## Notre Dame Journal of Formal Logic

### Definable Types Over Banach Spaces

José Iovino

#### Abstract

We study connections between asymptotic structure in a Banach space and model theoretic properties of the space. We show that, in an asymptotic sense, a sequence $(x_n)$ in a Banach space X generates copies of one of the classical sequence spaces $\ell_p$ or $c_0$ inside X (almost isometrically) if and only if the quantifier-free types approximated by $(x_n)$ inside X are quantifier-free definable. More precisely, if $(x_n)$ is a bounded sequence X such that no normalized sequence of blocks of $(x_n)$ converges, then the following two conditions are equivalent. (1) There exists a sequence $(y_n)$ of blocks of $(x_n)$ such that for every finite dimensional subspace E of X, every quantifier-free type over $E +\overline{\rm span}\{y_n\mid n\in \mathbb{N}\}$ is quantifier-free definable. (2) One of the following two conditions holds: (a) there exists $1\le p< \infty$ such that for every $\epsilon>0$ and every finite dimensional subspace E of X there exists a sequence of blocks of $(x_n)$ which is $(1+\epsilon)$equivalent over E to the standard unit basis of $\ell_p$; (b) for every $\epsilon>0$ and every finite dimensional subspace E of X there exists a sequence of blocks of $(x_n)$ which is $(1+\epsilon)$-equivalent over E to the standard unit basis of $c_0$. Several byproducts of the proof are analyzed.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 46, Number 1 (2005), 19-50.

Dates
First available in Project Euclid: 31 January 2005

https://projecteuclid.org/euclid.ndjfl/1107220672

Digital Object Identifier
doi:10.1305/ndjfl/1107220672

Mathematical Reviews number (MathSciNet)
MR2131545

Zentralblatt MATH identifier
1082.46010

Subjects
Primary: 03C
Secondary: 46B

#### Citation

Iovino, José. Definable Types Over Banach Spaces. Notre Dame J. Formal Logic 46 (2005), no. 1, 19--50. doi:10.1305/ndjfl/1107220672. https://projecteuclid.org/euclid.ndjfl/1107220672

#### References

• Beauzamy, B., and J.-T. Lapresté, Modèles étalés des espaces de Banach, Travaux en Cours [Works in Progress]. Hermann, Paris, 1984.
• Beauzamy, B., Introduction to Banach Spaces and Their Geometry, 2d edition, vol. 68 of North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 86.
• Brunel, A., and L. Sucheston, "On $B$"-convex Banach spaces, Mathematical Systems Theory, vol. 7 (1974), pp. 294–99.
• Bu, S. Q., "Deux remarques sur les espaces de B"anach stables, Compositio Mathematica, vol. 69 (1989), pp. 341–55.
• Chaatit, F., "Twisted types and uniform stability", pp. 183–99 in Functional Analysis (Austin, TX, 1987/1989), vol. 1470 of Lecture Notes in Mathematics, Springer, Berlin, 1991.
• Dellacherie, C., "Les dérivations en théorie descriptive des ensembles et le théorème de la borne", pp. 34–46 in Séminaire de Probabilités, XI (Universität Strasbourg, Strasbourg, 1975/1976), vol. 581 of Lecture Notes in Mathematics, Springer, Berlin, 1977. Erratum et addendum à “Les dérivations en théorie descriptive des ensembles et le théorème de la borne”, vol. 649, p. 523, 1978.
• Farmaki, V. A., "$c\sb 0$"-subspaces and fourth dual types", Proceedings of the American Mathematical Society, vol. 102 (1988), pp. 321–28.
• Guerre, S., "Types et suites symétriques dans $L\sp p,\;1\leq p<+\infty,\;p\not= 2$", Israel Journal of Mathematics, vol. 53 (1986), pp. 191–208.
• Haydon, R., and B. Maurey, "On B"anach spaces with strongly separable types, Journal of the London Mathematical Society. Second Series, vol. 33 (1986), pp. 484–98.
• Heinrich, S., "Ultraproducts in B"anach space theory, Journal für die Reine und Angewandte Mathematik, vol. 313 (1980), pp. 72–104.
• Henson, C. W., and J. Iovino, "Ultraproducts in analysis", pp. 1–110 in Analysis and Logic (Mons, 1997), vol. 262 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 2002.
• Iovino, J., "Indiscernible sequences in B"anach space geometry, preprint.
• Iovino, J., Stable Theories in Functional Analysis, Ph.D. thesis, University of Illinois at Urbana-Champaign, 1994.
• Iovino, J., "Types on stable B"anach spaces, Fundamenta Mathematicae, vol. 157 (1998), pp. 85–95.
• Iovino, J., "Stable B"anach spaces and Banach space structures. I. Fundamentals, pp. 77–95 in Models, Algebras, and Proofs (Bogotá, 1995), edited by C. Montenegro and X. Caicedo, vol. 203 of Lecture Notes in Pure and Applied Mathematics, Dekker, New York, 1999.
• Iovino, J., "Stable B"anach spaces and Banach space structures. II. Forking and compact topologies, pp. 97–117 in Models, Algebras, and Proofs (Bogotá, 1995), vol. 203 of Lecture Notes in Pure and Applied Mathematics, Dekker, New York, 1999.
• Krivine, J. L., "Sous-espaces de dimension finie des espaces de B"anach réticulés, Annals of Mathematics. Second Series, vol. 104 (1976), pp. 1–29.
• Krivine, J.-L., and B. Maurey, "Espaces de B"anach stables, Israel Journal of Mathematics, vol. 39 (1981), pp. 273–95.
• Lemberg, H., "Nouvelle démonstration d'un théorème de J".-L. Krivine sur la finie représentation de $l\sb{p}$ dans un espace de Banach, Israel Journal of Mathematics, vol. 39 (1981), pp. 341–48.
• Luxemburg, W. A. J., "A general theory of monads", pp. 18–86 in Applications of Model Theory to Algebra, Analysis, and Probability (International Symposium, Pasadena, 1967), Holt, Rinehart and Winston, New York, 1969.
• Maurey, B., "Types and $l\sb 1$"-subspaces, pp. 123–37 in Texas Functional Analysis Seminar 1982–1983 (Austin), Longhorn Notes, University of Texas Press, Austin, 1983.
• Odell, E., "On the types in T"sirelson's space, pp. 49–59 in Texas Functional Analysis Seminar 1982–1983 (Austin), Longhorn Notes, University of Texas Press, Austin, 1983.
• Raynaud, Y., "Séparabilité uniforme de l'espace des types d'un espace de B"anach. Quelques exemples, pp. 121–37 in Seminar on the Geometry of Banach Spaces, Vol. I, II (Paris, 1983), vol. 18 of Publications Mathématiques de l'Université Paris VII, University of Paris VII, Paris, 1984.
• Raynaud, Y., "Stabilité et séparabilité de l'espace des types d'un espace de B"anach: Quelques exemples, in Seminar on the Geometry of Banach Spaces, Tome II, 1983, vol. 18 of Publications Mathématiques de l'Université Paris VII, University of Paris VII, Paris, 1984.
• Raynaud, Y., "Almost isometric methods in some isomorphic embedding problems", pp. 427–45 in Banach Space Theory (Iowa City, 1987), vol. 85 of Contemporary Mathematics, American Mathematical Society, Providence, 1989.
• Rosenthal, H. P., "On a theorem of J". L. Krivine concerning block finite representability of $l\sp{p}$ in general Banach spaces, Journal of Functional Analysis, vol. 28 (1978), pp. 197–225.
• Rosenthal, H. P., "Double dual types and the M"aurey characterization of Banach spaces containing $l\sp 1$, pp. 1–37 in Texas Functional Analysis Seminar 1983–1984 (Austin), Longhorn Notes, University of Texas Press, Austin, 1984.
• Rosenthal, H. P., "The unconditional basic sequence problem", pp. 70–88 in Geometry of Normed Linear Spaces (Urbana-Champaign, 1983), vol. 52 of Contemporary Mathematics, American Mathematical Society, Providence, 1986.
• Shelah, S., Classification theory and the number of nonisomorphic models, 2d edition, vol. 92 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1990.
• Sims, B., “Ultra”-techniques in Banach space theory, vol. 60 of Queen's Papers in Pure and Applied Mathematics, Queen's University, Kingston, 1982.