Banach–Mazur compactum

The Banach - Mazur distance, named after Stefan Banach and Stanislaw Mazur, is a concept from the mathematical theory of Banach spaces. He defines a distance between two isomorphic normed spaces and is used particularly for finite dimensional spaces.

Motivation and definition

Are and two isomorphic normed spaces, so there is a bijective, continuous, linear map whose inverse is also limited. For the operator norm. is therefore

A number that measures how well the spaces and these are removed to be isometrically isomorphic. This number is called the Banach - Mazur distance between and. Are not isomorphic, then.

Apply the following simple rules:

It follows that such a metric behaves, where log any logarithm is, for example, the natural logarithm. This explains the name Banach - Mazur distance.

Comments

The Banach - Mazur distance depends on the underlying base, or from. There are a returning to Jean Bourgain example of a real Banach space with two complex Banach space structures, which are not isomorphic.

From not generally follows that and are isometrically isomorphic. For the following Aleksander Pelczynski and Czesław Bessaga back previous example are defined for the following norms on c0:

Substituting, as one can show that strictly convex, but not; therefore can not be isometrically isomorphic. Substituting

,

So is an isomorphism and it is so true.

This example must necessarily be infinite dimensional, finite because for two rooms, and one can show that if and only if and are isometrically isomorphic.

Minkowski compactum

It is the class of all n-dimensional Banach spaces. The isometric isomorphism is an equivalence relation on with designated. One can show that the Banach - Mazur distance a figure on the amount of induced and that is a compact metric space, called Minkowski compactum (after Hermann Minkowski ) or Banach - Mazur - compact set. Even if no metric is, but only the logarithm of such metric terms associated with the Minkowski compactum is often used with respect, this is especially true for the terms distance and diameter used in this paragraph.

Denote with the p-norm. Then this is easy for all: After Auerbach Auerbach basis of a lemma exists. For then applies and, therefore, and from which it follows.

Ornate is the inequality shown by Fritz John in 1948 for all. It immediately follows

For everyone.

Therefore, the diameter of the Minkowski compactum. ED Gluskin showed that the diameter can be estimated down by a constant times. There are known some specific distances, such as

If or.

In the event you know the following estimate:

.

Swell

  • 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 0470209828
  • Functional Analysis
Operatornorm Isometry Sequence space Strictly convex space Auerbach's lemma
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