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Difference between revisions of "Bunyakovskii inequality"

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An equality in mathematical analysis, established by V.Ya. Bunyakovskii [[#References|[1]]] for square-integrable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017770/b0177701.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017770/b0177702.png" />:
 
An equality in mathematical analysis, established by V.Ya. Bunyakovskii [[#References|[1]]] for square-integrable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017770/b0177701.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017770/b0177702.png" />:
  

Revision as of 07:57, 14 February 2013


An equality in mathematical analysis, established by V.Ya. Bunyakovskii [1] for square-integrable functions and :

This inequality is analogous to Cauchy's algebraic inequality

The Bunyakovskii inequality is also known as the Schwarz inequality; however, Bunyakovskii published his study as early as 1859, whereas in H.A. Schwarz' work this inequality appeared as late as 1884 (without any reference to the work of Bunyakovskii).

References

[1] W. [V.Ya. Bunyakovskii] Bounjakowsky, "Sur quelques inegalités concernant les intégrales aux différences finis" Mem. Acad. Sci. St. Petersbourg (7) , 1 (1859) pp. 9


Comments

In Western literature this inequality is often called the Cauchy inequality, or the Cauchy–Schwarz inequality. Its generalization to a function in and a function in , , is called the Hölder inequality.

Cauchy's algebraic inequality stated above holds for real numbers , . For complex numbers , , it reads

It has a generalization analogous to the Hölder inequality.

References

[a1] W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953)
How to Cite This Entry:
Bunyakovskii inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bunyakovskii_inequality&oldid=29421
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article