Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-26T12:30:41.339Z Has data issue: false hasContentIssue false
  • English
  • Français

Definability in functional analysis

Published online by Cambridge University Press:  12 March 2014

José Iovino
José Iovino*
Affiliation:
Department of Mathematics, Carnegie-Mellon University, Pittsburgh, PA 15213, USA, E-mail: iovino+@andrew.cmu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

The role played by real-valued functions in functional analysis is fundamental. One often considers metrics, or seminorms, or linear functionals, to mention some important examples.

We introduce the notion of definable real-valued function in functional analysis: a real-valued function f defined on a structure of functional analysis is definable if it can be “approximated” by formulas which do not involve f. We characterize definability of real-valued functions in terms of a purely topological condition which does not involve logic.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 2 , June 1997 , pp. 493 - 505
Copyright
Copyright © Association for Symbolic Logic 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Beauzamy, B. and Lapreste, J. T., Modèles Étalés des Espaces de Banach, Hermann, Paris, 1984.Google Scholar
[2]Dacuhna-Castelle, D. and Krivine, J.-L., Applications des ultraproduits à l'étude des espaces et de algèbres de Banach, Studia Mathematica, vol. 41 (1972), pp. 315–334.CrossRefGoogle Scholar
[3]Guerre-Delabrière, S., Classical sequences in Banach spaces, Marcel Dekker, New York, 1992.Google Scholar
[4]Heinrich, S., Ultraproducts in Banach space theory, Journal für die Reine und Angewandte Mathematik, vol. 313 (1980), pp. 72–104.Google Scholar
[5]Heinrich, S. and Henson, C. W., Banach space model theory, II: Isomorphic equivalence, Mathematische Nachrichten, vol. 125 (1986), pp. 301–317.CrossRefGoogle Scholar
[6]Henson, C. W., When do two Banach spaces have isometrically isomorphic nonstandard hulls?, Israel Journal of Mathematics, vol. 39 (1975), pp. 57–67.CrossRefGoogle Scholar
[7]Henson, C. W., Nonstandard hulls of Banach spaces, Israel Journal of Mathematics, vol. 25 (1976), pp. 108–144.CrossRefGoogle Scholar
[8]Henson, C. W. and Iovino, J., Banach space model theory, I: Basics, in preparation.Google Scholar
[9]Henson, C. W. and Moore, L. C. Jr., Nonstandard analysis and the theory of Banach spaces, Lecture Notes in Mathematics, no. 983, pp. 27–112, Lecture Notes in Mathematics, no. 983, Springer-Verlag, Berlin, 1983, pp. 27–112.Google Scholar
[10]Iovino, J., Stable Banach spaces, I, submitted.Google Scholar
[11]Krivine, J.-L. and Maurey, B., Espaces de Banach stables, Israel Journal of Mathematics, vol. 39 (1981), pp. 273–295.CrossRefGoogle Scholar
[12]Maurey, B., Double dual types and the characterization of Banach spaces containing ℓ1, Longhorn Notes, The University of Texas, Texas Functional Analysis Seminar, 1983–1984.Google Scholar
[13]Rosenthal, H., Some remarks concerning unconditional basic sequences, Longhorn Notes, The University of Texas, Texas Functional Analysis Seminar, 1982–1983.Google Scholar
[14]Rosenthal, H., Types and ℓ1-subspaces, Longhorn Notes, The University of Texas, Texas Functional Analysis Seminar, 1982–1983.Google Scholar
[15]Shelah, S., Classification theory and the number of non-isomorphic models, North-Holland, Amsterdam, 1990.Google Scholar