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Finitely representible BS
A translation from German, in progress:

In functional analysis, finitely representablility is a concept in the theory of of Banach spaces. The idea is to study a Banach space via its finite-dimensional subspaces.

Definition
A normed space $$ F $$ is called finitely representable in a normed space $$ E $$, if for every finite-dimensional subspace $$ U \subset{F} $$ and for every $$ \epsilon > 0 $$ there exists a subspace $$ V \subset E $$ and a linear isomorphism $$ T: U \rightarrow V $$ such that $$ \| T \| \cdot \| T^{(-1)} \| < 1 + \epsilon. $$

These are calculated as the operator norm $$ \| T \| $$ and $$ \| T^{(-1)} \| $$ with respect to the U and $$ V $$-induced sub-space norms.

The space $$ F $$ is finitely representable in $$ E $$ if every finite-dimensional subspace of $$ F $$ to a $$ \ epsilon $$ and in $$ E $$ occurs. With the concept of Banach-Mazur distance, you put it that way so that at any finite-dimensional subspace $$ U \subset F $$ finite-dimensional subspaces in E < /math> find any small Banach-Mazur distance to $$ U $$.

subspaces of Banach spaces in these finally presentable. The property of finite presentable is transitive, which states: $$ F $$ is finally presentable in $$ E $$ and $$ G $$ last presentable in $$ F $$, so $$ G $$ is finally presentable in $$ E $$.

Examples

 * The Lebesgue space of $p$th integrable functions $L^p$ is finitely presentable in the spaces of $p$th–summable sequences $ \ell^p, \, 1 \le p < \infty $.
 * the Lebesgue space $$ L^p, \, p > 2 $$ is not finitely presentable in $$ L^q (X, \mu), \, 1 \le q \le 2 $$.
 * The space of continuous functions $C([0,1])$ is finitely presentable in the space of sequences that converge to zero $c_0$ and vice versa.

Dvoretzky theorem
After the Banach-Mazur theorem, every separable Banach space is isometrically isomorphic to a subspace of $$ C ([0,1]) $$. Therefore, every Banach space is finally presentable in $$ C ([0,1]) $$, which means that $$ C ([0,1]) $$ is a maximum of about finite presentable. Aryeh Dvoretzky proved that that Hilbert spaces are minimal with respect to finite presentability:


 * Dvoretzky theorem: Every Hilbert space is in every infinite dimensional Banach space finally presentable.

The property, in every infinite dimensional Banach space to be presentable finally characterized the Hilbert spaces. In fact, if $$ E $$ in any Banach space finally presentable, so in $$ \ell^2 $$, and it is easy to show that in $$ E $$ parallelogram must be applied; therefore $$ E $$ is also a Hilbert space, by the characterization of inner-product spaces by the parallelogram law, which is due to Jordan and von Neumann.

Original formulation
From English article on Dvoretzky's theorem: Much clearer, imho:

For every $$k \in \mathbf{N}$$ and every $$\epsilon > 0$$ there exists $$N(k, \epsilon) \in \mathbf{N}$$ such that if $$(X, \| \cdot \|)$$ is a Banach space of dimension $$\geq N(k, \epsilon)$$, there exist a subspace $$E \subset X$$ of dimension $$k$$ and a positive quadratic form $$Q$$ on $$E$$ such that the corresponding Euclidean norm


 * $$| \cdot | = \sqrt{Q(\cdot)} $$

on $$E$$ satisfies:


 * $$ |x| \leq \|x\| \leq (1+\epsilon)|x| \quad \text{for every} \quad x \in E.$$

Super-properties
Let P be a property that can have a Banach space. They say that a Banach space E  math (or have) super-P if every Banach space, in the $$ E $$ is finally presentable, also has the property P. If a Banach space has a super-property, then by the theorem of Dvoretzky have any Hilbert space this property.

Is $$ E $$ a uniformly convex space and $$ F $$ finally presentable in $$ E $$, so is $$ F $$ 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 after the theorem of Milman is reflexive and as uniform convexity is a super-property, are uniformly convex spaces super-reflexive. Reflexivity itself is not a super-property, that is super-reflexivity and reflexivity are not equivalent. Super-reflexivity is characterized by the following theorem of Per Enflo


 * A Banach space is super-reflexive if and only if there is a #equivalent|equivalent norm, which makes him a uniformly convex space.

Since uniformly convex spaces have] a theorem of Shizuo Kakutani The [[Banach-Saks property], it follows:


 * Super-reflexive spaces have the Banach-Saks property.

Therefore, super-reflexivity implies the Super-Banach Saks property, which further implies:


 * Super-reflexivity and the super-Banach-Saks property are equivalent.

Principle of local reflexivity
After a theorem of Joram Lindenstrauss and Haskell Rosenthal, the bidual of a Banach space $$ E $$ is always finite presentable in $$ E $$. This so-calledprinciple of local reflexivitywill be strengthened to the following detailed statement:


 * Let $$ E $$ a Banach space, $$ U \subset E \,''$$ and $$ V \subset \ E, '$$ are finite-dimensional subspaces and was $$ \epsilon > 0 $$. Then there is a injective, constant,  linear operator $$ T: U \ rightarrow E $$ with:
 * 1) $$ \| T \| \cdot \| T^{(-1)} \|_{(T (U)}) \| \, < \, 1 + \epsilon $$
 * 2) $$ T |_(U \cap E) = ( \mathrm id)_(U \cap E) $$
 * 3)  F (u) \, = \, u (f) for all $$ u \ in U \ f V $$

Literature
given Per Enflo:Banach spaces which can be an equivalent uniformly convex norm. In:Israel Journal of Mathematics. Volume 13, 1972, p. 281-288.
 * Bernard Beauzamy:Introduction to Banach Spaces and their Geometry. 2nd Edition. North-Holland, Amsterdam etc. 1985, ISBN 0-444-87878-5.
 * Joseph Diestel: Sequences and Seriesin Banach Spaces. Springer, New York etc. 1984, ISBN 0-387-90859-5.
 * Joram Lindenstrauss, Haskell Paul Rosenthal:The L p -spaces. In:Israel Journal of Mathematics. Volume 7, 1969, p. 325-349.

Category: Functional analysis