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PROOF MINING IN Lp SPACES

Published online by Cambridge University Press:  29 August 2019

ANDREI SIPOŞ*
Affiliation:
DEPARTMENT OF MATHEMATICS TECHNISCHE UNIVERSITÄT DARMSTADT SCHLOSSGARTENSTRASSE 7, 64289 DARMSTADT, GERMANY and SIMION STOILOW INSTITUTE OF MATHEMATICS OF THE ROMANIAN ACADEMY CALEA GRIVIŢEI 21, 010702BUCHAREST, ROMANIA E-mail: sipos@mathematik.tu-darmstadt.de Current address: DEPARTMENT OF MATHEMATICS TECHNISCHE UNIVERSITÄT DARMSTADT SCHLOSSGARTENSTRASSE 7, 64289DARMSTADT, GERMANY

Abstract

We obtain an equivalent implicit characterization of Lp Banach spaces that is amenable to a logical treatment. Using that, we obtain an axiomatization for such spaces into a higher order logical system, the kind of which is used in proof mining, a research program that aims to obtain the hidden computational content of mathematical proofs using tools from mathematical logic. As an aside, we obtain a concrete way of formalizing Lp spaces in positive-bounded logic. The axiomatization is followed by a corresponding metatheorem in the style of proof mining. We illustrate its use with the derivation for this class of spaces of the standard modulus of uniform convexity.

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Articles
Copyright
Copyright © The Association for Symbolic Logic 2019 

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References

REFERENCES

Avigad, J. and Iovino, J., Ultraproducts and metastability. New York Journal of Mathematics, vol. 19 (2013), pp. 713727.Google Scholar
Ben Yaacov, I., Berenstein, A., Henson, C. W., and Usvyatsov, A., Model theory for metric structures, Model Theory with Applications to Algebra and Analysis, vol. 2 (Chatzidakis, Z., Macpherson, D., Pillay, A., and Wilkie, A., editors), London Mathematical Society Lecture Note Series, vol. 350, Cambridge University Press, Cambridge, 2008, pp. 315427.CrossRefGoogle Scholar
Chang, C. C. and Keisler, H. J., Continuous Model Theory, Princeton University Press, Princeton, NJ, 1966.CrossRefGoogle Scholar
Clarkson, J. A., Uniformly convex spaces. Transactions of the American Mathematical Society, vol. 40 (1936), no. 3, pp. 396414.CrossRefGoogle Scholar
Gerhardy, P. and Kohlenbach, U., Strongly uniform bounds from semi-constructive proofs. Annals of Pure and Applied Logic, vol. 141 (2006), pp. 89107.CrossRefGoogle Scholar
Gerhardy, P. and Kohlenbach, U., General logical metatheorems for functional analysis. Transactions of the American Mathematical Society, vol. 360 (2008), pp. 26152660.CrossRefGoogle Scholar
Günzel, D. and Kohlenbach, U., Logical metatheorems for abstract spaces axiomatized in positive bounded logic. Advances in Mathematics, vol. 290 (2016), pp. 503551.CrossRefGoogle Scholar
Hanner, O., On the uniform convexity of Lp and ℓp. Arkiv för Matematik, vol. 3 (1956), pp. 239244.CrossRefGoogle Scholar
Henson, C. W., Nonstandard hulls of Banach spaces. Israel Journal of Mathematics, vol. 25 (1976), pp. 108144.CrossRefGoogle Scholar
Henson, C. W. and Iovino, J., Ultraproducts in analysis, Analysis and Logic (Finet, C. and Michaux, C., editors), London Mathematical Society Lecture Notes Series, vol. 262, Cambridge University Press, Cambridge, 2002, pp. 1113.Google Scholar
Henson, C. W. and Raynaud, Y., On the theory of Lp (Lq)-Banach lattices. Positivity, vol. 11 (2007), no. 2, pp. 201230.CrossRefGoogle Scholar
Hilbert, D. and Bernays, P., Grundlagen der Mathematik. I, Zweite Auflage. Die Grundlehren der mathematischen Wissenschaften, vol. 40, Springer-Verlag, Berlin-New York, 1968.CrossRefGoogle Scholar
Kohlenbach, U., Some logical metatheorems with applications in functional analysis. Transactions of the American Mathematical Society, vol. 357 (2005), no. 1, pp. 89128.CrossRefGoogle Scholar
Kohlenbach, U., Applied Proof Theory: Proof Interpretations and their Use in Mathematics, Springer Monographs in Mathematics, Springer-Verlag, Berlin-Heidelberg, 2008.Google Scholar
Kohlenbach, U., Recent progress in proof mining in nonlinear analysis. IFCoLog Journal of Logics and their Applications, vol. 10 (2017), pp. 33573406.Google Scholar
Kohlenbach, U., Proof-theoretic methods in nonlinear analysis, Proceedings of the International Congress of Mathematicians 2018 (Sirakov, B., Ney de Souza, P., and Viana, M., editors), vol. 2, World Scientific, Singapore, 2019, pp. 6182.Google Scholar
Kohlenbach, U. and Leuştean, L., On the computational content of convergence proofs via Banach limits. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 370 (2012), no. 1971, pp. 34493463.CrossRefGoogle ScholarPubMed
Krivine, J.-L., Langages à valeurs réelles et applications. Fundamenta Mathematicae, vol. 81 (1974), pp. 213253.CrossRefGoogle Scholar
Lacey, H. E., The Isometric Theory of Classical Banach Spaces, Die Grundlehren der mathematischen Wissenschaften, Band 208, Springer-Verlag, New York-Heidelberg, 1974.CrossRefGoogle Scholar
Lindenstrauss, J. and Pełczyński, A., Absolutely summing operators in ${\cal L}_p $ spaces and their applications . Studia Mathematica , vol. 29 (1968), pp. 275326.CrossRefGoogle Scholar
Lindenstrauss, J. and Tzafriri, L., Classical Banach Spaces, Lecture Notes in Mathematics, vol. 338, Springer-Verlag, Berlin-New York, 1973.Google Scholar
Pełczyński, A. and Rosenthal, H. P., Localization techniques in Lp spaces. Studia Mathematica, vol. 52 (1974/75), pp. 263289.Google Scholar
Spector, C., Provably recursive functionals of analysis: A consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics, Proceedings of Symposia in Pure Mathematics, vol. 5 (Dekker, J. C. E., editor), American Mathematical Society, Providence, RI, 1962, pp. 127.Google Scholar
Tzafriri, L., Remarks on contractive projections in Lp-spaces. Israel Journal of Mathematics, vol. 7 (1969), pp. 915.CrossRefGoogle Scholar

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