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
![(E, \ | \ cdot \ | _ {E})](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b6da6c16b5c7bcaae0674655d586dd7e0dfc117)
![T: E \ rightarrow F](https://wikimedia.org/api/rest_v1/media/math/render/svg/080a5e1d212d2a44ec6fcfe4cb6eb55f21629fb7)
![{\ displaystyle 1 = \ | \ mathrm {id} _ {E} \ | = \ | T ^ {- 1} \ circ T \ | \ leq \ | T ^ {- 1} \ | \ cdot \ | T \ |}](https://wikimedia.org/api/rest_v1/media/math/render/svg/497cdc53bfddf977ad8fa4d289a6c5d4bc796f0d)
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 .
![\ geq 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/0023e7ae28acaa679f1a9dea66f45ce1affca3c9)
![(E, \ | \ cdot \ | _ {E})](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b6da6c16b5c7bcaae0674655d586dd7e0dfc117)
![(F, \ | \ cdot \ | _ {F})](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0ae42922c76a77edfe1d6a2e1cc379e62064ca2)
![(E, \ | \ cdot \ | _ {E})](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b6da6c16b5c7bcaae0674655d586dd7e0dfc117)
![(F, \ | \ cdot \ | _ {F})](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0ae42922c76a77edfe1d6a2e1cc379e62064ca2)
![E.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b)
![F.](https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57)
![\ delta (E, F) = \ infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/00b5243d0c49e3cfefe047c5f01c6f57015a2217)
The following simple rules apply:
-
; more general , if and isometrically isomorphic,![\ delta (E, F) \, = \, 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b1e0c72bad1d214da7d270512cc9ef566a89826)
![E.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b)
![F.](https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57)
-
for standardized spaces and ,![E.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b)
![F.](https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57)
-
for standardized spaces and .![E, F](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bb9a1b35f59889d075043aa767ee6941df5cf91)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
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 .
![\ log \ circ \ delta (\ cdot, \ cdot)](https://wikimedia.org/api/rest_v1/media/math/render/svg/d723aaeca67a5ceb8dc633b62076f1b8e9ff3a3f)
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.
![\ delta (E, F)](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c044db6f4af22ce2d941229d0382b42af034082)
![\ mathbb {R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc)
![{\ displaystyle \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7)
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:
![\ delta (E, F) \, = \, 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b1e0c72bad1d214da7d270512cc9ef566a89826)
![E.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b)
![F.](https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57)
![i \ in \ {0.1 \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24f7fdf564e6c9c0fdb61378f08ffc41f3bd2837)
If one sets , one can show that is strictly convex , but not; therefore, and cannot be isometric isomorphic. If you set
![E_ {i}: = (c_ {0}, \ | \ cdot \ | _ {i})](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fceaf8655bbe49a19a4fdbf6a7118546b42622d)
![E_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac42446bcd2cbb76ec8fe2895635d328da22e26)
![E_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/411d268de7b1cf300d7481e3fe59f3b20887e0d0)
![E_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac42446bcd2cbb76ec8fe2895635d328da22e26)
,
so is an isomorphism and it is , so holds .
![T_ {n}: E_ {0} \ rightarrow E_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/227189b234318ed6edd821b14895735336e5f5ab)
![\ lim _ {{n \ to \ infty}} \ | T_ {n} ^ {{- 1}} \ | \ | T_ {n} \ | = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cf22ffa8f6c8ac027e8e871776028a93ac303dc)
![\ delta (E_ {0}, E_ {1}) = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/b82b639ff659d41c6a02da48403a99d4e9f1802f)
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.
![E.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b)
![F.](https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57)
![\ delta (E, F) = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cd5b5113c71a0d42bda6cc0bef4f020b51bd504)
![E.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b)
![F.](https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57)
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 .
![{{\ mathcal Q}} _ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d7cff23e4fceac26b3a49f2e86b968c615f0d39)
![\ sim](https://wikimedia.org/api/rest_v1/media/math/render/svg/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173)
![{{\ mathcal Q}} _ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d7cff23e4fceac26b3a49f2e86b968c615f0d39)
![Q_ {n}: = {{\ mathcal Q}} _ {n} / \ sim](https://wikimedia.org/api/rest_v1/media/math/render/svg/4255b5edd2b133b4e5b2afb84f03c3770d2ef0a1)
![\delta](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a)
![\delta](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a)
![\delta](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a)
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.
![\ ell _ {p} ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b683a16aec7d24818c2e27d37071bea12e02c07)
![\ mathbb {R} ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d)
![\ delta (E, \ ell _ {1} ^ {n}) \ leq n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0a24823eaf1a1d354c910c5cd59925d33258cdb)
![(e_ {i}, e_ {i} ') _ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cf34a3afea008431696c704e613fe926063c6df)
![E.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b)
![T: \ ell _ {1} ^ {n} \ rightarrow E, \, T ((t_ {i}) _ {i}): = \ sum _ {{i = 1}} ^ {n} t_ {i }egg}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4f8137c0da975f8b64b6ccaf298b5325611ddac)
![T ^ {{- 1}} x \, = \, (e_ {i} '(x)) _ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0388fa89d88db8d1146be154ed7bd66ad85ca96)
![\ | T \ | = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/aacd7111473d996c55fe5be8960fb7849a785783)
![\ | T ^ {{- 1}} \ | \ leq n](https://wikimedia.org/api/rest_v1/media/math/render/svg/b66cb08fa4d59e58b1b53ca2962e56faf5c566a2)
![\ delta (E, \ ell _ {1} ^ {n}) \ leq n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0a24823eaf1a1d354c910c5cd59925d33258cdb)
The inequality shown by Fritz John in 1948 is more complex for everyone . It follows immediately
![\ delta (E, \ ell _ {2} ^ {n}) \ leq {\ sqrt {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/667709a4dd747b324866e0b3fdee4a31efe0d440)
![E \ in {{\ mathcal Q}} _ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fb651ae287668dc802942d0ae5ea7f34338d2ad)
for everyone .
![E, F \ in {{\ mathcal Q}} _ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c672039c74c2245f4c642e9526587e283b48bfca)
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
![\ leq n](https://wikimedia.org/api/rest_v1/media/math/render/svg/61fbd99c9375a5553cc47f02b7da90cb8becd4ca)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
if or .
![1 \ leq p \ leq q \ leq 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/22951bd8d2d9029a4f7d8f4f16304c83f7ba9e2d)
![2 \ leq p \ leq q \ leq \ infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/f321c970b6b26de279dc8de01573ceafb8f3b177)
The following estimate is known for this case :
![1 \ leq p <2 <q \ leq \ infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/1499920c7ff4e79d41ae4a22fd3330d15412873e)
.
swell