapping , because, according to a result from the topology , every compact metrizable space is a continuous image of Cantor's discontinuum. If one now defines through , then is obviously linear and because of ${\displaystyle E_{1}'}$ ${\displaystyle E}$ ${\displaystyle \phi \colon \Delta \rightarrow E_{1}'}$ ${\displaystyle T\colon E\rightarrow C(\Delta )}$ ${\displaystyle (Tx)(\delta ):=(\phi (\delta ))(x),\ x\in E,\ \delta \in \Delta }$ ${\displaystyle T}$

${\displaystyle \|Tx\|_{\infty }:=\sup _{\delta \in \Delta }|(Tx)(\delta )|=\sup _{\delta \in \Delta }|\phi (\delta )(x)|=\sup _{f\in E_{1}'}|f(x)|=\|x\|}$

also isometric, whereby the last equality follows from Hahn-Banach's theorem and the penultimate from the surjectivity of . ${\displaystyle \phi \,}$

Second version

As a consequence, the following version is obtained:

For every separable Banach space there is an isometric, linear operator from to . ${\displaystyle E}$ ${\displaystyle E}$ ${\displaystyle C([0,1])}$

For each one defined as a continuous function that, on with matches and the intervals from linear. The figure then defines an isometric embedding of to and the claim follows from the above first version of the Banach-Mazur theorem. ${\displaystyle f\in C(\Delta )}$ ${\displaystyle {\tilde {f}}\colon [0,1]\rightarrow \mathbb {R} }$ ${\displaystyle \Delta }$ ${\displaystyle f}$ ${\displaystyle [0,1]\setminus \Delta }$ ${\displaystyle f\mapsto {\tilde {f}}}$ ${\displaystyle C(\Delta )}$ ${\displaystyle C([0,1])}$

Remarks

• Along with the fact that it has a shudder basis , there are applications in the theory of basis sequences in separable Banach spaces; Examples of this can be found in the book by Terry J. Morrison given below. ${\displaystyle C([0,1])}$

• The property of having a shudder base is not inherited by subspaces, because, as is well known, has a shudder base and there are separable Banach spaces without a shudder base, and these can be obtained as subspaces of according to the Banach-Mazur theorem . For the same reason, the approximation property cannot be passed on to subspaces. ${\displaystyle C([0,1])}$ ${\displaystyle C([0,1])}$

• ${\displaystyle C([0,1])}$is a universal separable Banach space with respect to subspace formation in the class of all separable Banach spaces, that is precisely the content of the Banach-Mazur theorem. There are also universal separable Banach spaces with regard to the formation of quotients : One can show that every separable Banach space is isometrically isomorphic to a quotient of the sequence space . ${\displaystyle \ell ^{1}}$

• Aleksander Pełczyński showed in 1962 that the following statements about a separable Banach space are equivalent: ${\displaystyle E}$

1. ${\displaystyle E}$ is a universal separable Banach space with respect to subspace formation.
2. ${\displaystyle E}$contains a subspace that is too isometrically isomorphic. ${\displaystyle C(\Delta )}$
3. ${\displaystyle E}$contains a subspace that is too isometrically isomorphic. ${\displaystyle C([0,1])}$
4. There are elements for and such that and for all scalars . ${\displaystyle x_{n,k}\in E}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle k=0,1,\ldots 2^{n}-1}$ ${\displaystyle x_{n,k}\,=\,x_{n+1,2k}+x_{n+1,2k+1}}$ ${\displaystyle \left\|\sum _{k=0}^{2^{n}-1}t_{k}x_{n,k}\right\|=\max _{k=0,\ldots 2^{n}-1}\left|t_{k}\right|}$ ${\displaystyle t_{k}\in \mathbb {R} }$

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• S. Banach, S. Mazur: On the theory of the linear dimension , Studia Mathematica (1933), Volume 4, pages 100-112
• A. Pełczyński: On the universality of some Banach spaces (Russian), Vestnik Leningrad. Univ. Ser. Mat. Meh. Astr. 13 (1962), pages 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, An Introduction to Banach Space Theory , Wiley-Verlag (2001) ISBN 0471372145