Finite presentability (Banach space)
The finite presentability is a mathematical concept that is used in the investigation of the Banach spaces . The basic idea is to examine a Banach space via the finite-dimensional subspaces it contains.
definition
A normed space is finitely presentable in a normed space , if for every finite-dimensional subspace and every part of space and a linear isomorphism are with .
The operator norms and with respect to the and induced subspace norms are calculated .
is therefore finitely presentable in , if every finite-dimensional subspace also occurs from but to one in . With the concept of the Banach-Mazur distance one can also formulate this in such a way that for every finite-dimensional subspace one can find finite-dimensional subspaces with arbitrarily small Banach-Mazur distance .
Sub-spaces of Banach spaces can finally be presented in these. The property of finite presentability is transitive , that is: is finitely presentable in and finitely presentable in , then finitely presentable in .
Examples
- L p ([0,1]) is finitely presentable in the sequence space .
- is not finite presentable in .
- The function space is finitely presentable in c 0 and vice versa.
Dvoretzky's theorem
According to the Banach-Mazur theorem , every separable Banach space is isometrically isomorphic to a subspace of . Therefore every Banach space is finitely presentable in , that is, is maximal with respect to finite presentability. The set of Dvoretzky (after Aryeh Dvoretzky ) indicates that Hilbert spaces are minimal with respect to finite presentability:
- Theorem of Dvoretzky: Every Hilbert space is finitely presentable in every infinite-dimensional Banach space.
The property of being finite presentable in every infinite-dimensional Banach space characterizes the Hilbert spaces. Namely, is finite presentable in every Banach space, so also in , and one easily shows that in the parallelogram equation must hold; Therefore, after the set of Jordan-von Neumann also a Hilbert space.
Super properties
Let P be a property that a Banach space can have. One says that a Banach space is (or has) super-P if every Banach space that is finitely presentable in also has the property P. If a Banach space has a super property, then by Dvoretzky's theorem, every Hilbert space must have this property too.
If a space is uniformly convex and finitely presentable in , then it is also uniformly convex. Uniform convexity is therefore a super property, that is, a uniformly convex space is already super-uniformly convex.
Super reflexivity
Since uniformly convex spaces are reflexive according to Milman's theorem, and since uniform convexity is a super property, uniformly convex spaces are super-reflexive. Reflexivity itself is not a super-property, that is, reflexivity and super-reflexivity are not equivalent. Super-reflexivity is characterized by the following Per Enflo theorem :
- A Banach space is super-reflexive if and only if there is an equivalent norm that makes it a uniformly convex space.
Since uniformly convex spaces have the Banach-Saks property according to a theorem of Shizuo Kakutani , it follows from this:
- Super-reflective spaces have the Banach-Saks property.
Hence the super-Banach-Saks quality follows from super-reflexivity; one can even show:
- Super-reflexivity and the super-Banach-Saks property are equivalent.
Principle of local reflexivity
According to a sentence by Joram Lindenstrauss and Haskell Rosenthal , the bidual of a Banach space can always be finally presented in . This so-called principle of local reflexivity is tightened to the following more precise statement:
- Let be a Banach space, and let be finite-dimensional subspaces and let it be . Then there is an injective , continuous, linear operator with:
- for all
literature
- Bernard Beauzamy: Introduction to Banach Spaces and their Geometry . 2nd Edition. North-Holland, Amsterdam et al. 1985, ISBN 0-444-87878-5 .
- Joseph Diestel: Sequences and Series in Banach Spaces . Springer, New York et al. 1984, ISBN 0-387-90859-5 .
- Per Enflo : Banach spaces which can be given an equivalent uniformly convex norm . In: Israel Mathematical Journal . Volume 13, 1972, pp. 281-288.
- Joram Lindenstrauss , Haskell Paul Rosenthal: The L p -spaces . In: Israel Mathematical Journal . Volume 7, 1969, pp. 325-349.