Skip to main content Accessibility help
×
Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-24T22:52:24.379Z Has data issue: false hasContentIssue false

Computable versions of the uniform boundedness theorem

Published online by Cambridge University Press:  31 March 2017

Zoé Chatzidakis
Affiliation:
Université de Paris VII (Denis Diderot)
Peter Koepke
Affiliation:
Rheinische Friedrich-Wilhelms-Universität Bonn
Wolfram Pohlers
Affiliation:
Westfälische Wilhelms-Universität Münster, Germany
Get access
Type
Chapter
Information
Logic Colloquium '02 , pp. 130 - 151
Publisher: Cambridge University Press
Print publication year: 2006

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

[1] Stefan, Banach, Sur les opérations dans les ensembles abstraits et leur application aux equations integrales Fundamenta Mathematicae, vol. 3 (1922), pp. 133–181.
[2] Stefan, Banach and Stanisław, Mazur, Sur les fonctions calculables Annales de la Société Polonaise de Mathématique, vol. 16 (1937), p. 223.
[3] Stefan, Banach and Hugo, Steinhaus, Sur le principe de la condensation des singularités Fundamenta Mathematicae, vol. 9 (1927), pp. 50–61.
[4] Errett, Bishop and Douglas, Bridges, Constructive Analysis, Grundlehren der Mathematischen Wissenschaften, vol. 279, Springer-Verlag, Berlin, 1985.
[5] Vasco, Brattka, Computability and Complexity in Analysis, Informatik Berichte, vol. 286, Fern Universität Hagen, Fachbereich Informatik, Hagen, 2001.
[6] Vasco, Brattka, Computable versions of Baire's category theorem Mathematical Foundations of Computer Science, 2001 (Mariánské Lázně) (Jiří Sgall, Aleš Pultr, and Petr Kolman, editors), Lecture Notes in Computer Science, vol. 2136, Springer, Berlin, 2001, pp. 224–235.
[7] Vasco, Brattka, Computing uniform bounds CCA 2002 Computability and Complexity in Analysis (Vasco Brattka, Matthias Schröder, and Klaus, Weihrauch, editors), Electronic Notes in Theoretical Computer Science, vol. 66, Elsevier, Amsterdam, 2002, 5th InternationalWorkshop, CCA 2002, Málaga, Spain, July 12–13, 2002.
[8] Vasco, Brattka, Computability over topological structures Computability and Models (S. Barry, Cooper and Sergey S., Goncharov, editors), Kluwer/Plenum, New York, 2003, pp. 93–136.
[9] Vasco, Brattka, Computability on non-separable Banach spaces and Landau's theorem From Sets and Types to Topology and Analysis: Towards Practicable Foundations for Constructive Mathematics (Laura Crosilla and Peter, Schuster, editors), Oxford University Press, 2005, pp. 316–333.
[10] Vasco, Brattka and Gero, Presser, Computability on subsets of metric spaces Theoretical Computer Science, vol. 305 (2003), no. 1-3, pp. 43–76.
[11] Vasco, Brattka and Matthias, Schröder, Computing with sequences, weak topologies and the axiom of choice Computer Science Logic (Luke Ong, editor), Lecture Notes in Computer Science, vol. 3634, Springer, 2005, pp. 462–476.
[12] Vasco, Brattka and Klaus, Weihrauch, Computability on subsets of Euclidean space. I. Closed and compact subsets Theoretical Computer Science, vol. 219 (1999), no. 1-2, pp. 65–93.
[13] Douglas K., Brown and Stephen G., Simpson, The Baire category theorem in weak subsystems of second-order arithmetic The Journal of Symbolic Logic, vol. 58 (1993), no. 2, pp. 557–578.
[14] Casper, Goffman and George, Pedrick, First Course in Functional Analysis, Prentice-Hall Inc., Englewood Cliffs, N.J., 1965.
[15] Andrzej, Grzegorczyk, Onthe definitions of computable real continuous functions Polska Akademia Nauk. Fundamenta Mathematicae, vol. 44 (1957), pp. 61–71.
[16] Hajime, Ishihara, Sequential continuity of linear mappings in constructive mathematics The Journal of Universal Computer Science, vol. 3 (1997), no. 11, pp. 1250–1254.
[17] Hajime, Ishihara, Sequentially continuity in constructive mathematics Combinatorics, computability and logic (Constant,a, 2001) (C. S. Calude,M. J. Dinneen, and S. Sburlan, editors), Springer Ser. Discrete Math. Theor. Comput. Sci., Springer, London, 2001, pp. 5–12.
[18] Ker-IKo, Complexity Theory of Real Functions, Progress in Theoretical Computer Science, Birkhäuser, Boston, MA, 1991.
[19] Boris, Abramovich Kušner, Lectures on Constructive Mathematical Analysis, Translations of Mathematical Monographs, vol. 60, American Mathematical Society, Providence, RI, 1984.Google Scholar
[20] Daniel, Lacombe, Extension de la notion de fonction récursive aux fonctions d'une ou plusieurs variables réelles. I, II, III Comptes Rendus Mathematique. Académie des Sciences. Paris, vol. 240, 241 (1955), pp. 2478–2480, 13–14, 151–153.
[21] Daniel, Lacombe, Quelques procédés de definition en topologie recursive Constructivity in Mathematics: Proceedings of the Colloquium Held at Amsterdam, 1957 (A. Heyting, editor), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1959, pp. 129–158.
[22] George, Metakides, Anil, Nerode, and R. A., Shore, Recursive limits on the Hahn-Banach theorem Errett Bishop: Reflections on Him and His Research (San Diego, Calif., 1983) (Murray Rosenblatt, editor), Contemporary Mathematics, vol. 39, AMS, Providence, RI, 1985, pp. 85–91.Google Scholar
[23] Yiannis, Nicholas Moschovakis, Recursive metric spaces Polska Akademia Nauk. Fundamenta Mathematicae, vol. 55 (1964), pp. 215–238.
[24] Marian B., Pour-El and J., Ian Richards, Computability in Analysis and Physics, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1989.
[25] Matthias, Schröder, Extended admissibility Theoretical Computer Science, vol. 284 (2002), no. 2, pp. 519–538.Google Scholar
[26] Stephen G., Simpson, Subsystems of Second Order Arithmetic, Perspectives inMathematical Logic, Springer-Verlag, Berlin, 1999.
[27] Dieter, Spreen, On effective topological spaces The Journal of Symbolic Logic, vol. 63 (1998), no. 1, pp. 185–221.
[28] A. S., Troelstra, Comparing the theory of representations and constructive mathematics Computer Science Logic (Berne, 1991) (E. Börger et al., editors), Lecture Notes in Computer Science, vol. 626, Springer, Berlin, 1992, pp. 382–395.
[29] Alan M., Turing, On computable numbers, with an application to the “Entscheidungsproblem” Proceedings of the London Mathematical Society, Vol. 42, no. 2, 1936, pp. 230–265.
[30] Klaus, Weihrauch, Computability on computable metric spaces Theoretical Computer Science, vol. 113 (1993), no. 2, pp. 191–210.
[31] Klaus, Weihrauch, Computable Analysis, Springer-Verlag, Berlin, 2000.
[32] Klaus, Weihrauch and Ning, Zhong, Is wave propagation computable or can wave computers beat the Turing machine? Proceedings of the London Mathematical Society. Third Series, vol. 85 (2002), no. 2, pp. 312–332.
[33] Mariko, Yasugi, Takakazu, Mori, and Yoshiki, Tsujii, Effective properties of sets and functions in metric spaces with computability structure Theoretical Computer Science, vol. 219 (1999), no. 1-2, pp. 467–486.
[34] Ning, Zhong, Computability structure of the Sobolev spaces and its applications Theoretical Computer Science, vol. 219 (1999), no. 1-2, pp. 487–510.

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×