Abstract
An elementary algorithm for computing the infimum of two projections in a Hilbert space is examined constructively. It is shown that in order to obtain a constructive convergence proof for the algorithm, one must add some hypotheses such as Markov’s principle or the locatedness of a certain range; and that in the finite-dimensional case, the existence of both the infimum and the supremum of the two projections suffices for the convergence of the algorithm.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aczel, P., Rathjen, M.: Notes on Constructive Set Theory, Report No. 40, Institut Mittag-Leffler, Royal Swedish Academy of Sciences (2001)
Aczel, P., Rathjen, M.: Constructive Set Theory (in preparation)
Bishop, E.A., Bridges, D.S.: Constructive Analysis. Grundlehren der Math. Wissenschaften, vol. 279. Springer, Heidelberg (1985)
Bridges, D.S., Ishihara, H.: Locating the range of an operator on a Hilbert space. Bull. London Math. Soc. 24, 599–605 (1992)
Bridges, D.S., Richman, F.: Varieties of Constructive Mathematics. London Math. Soc., Lecture Notes in Mathematics, vol. 97. Cambridge Univ. Press (1987)
Bridges, D.S., Vîţă, L.S.: Techniques of Constructive Analysis. Universitext. Springer, Heidelberg (2006)
Bridges, D.S., Julian, W.H., Mines, R.: A constructive treatment of open and unopen mapping theorems. Zeit. Math. Logik Grundlagen Math. 35, 29–43 (1989)
(Vîţă) Dediu, L.S.: The Constructive Theory of Operator Algebras, Ph.D. thesis, University of Canterbury, New Zealand (2000)
Halmos, P.R.: A Hilbert Space Problem Book. Grad. Texts in Math., vol. 19. Springer, Heidelberg (1974)
Ishihara, H.: Locating subsets of a Hilbert space. Proc. Amer. Math. Soc. 129(5), 1385–1390 (2001)
Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras, vol. I. Academic Press, New York (1983)
Martin-Löf, P.: An Intuitionistic Theory of Types: Predicative Part. In: Rose, H.E., Shepherdson, J.C. (eds.) Logic Colloquium 1973, pp. 73–118. North-Holland, Amsterdam (1975)
Myhill, J.: Constructive set theory. J. Symbolic Logic 40(3), 347–382 (1975)
Richman, F.: Adjoints and the image of the unit ball. Proc. Amer. Math. Soc. 129(4), 1189–1193 (2001)
Specker, E.: Nicht konstruktiv beweisbare Sätze der Analysis. J. Symbolic Logic 14, 145–158 (1949)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Bridges, D.S., Vîţă, L.S. (2012). Constructing the Infimum of Two Projections. In: Dinneen, M.J., Khoussainov, B., Nies, A. (eds) Computation, Physics and Beyond. WTCS 2012. Lecture Notes in Computer Science, vol 7160. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27654-5_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-27654-5_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-27653-8
Online ISBN: 978-3-642-27654-5
eBook Packages: Computer ScienceComputer Science (R0)