Banach-Mazur distance

The Banach-Mazur distance , named after Stefan Banach and Stanisław Mazur , is a term from the mathematical theory of Banach spaces . It defines a distance between two isomorphic normalized spaces and is used especially for finite-dimensional spaces.

Motivation and Definition

If and are two isomorphic normalized spaces, there is a bijective, continuous, linear mapping , the inversion of which is also restricted. The following applies to the operator norm . thats why ${\displaystyle (E,\|\cdot \|_{E})}$ ${\displaystyle (F,\|\cdot \|_{F})}$ ${\displaystyle T:E\rightarrow F}$ ${\displaystyle 1=\|\mathrm {id} _{E}\|=\|T^{-1}\circ T\|\leq \|T^{-1}\|\cdot \|T\|}$

${\displaystyle \delta (E,F):=\inf\{\|T^{-1}\|\cdot \|T\|;\,T:E\rightarrow F\,\,{\text{Isomorphismus}}\}}$

a number that measures how far the spaces are from and from being isometrically isomorphic. This number is called the Banach-Mazur distance between and . Are and not isomorphic, so is . ${\displaystyle \geq 1}$ ${\displaystyle (E,\|\cdot \|_{E})}$ ${\displaystyle (F,\|\cdot \|_{F})}$ ${\displaystyle (E,\|\cdot \|_{E})}$ ${\displaystyle (F,\|\cdot \|_{F})}$ ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle \delta (E,F)=\infty }$

The following simple rules apply:

1. ${\displaystyle \delta (E,E)\,=\,1}$; more general , if and isometrically isomorphic, ${\displaystyle \delta (E,F)\,=\,1}$ ${\displaystyle E}$ ${\displaystyle F}$
2. ${\displaystyle \delta (E,F)\,=\,\delta (F,E)}$for standardized spaces and , ${\displaystyle E}$ ${\displaystyle F}$
3. ${\displaystyle \delta (E,F)\leq \delta (E,G)\cdot \delta (G,F)}$for standardized spaces and . ${\displaystyle E,F}$ ${\displaystyle G}$

It follows that it behaves like a metric , where log is any logarithm function , for example the natural logarithm. That explains the name Banach-Mazur distance . ${\displaystyle \log \circ \delta (\cdot ,\cdot )}$

Remarks

The Banach-Mazur distance depends on the underlying body, or , from. There is an example going back to Jean Bourgain of a real Banach space with two complex Banach space structures that are not isomorphic. ${\displaystyle \delta (E,F)}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$

In general, it does not follow that and are isometrically isomorphic. For the year following Aleksander Pełczyński and Czeslaw Bessaga declining as are for the following standards on c ₀ defined: ${\displaystyle \delta (E,F)\,=\,1}$ ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle i\in \{0,1\}}$

${\displaystyle \|x\|_{i}:=\sup _{j\in \mathbb {N} }|x(j)|+(\sum _{j=1}^{\infty }2^{-2j}|x(j+i)|^{2})^{\frac {1}{2}}}$

If one sets , one can show that is strictly convex , but not; therefore, and cannot be isometric isomorphic. If you set ${\displaystyle E_{i}:=(c_{0},\|\cdot \|_{i})}$ ${\displaystyle E_{0}}$ ${\displaystyle E_{1}}$ ${\displaystyle E_{0}}$ ${\displaystyle E_{1}}$

${\displaystyle T_{n}:E_{0}\rightarrow E_{1}:\quad (x(1),x(2),\ldots )\mapsto (x(n),x(1),\ldots ,x(n-1),x(n+1),\ldots )}$,

so is an isomorphism and it is , so holds . ${\displaystyle T_{n}:E_{0}\rightarrow E_{1}}$ ${\displaystyle \lim _{n\to \infty }\|T_{n}^{-1}\|\|T_{n}\|=1}$ ${\displaystyle \delta (E_{0},E_{1})=1}$

This example must necessarily be infinite-dimensional, because for two finite-dimensional spaces and one can show that if and only if and are isometrically isomorphic. ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle \delta (E,F)=1}$ ${\displaystyle E}$ ${\displaystyle F}$

Minkowski compact

Let it be the class of all n -dimensional Banach spaces. The isometric isomorphism is having called equivalence relation on . One can show that the Banach-Mazur distance induces a mapping on the set and that it is a compact metric space , the so-called Minkowski compact (after Hermann Minkowski ) or Banach-Mazur compact . Even if there is no metric, but only the logarithm of , metric terms in connection with the Minkowski compact are often used with reference to , this applies in particular to the terms distance and diameter used in this paragraph . ${\displaystyle {\mathcal {Q}}_{n}}$ ${\displaystyle \sim }$ ${\displaystyle {\mathcal {Q}}_{n}}$ ${\displaystyle Q_{n}:={\mathcal {Q}}_{n}/\sim }$ ${\displaystyle (Q_{n},\log \circ \delta )}$ ${\displaystyle \delta }$ ${\displaystyle \delta }$ ${\displaystyle \delta }$

Let it denote the one with the p norm . Then one easily shows for everyone : According to the Auerbach lemma, there is an Auerbach basis of . For then and therefore and , from which it follows. ${\displaystyle \ell _{p}^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \delta (E,\ell _{1}^{n})\leq n}$ ${\displaystyle E\in {\mathcal {Q}}_{n}}$ ${\displaystyle (e_{i},e_{i}')_{i}}$ ${\displaystyle E}$ ${\displaystyle T:\ell _{1}^{n}\rightarrow E,\,T((t_{i})_{i}):=\sum _{i=1}^{n}t_{i}e_{i}}$ ${\displaystyle T^{-1}x\,=\,(e_{i}'(x))_{i}}$ ${\displaystyle \|T\|=1}$ ${\displaystyle \|T^{-1}\|\leq n}$ ${\displaystyle \delta (E,\ell _{1}^{n})\leq n}$

The inequality shown by Fritz John in 1948 is more complex for everyone . It follows immediately ${\displaystyle \delta (E,\ell _{2}^{n})\leq {\sqrt {n}}}$ ${\displaystyle E\in {\mathcal {Q}}_{n}}$

${\displaystyle \delta (E,F)\leq \delta (E,\ell _{2}^{n})\cdot \delta (\ell _{2}^{n},F)\leq {\sqrt {n}}^{2}=n}$for everyone . ${\displaystyle E,F\in {\mathcal {Q}}_{n}}$

Hence the diameter of the Minkowski compact . ED Gluskin was able to show that the downward diameter can be estimated by a constant times . Some specific distances are still known, for example ${\displaystyle \leq n}$ ${\displaystyle n}$

${\displaystyle \delta (\ell _{p}^{n},\ell _{q}^{n})=n^{{\frac {1}{p}}-{\frac {1}{q}}}}$if or . ${\displaystyle 1\leq p\leq q\leq 2}$ ${\displaystyle 2\leq p\leq q\leq \infty }$

The following estimate is known for this case : ${\displaystyle 1\leq p<2<q\leq \infty }$

${\displaystyle {\frac {1}{\sqrt {2}}}\max\{n^{{\frac {1}{p}}-{\frac {1}{2}}},n^{{\frac {1}{2}}-{\frac {1}{q}}}\}\leq \delta (\ell _{p}^{n},\ell _{q}^{n})\leq {\frac {1}{{\sqrt {2}}-1}}\max\{n^{{\frac {1}{p}}-{\frac {1}{2}}},n^{{\frac {1}{2}}-{\frac {1}{q}}}\}}$.

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• Albrecht Pietsch : History of Banach Spaces and Linear Operators , Birkhäuser Boston (2007), ISBN 978-0-8176-4367-6
• Nicole Tomczak-Jaegermann : Banach-Mazur-Distances and Finite Dimensional Operator Ideals , Pitman monographs and Surveys in Pure and Applied Mathematics 38 (1988) ISBN 0-47020-982-8