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 . ${\displaystyle F}$ ${\displaystyle E}$ ${\displaystyle U\subset F}$ ${\displaystyle \epsilon >0}$ ${\displaystyle V\subset E}$ ${\displaystyle T:U\rightarrow V}$ ${\displaystyle \|T\|\cdot \|T^{-1}\|<1+\epsilon }$

The operator norms and with respect to the and induced subspace norms are calculated . ${\displaystyle \|T\|}$ ${\displaystyle \|T^{-1}\|}$ ${\displaystyle U}$ ${\displaystyle V}$

${\displaystyle F}$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 . ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle \epsilon }$ ${\displaystyle E}$ ${\displaystyle U\subset F}$ ${\displaystyle E}$ ${\displaystyle U}$

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 . ${\displaystyle F}$ ${\displaystyle E}$ ${\displaystyle G}$ ${\displaystyle F}$ ${\displaystyle G}$ ${\displaystyle E}$

Examples

• L ^p ([0,1]) is finitely presentable in the sequence space . ${\displaystyle \ell ^{p},\,1\leq p<\infty }$
• ${\displaystyle \ell ^{p},\,p>2}$is not finite presentable in . ${\displaystyle L^{q}(X,\mu ),\,1\leq q\leq 2}$
• The function space is finitely presentable in c ₀ and vice versa. ${\displaystyle C([0,1])}$

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: ${\displaystyle C([0,1])}$ ${\displaystyle C([0,1])}$ ${\displaystyle C([0,1])}$

• 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. ${\displaystyle E}$ ${\displaystyle \ell ^{2}}$ ${\displaystyle E}$ ${\displaystyle E}$

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. ${\displaystyle E}$ ${\displaystyle E}$

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. ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle E}$ ${\displaystyle F}$

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: ${\displaystyle E}$ ${\displaystyle E}$

• Let be a Banach space, and let be finite-dimensional subspaces and let it be . Then there is an injective , continuous, linear operator with: ${\displaystyle E}$ ${\displaystyle U\subset E\,''}$ ${\displaystyle V\subset E\,'}$ ${\displaystyle \epsilon >0}$ ${\displaystyle T:U\rightarrow E}$

1. ${\displaystyle \|T\|\cdot \|T^{-1}|_{T(U)}\|\,<\,1+\epsilon }$
2. ${\displaystyle T|_{U\cap E}=\mathrm {id} _{U\cap E}}$
3. ${\displaystyle f(Tu)\,=\,u(f)}$ for all ${\displaystyle u\in U,f\in V}$

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.