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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: [email protected]
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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

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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