Some logical metatheorems with applications in functional analysis

Kohlenbach, Ulrich

doi:10.1090/S0002-9947-04-03515-9

Some logical metatheorems with applications in functional analysis
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Trans. Amer. Math. Soc. 357 (2005), 89-128 Request permission

Abstract:

In previous papers we have developed proof-theoretic techniques for extracting effective uniform bounds from large classes of ineffective existence proofs in functional analysis. Here ‘uniform’ means independence from parameters in compact spaces. A recent case study in fixed point theory systematically yielded uniformity even w.r.t. parameters in metrically bounded (but noncompact) subsets which had been known before only in special cases. In the present paper we prove general logical metatheorems which cover these applications to fixed point theory as special cases but are not restricted to this area at all. Our theorems guarantee under general logical conditions such strong uniform versions of non-uniform existence statements. Moreover, they provide algorithms for actually extracting effective uniform bounds and transforming the original proof into one for the stronger uniformity result. Our metatheorems deal with general classes of spaces like metric spaces, hyperbolic spaces, CAT(0)-spaces, normed linear spaces, uniformly convex spaces, as well as inner product spaces.

References

Michael J. Beeson, Foundations of constructive mathematics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 6, Springer-Verlag, Berlin, 1985. Metamathematical studies. MR 786465, DOI 10.1007/978-3-642-68952-9
Gianluigi Bellin, Ramsey interpreted: a parametric version of Ramsey’s theorem, Logic and computation (Pittsburgh, PA, 1987) Contemp. Math., vol. 106, Amer. Math. Soc., Providence, RI, 1990, pp. 17–37. MR 1057813, DOI 10.1090/conm/106/1057813
Marc Bezem, Strongly majorizable functionals of finite type: a model for bar recursion containing discontinuous functionals, J. Symbolic Logic 50 (1985), no. 3, 652–660. MR 805674, DOI 10.2307/2274319
Jonathan Borwein, Simeon Reich, and Itai Shafrir, Krasnosel′ski-Mann iterations in normed spaces, Canad. Math. Bull. 35 (1992), no. 1, 21–28. MR 1157459, DOI 10.4153/CMB-1992-003-0
Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
F. E. Browder and W. V. Petryshyn, The solution by iteration of linear functional equations in Banach spaces, Bull. Amer. Math. Soc. 72 (1966), 566–570. MR 190744, DOI 10.1090/S0002-9904-1966-11543-4
F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5–251 (French). MR 327923
Delzell, C., Kreisel’s unwinding of Artin’s proof-Part I. In: Odifreddi, P., Kreiseliana, 113-246, A K Peters, Wellesley, MA (1996).
Michael Edelstein and Richard C. O’Brien, Nonexpansive mappings, asymptotic regularity and successive approximations, J. London Math. Soc. (2) 17 (1978), no. 3, 547–554. MR 500642, DOI 10.1112/jlms/s2-17.3.547
Wolfgang Friedrich, Spielquantorinterpretation unstetiger Funktionale der höheren Analysis, Arch. Math. Logik Grundlag. 24 (1984), no. 1-2, 73–99 (German). MR 755585, DOI 10.1007/BF02007142
Wolfgang Friedrich, Gödelsche Funktionalinterpretation für eine Erweiterung der klassischen Analysis, Z. Math. Logik Grundlag. Math. 31 (1985), no. 1, 3–29 (German). MR 781525, DOI 10.1002/malq.19850310102
Kazimierz Goebel and W. A. Kirk, Iteration processes for nonexpansive mappings, Topological methods in nonlinear functional analysis (Toronto, Ont., 1982) Contemp. Math., vol. 21, Amer. Math. Soc., Providence, RI, 1983, pp. 115–123. MR 729507, DOI 10.1090/conm/021/729507
Kazimierz Goebel and Simeon Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Monographs and Textbooks in Pure and Applied Mathematics, vol. 83, Marcel Dekker, Inc., New York, 1984. MR 744194
Kurt Gödel, Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes, Dialectica 12 (1958), 280–287 (German, with English summary). MR 102482, DOI 10.1111/j.1746-8361.1958.tb01464.x
Kurt Gödel, Collected works. Vol. II, The Clarendon Press, Oxford University Press, New York, 1990. Publications 1938–1974; Edited and with a preface by Solomon Feferman. MR 1032517
C. W. Groetsch, A note on segmenting Mann iterates, J. Math. Anal. Appl. 40 (1972), 369–372. MR 341204, DOI 10.1016/0022-247X(72)90056-X
Henson, C.W., Iovino, J., Ultraproducts in analysis. In: Analysis and Logic. London Mathematical Society LNS 262, pp. 1-112. Cambridge University Press (2002).
W. A. Howard, Appendix: Hereditarily majorizable function of finite type, Metamathematical investigation of intuitionistic arithmetic and analysis, Lecture Notes in Math., Vol. 344, Springer, Berlin, 1973, pp. 454–461. MR 0469712
W. A. Howard and G. Kreisel, Transfinite induction and bar induction of types zero and one, and the role of continuity in intuitionistic analysis, J. Symbolic Logic 31 (1966), 325–358. MR 209123, DOI 10.2307/2270450
Shiro Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc. 59 (1976), no. 1, 65–71. MR 412909, DOI 10.1090/S0002-9939-1976-0412909-X
Shigeru Itoh, Some fixed-point theorems in metric spaces, Fund. Math. 102 (1979), no. 2, 109–117. MR 525934, DOI 10.4064/fm-102-2-109-117
W. A. Kirk, Krasnosel′skiĭ’s iteration process in hyperbolic space, Numer. Funct. Anal. Optim. 4 (1981/82), no. 4, 371–381. MR 673318, DOI 10.1080/01630568208816123
W. A. Kirk, Nonexpansive mappings and asymptotic regularity, Nonlinear Anal. 40 (2000), no. 1-8, Ser. A: Theory Methods, 323–332. Lakshmikantham’s legacy: a tribute on his 75th birthday. MR 1768416, DOI 10.1016/S0362-546X(00)85019-1
Kirk, W.A., Geodesic geometry and fixed point theory. Preprint 23pp. (2003).
W. A. Kirk and Carlos Martínez-Yañez, Approximate fixed points for nonexpansive mappings in uniformly convex spaces, Ann. Polon. Math. 51 (1990), 189–193. MR 1093989, DOI 10.4064/ap-51-1-189-193
Kohlenbach, U., Theorie der majorisierbaren und stetigen Funktionale und ihre Anwendung bei der Extraktion von Schranken aus inkonstruktiven Beweisen: Effektive Eindeutigkeitsmodule bei besten Approximationen aus ineffektiven Beweisen. Ph.D. Thesis, Frankfurt am Main, xxii+278pp. (1990).
Ulrich Kohlenbach, Pointwise hereditary majorization and some applications, Arch. Math. Logic 31 (1992), no. 4, 227–241. MR 1155034, DOI 10.1007/BF01794980
Ulrich Kohlenbach, Effective moduli from ineffective uniqueness proofs. An unwinding of de la Vallée Poussin’s proof for Chebycheff approximation, Ann. Pure Appl. Logic 64 (1993), no. 1, 27–94. MR 1241250, DOI 10.1016/0168-0072(93)90213-W
Ulrich Kohlenbach, New effective moduli of uniqueness and uniform a priori estimates for constants of strong unicity by logical analysis of known proofs in best approximation theory, Numer. Funct. Anal. Optim. 14 (1993), no. 5-6, 581–606. MR 1248130, DOI 10.1080/01630569308816541
Ulrich Kohlenbach, Analysing proofs in analysis, Logic: from foundations to applications (Staffordshire, 1993) Oxford Sci. Publ., Oxford Univ. Press, New York, 1996, pp. 225–260. MR 1428007
Ulrich Kohlenbach, Mathematically strong subsystems of analysis with low rate of growth of provably recursive functionals, Arch. Math. Logic 36 (1996), no. 1, 31–71. MR 1462200, DOI 10.1007/s001530050055
Ulrich Kohlenbach, Arithmetizing proofs in analysis, Logic Colloquium ’96 (San Sebastián), Lecture Notes Logic, vol. 12, Springer, Berlin, 1998, pp. 115–158. MR 1674949, DOI 10.1007/978-3-662-22110-5_{5}
Ulrich Kohlenbach, A note on Spector’s quantifier-free rule of extensionality, Arch. Math. Logic 40 (2001), no. 2, 89–92. MR 1816479, DOI 10.1007/s001530000048
Ulrich Kohlenbach, On the computational content of the Krasnoselski and Ishikawa fixed point theorems, Computability and complexity in analysis (Swansea, 2000) Lecture Notes in Comput. Sci., vol. 2064, Springer, Berlin, 2001, pp. 119–145. MR 1893075, DOI 10.1007/3-540-45335-0_{9}
Ulrich Kohlenbach, A quantitative version of a theorem due to Borwein-Reich-Shafrir, Numer. Funct. Anal. Optim. 22 (2001), no. 5-6, 641–656. MR 1849571, DOI 10.1081/NFA-100105311
Kohlenbach, U., Uniform asymptotic regularity for Mann iterates. J. Math. Anal. Appl. 279, pp. 531-544 (2003).
Kohlenbach, U., Leu¸stean, L., Mann iterates of directionally nonexpansive mappings in hyperbolic spaces. Abstract and Applied Analysis, vol. 2003, issue 8, pp. 449-477 (2003).
Kohlenbach, U., Oliva, P., Effective bounds on strong unicity in $L_1$-approximation. Ann. Pure Appl. Logic 121, pp. 1-38 (2003).
Kohlenbach, U., Oliva, P., Proof mining: a systematic way of analysing proofs in mathematics. Proc. Steklov Inst. Math. 242, pp. 136-164 (2003).
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
G. Kreisel, Finiteness theorems in arithmetic: an application of Herbrand’s theorem for $\Sigma _{2}$-formulas, Proceedings of the Herbrand symposium (Marseilles, 1981) Stud. Logic Found. Math., vol. 107, North-Holland, Amsterdam, 1982, pp. 39–55. MR 757021, DOI 10.1016/S0049-237X(08)71876-7
G. Kreisel, Proof theory and the synthesis of programs: potential and limitations, EUROCAL ’85, Vol. 1 (Linz, 1985) Lecture Notes in Comput. Sci., vol. 203, Springer, Berlin, 1985, pp. 136–150. MR 826549, DOI 10.1007/3-540-15983-5_{1}2
G. Kreisel and A. Macintyre, Constructive logic versus algebraization. I, The L. E. J. Brouwer Centenary Symposium (Noordwijkerhout, 1981) Stud. Logic Found. Math., vol. 110, North-Holland, Amsterdam, 1982, pp. 217–260. MR 717246, DOI 10.1016/S0049-237X(09)70130-2
Horst Luckhardt, Extensional Gödel functional interpretation. A consistency proof of classical analysis, Lecture Notes in Mathematics, Vol. 306, Springer-Verlag, Berlin-New York, 1973. MR 0337512
H. Luckhardt, Herbrand-Analysen zweier Beweise des Satzes von Roth: Polynomiale Anzahlschranken, J. Symbolic Logic 54 (1989), no. 1, 234–263 (German, with English summary). MR 987335, DOI 10.2307/2275028
Luckhardt, H., Bounds extracted by Kreisel from ineffective proofs. In: Odifreddi, P., Kreiseliana, 289–300, A K Peters, Wellesley, MA (1996).
Hilton Vieira Machado, A characterization of convex subsets of normed spaces, K\B{o}dai Math. Sem. Rep. 25 (1973), 307–320. MR 326359
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Simeon Reich and Itai Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal. 15 (1990), no. 6, 537–558. MR 1072312, DOI 10.1016/0362-546X(90)90058-O
Simeon Reich and Alexander J. Zaslavski, Generic aspects of metric fixed point theory, Handbook of metric fixed point theory, Kluwer Acad. Publ., Dordrecht, 2001, pp. 557–575. MR 1904287, DOI 10.1007/978-94-017-1748-9_{1}6
Stephen G. Simpson, Subsystems of second order arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1999. MR 1723993, DOI 10.1007/978-3-642-59971-2
Clifford Spector, Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics, Proc. Sympos. Pure Math., Vol. V, American Mathematical Society, Providence, R.I., 1962, pp. 1–27. MR 0154801
Wataru Takahashi, A convexity in metric space and nonexpansive mappings. I, K\B{o}dai Math. Sem. Rep. 22 (1970), 142–149. MR 267565
A. S. Troelstra (ed.), Metamathematical investigation of intuitionistic arithmetic and analysis, Lecture Notes in Mathematics, Vol. 344, Springer-Verlag, Berlin-New York, 1973. MR 0325352
Joachim Weidmann, Linear operators in Hilbert spaces, Graduate Texts in Mathematics, vol. 68, Springer-Verlag, New York-Berlin, 1980. Translated from the German by Joseph Szücs. MR 566954