HEXAGONbeta

Improved Cauchy-Schwarz-Kaplansky Inequality

Primarily human-written textHuman understanding: all partsmath.OA — Operator Algebrasmath.FA — Functional Analysis

Contributed by K. MAHESH KRISHNA ↗

Version 1 / Oct 27, 2026 / CC BY 4.0

Abstract

We make an improvement and give a new proof for Cauchy-Schwarz-Kaplansky inequality derived by Kaplansky [\textit{Am. J. Math., 1953}].

References

  1. E. F. Beckenbach and R. Bellman. Inequalities , volume 30 of Ergeb. Math. Grenzgeb. Springer-Verlag, Berlin, 1983.
  2. Arthur Engel. Problem-solving strategies . Probl. Books Math. New York, NY: Springer, 1998.
  3. M. Fernandez-Guasti. Lagrange's identity obtained from product identity. Int. Math. Forum , 7(49-52):2555--2559, 2012.
  4. Marian Gidea and Constantin P. Niculescu. A brief account on Lagrange 's algebraic identity. Math. Intell. , 34(3):55--61, 2012.
  5. G. H. Hardy, J. E. Littlewood, and G. P ó lya. Inequalities . Camb. Math. Libr. Cambridge University Press, 2 edition, 1988.
  6. Irving Kaplansky. Modules over operator algebras. Am. J. Math. , 75:839--858, 1953.
  7. Lech Maligranda. Why H ö lder 's inequality should be called Rogers ' inequality. Math. Inequal. Appl. , 1(1):69--83, 1998.
  8. D. S. Mitrinovi ć . Analytic inequalities , volume 165 of Grundlehren Math. Wiss. Berlin: Springer-Verlag, 1970.
  9. D. S. Mitrinovi ć , J. E. Pe č ari ć , and A. M. Fink. Classical and new inequalities in analysis , volume 61 of Math. Appl., East Eur. Ser. Dordrecht: Kluwer Academic Publishers, 1993.
  10. William L. Paschke. Inner product modules over B ∗^* -algebras. Trans. Am. Math. Soc. , 182:443--468, 1973.
Show all 14 referencesShow fewer references
  1. Marc A. Rieffel. Induced representations of C ∗^* -algebras. Adv. Math. , 13:176--257, 1974.
  2. J. Michael Steele. The Cauchy - Schwarz master class. An introduction to the art of mathematical inequalities . Cambridge University Press, 2004.
  3. B. J. Venkatachala. Inequalities. An approach through problems , volume 49 of Texts Read. Math. New Delhi: Hindustan Book Agency, 2009.
  4. Tommy Wright. From Cauchy - Schwarz to the House of Representatives : applications of Lagrange 's identity. Math. Mag. , 94(4):244--256, 2021.

Version history

  1. v1Submitted by K. MAHESH KRISHNAInitial depositCurrentOct 27, 2026