Skip to main content
Log in

Complementary inequalities III: Inequalities complementary to Schwarz's inequality in Hilbert space

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Bibliography

  1. Beckenbach, E. F., andRichard Bellman: Inequalities. Berlin-Göttingen-Heidelberg: Springer-Verlag 1961.

    Google Scholar 

  2. Berberian, S. K.: Introduction to Hilbert space. New York: Oxford University Press 1961.

    Google Scholar 

  3. Cauchy, A. L.: Cours d'analyse de l'Ecole Royale Polytechnique, Ire partie, analyse algébrique. Paris, 1821 (Oeuvres complètes, IIe série, III).

  4. Cordes, H. O.: A matrix inequality. Proc. Am. Math. Soc.11, 206–210 (1960).

    Google Scholar 

  5. Diaz, J. B., andF. T. Metcalf: Complementary inequalities I: Inequalities complementary to Cauchy's inequality for sums of real numbers. J. Math. Anal. and Appls.9, 59–74 (1964).

    Google Scholar 

  6. —— —— Complementary inequalities II: Inequalities complementary to the Buniakowsky-Schwarz inequality for integrals. J. Math. Anal. and Appls.9, 278–293 (1964).

    Google Scholar 

  7. —— —— Stronger forms of a class of inequalities ofG. Pólya-G. Szegö, andL. V. Kantorovich. Bull. Am. Math. Soc.69, 415–418 (1963).

    Google Scholar 

  8. Dixmier, J.: Sur une inéqualité deE. Heinz. Math. Ann.126, 75–78 (1953).

    Google Scholar 

  9. Greub, W., andW. Rheinboldt: On a generalization of an inequality ofL. V. Kantorovich. Proc. Am. Math. Soc.10, 407–415 (1959).

    Google Scholar 

  10. Hardy, G. H., J. E. Littlewood, andG. Pólya: Inequalities. Cambridge 1952.

  11. Heinz, E.: On an inequality for linear operators in a Hilbert space. Report of an International Conference on Operator Theory and Group Representations, Arden House, Harriman, N.Y., 1955, 27–29. See also Math. Rev.18, 35 (1957).

  12. —— Beiträge zur Störungstheorie der Spektralzerlegung. Math. Ann.123, 415–438 (1951).

    Google Scholar 

  13. Kantorovich, L. V.: Functional analysis and applied mathematics. Uspehi Mat. Nauk3, 89–185 (1948); in particular pp. 142–144 (also translated from the Russian byC. D. Benster, Nat. Bur. Standards Report No. 1509, March 7, 1952; in particular pp. 106–109).

    Google Scholar 

  14. Kato, Tosio: Notes on some inequalities for linear operators. Math. Ann.125, 208–212 (1952).

    Google Scholar 

  15. —— A generalization of the Heinz inequality. Proc. Japan Acad.37, 305–308 (1961).

    Google Scholar 

  16. Krasnoselskii, M. A., andS. G. Krein: An iteration process with minimal residuals (in Russian), Mat. Sb., N.S.31 (73), 315–334 (1952).

    Google Scholar 

  17. Petryshyn, W. V.: Direct and iterative methods for the solution of linear operator equations in Hilbert space. Trans. Am. Math. Soc.105, 136–175 (1962).

    Google Scholar 

  18. Riesz, F., andB. Sz.-Nagy: Functional analysis. New York: Ungar 1955.

    Google Scholar 

  19. —— Über die linearen Transformationen des komplexen Hilbertschen Raumes. Acta Litterarum Ac Scientiarum Szeged5, 23–54 (1930).

    Google Scholar 

  20. Strang, W. G.: On the Kantorovich inequality. Proc. Am. Math. Soc.11, 468 (1960).

    Google Scholar 

  21. Taylor, A. E.: Introduction to Functional Analysis. New York 1958.

Download references

Author information

Authors and Affiliations

Authors

Additional information

Dedicated to ProfessorGottfried Köthe on the occasion of his sixtieth birthday, December 25, 1965

Presented to the American Mathematical Society at the Seventieth Annual Meeting, Miami, Florida, January 23–27, 1964 (See abstract 608–121, on page 92 of Notices of the American Mathematical Society, volume 11, No. 1, part 1, January 1964).

The research of the authors was supported by the Air Force Office of Scientific Research — Grant AFOSR 400–63 to the University of Maryland; and also by the U. S. Naval Ordnance Laboratory, White Oak, Maryland.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Diaz, J.B., Metcalf, F.T. Complementary inequalities III: Inequalities complementary to Schwarz's inequality in Hilbert space. Math. Ann. 162, 120–139 (1965). https://doi.org/10.1007/BF01361939

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01361939

Keywords

Navigation