# Types on stable Banach spaces

Fundamenta Mathematicae (1998)

• Volume: 157, Issue: 1, page 85-95
• ISSN: 0016-2736

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## Abstract

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We prove a geometric characterization of Banach space stability. We show that a Banach space X is stable if and only if the following condition holds. Whenever $\stackrel{^}{X}$ is an ultrapower of X and B is a ball in $\stackrel{^}{X}$, the intersection B ∩ X can be uniformly approximated by finite unions and intersections of balls in X; furthermore, the radius of these balls can be taken arbitrarily close to the radius of B, and the norm of their centers arbitrarily close to the norm of the center of B.  The preceding condition can be rephrased without any reference to ultrapowers, in the language of types, as follows. Whenever τ is a type of X, the set ${\tau }^{-1}\left[0,r\right]$ can be uniformly approximated by finite unions and intersections of balls in X; furthermore, the radius of these balls can be taken arbitrarily close to r, and the norm of their centers arbitrarily close to τ(0).  We also provide a geometric characterization of the real-valued functions which satisfy the above condition.

## How to cite

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Iovino, José. "Types on stable Banach spaces." Fundamenta Mathematicae 157.1 (1998): 85-95. <http://eudml.org/doc/212280>.

@article{Iovino1998,
abstract = { We prove a geometric characterization of Banach space stability. We show that a Banach space X is stable if and only if the following condition holds. Whenever $\widehat\{X\}$ is an ultrapower of X and B is a ball in $\widehat\{X\}$, the intersection B ∩ X can be uniformly approximated by finite unions and intersections of balls in X; furthermore, the radius of these balls can be taken arbitrarily close to the radius of B, and the norm of their centers arbitrarily close to the norm of the center of B.  The preceding condition can be rephrased without any reference to ultrapowers, in the language of types, as follows. Whenever τ is a type of X, the set $τ^\{-1\}[0,r]$ can be uniformly approximated by finite unions and intersections of balls in X; furthermore, the radius of these balls can be taken arbitrarily close to r, and the norm of their centers arbitrarily close to τ(0).  We also provide a geometric characterization of the real-valued functions which satisfy the above condition.},
author = {Iovino, José},
journal = {Fundamenta Mathematicae},
keywords = {geometric characterization; Banach space stability; ultrapower},
language = {eng},
number = {1},
pages = {85-95},
title = {Types on stable Banach spaces},
url = {http://eudml.org/doc/212280},
volume = {157},
year = {1998},
}

TY - JOUR
AU - Iovino, José
TI - Types on stable Banach spaces
JO - Fundamenta Mathematicae
PY - 1998
VL - 157
IS - 1
SP - 85
EP - 95
AB -  We prove a geometric characterization of Banach space stability. We show that a Banach space X is stable if and only if the following condition holds. Whenever $\widehat{X}$ is an ultrapower of X and B is a ball in $\widehat{X}$, the intersection B ∩ X can be uniformly approximated by finite unions and intersections of balls in X; furthermore, the radius of these balls can be taken arbitrarily close to the radius of B, and the norm of their centers arbitrarily close to the norm of the center of B.  The preceding condition can be rephrased without any reference to ultrapowers, in the language of types, as follows. Whenever τ is a type of X, the set $τ^{-1}[0,r]$ can be uniformly approximated by finite unions and intersections of balls in X; furthermore, the radius of these balls can be taken arbitrarily close to r, and the norm of their centers arbitrarily close to τ(0).  We also provide a geometric characterization of the real-valued functions which satisfy the above condition.
LA - eng
KW - geometric characterization; Banach space stability; ultrapower
UR - http://eudml.org/doc/212280
ER -

## References

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1. [1] D. Aldous, Subspaces of ${L}_{1}$ via random measures, Trans. Amer. Math. Soc. 267 (1981), 445-463.
2. [2] S. Guerre-Delabrière, Classical Sequences in Banach Spaces, Marcel Dekker, New York, 1992. Zbl0756.46007
3. [3] S. Heinrich, Ultraproducts in Banach space theory, J. Reine Angew. Math. 313 (1980), 72-104. Zbl0412.46017
4. [4] J. Iovino, Stable theories in functional analysis, PhD thesis, Univ. of Illinois at Urbana-Champaign, 1994.
5. [5] J.-L. Krivine et B. Maurey, Espaces de Banach stables, Israel J. Math. 39 (1981), 273-295. Zbl0504.46013
6. [6] E. Odell, On the types in Tsirelson's space, in: Longhorn Notes, Texas Functional Analysis Seminar, 1982-1983.
7. [7] A. Pillay, Geometric Stability Theory, Clarendon Press, Oxford, 1996. Zbl0871.03023
8. [8] Y. Raynaud, Stabilité et séparabilité de l'espace des types d'un espace de Banach: Quelques exemples, in: Séminarie de Géométrie des Espaces de Banach, Paris VII, Tome II, 1983.

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