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Schwarz's Inequality

Schwarz's inequality for an inner product states that

|<u,v>|^2<=<u,u><v,v>,
(1)

with equality iff u and v are linearly dependent vectors, including the case where either is zero (Axler 2026, p. 218). For square integrable real functions psi_1 and psi_2 on [a,b], this gives

[int_a^bpsi_1(x)psi_2(x)dx]^2<=int_a^b[psi_1(x)]^2dxint_a^b[psi_2(x)]^2dx.
(2)

Here equality means linear dependence as elements of L2, identifying functions that agree almost everywhere. If neither is zero almost everywhere, equality is equivalent to psi_1=alphapsi_2 almost everywhere for a constant alpha. Schwarz's inequality is sometimes also called the Cauchy-Schwarz inequality (Gradshteyn and Ryzhik 2000, p. 1099) or Buniakowsky inequality (Hardy et al. 1952, p. 16).

To derive the integral inequality for complex functions, let f and g be square integrable on [a,b] and put psi=f+lambdag for a complex number lambda. All integrals below are over [a,b]; products are integrable since 2|fg|<=|f|^2+|g|^2. With z^_ denoting the complex conjugate of z,

intpsi^_psidx=intf^_fdx+lambdaintf^_gdx+lambda^_intg^_fdx+lambdalambda^_intg^_gdx>=0,
(3)

with equality iff psi=0 almost everywhere. Write A=int|f|^2dx, B=int|g|^2dx, and C=intfg^_dx. The expanded inequality is

A+lambdaC^_+lambda^_C+|lambda|^2B>=0.
(4)

If B=0, then g=0 almost everywhere and the result is immediate. Otherwise, choose

lambda=-C/B.
(5)

Substitution gives

0<=A-(|C|^2)/B,
(6)

and therefore

|int_a^bf(x)g^_(x)dx|^2<=int_a^b|f(x)|^2dxint_a^b|g(x)|^2dx.
(7)

The same argument, with an inner product linear in its first argument in place of the integral, proves the general form and its equality condition.

Bessel's inequality follows from Schwarz's inequality.

See also

Bessel's Inequality, Cauchy's Inequality, Hölder's Inequalities

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972.Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 527-529, 1985.Axler, S. Measure, Integration & Real Analysis. Author-hosted revision, June 12, 2026. https://measure.axler.net/MIRA.pdf.Buniakowsky, V. "Sur quelques inégalités concernant les intégrales ordinaires et les intégrales aux différences finies." Mémoires de l'Acad. de St. Pétersbourg (VII) 1, No. 9, p. 4, 1959.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1099, 2000.Hardy, G. H.; Littlewood, J. E.; and Pólya, G. "Further Remarks on Method: The Inequality of Schwarz." §6.5 in Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 132-134, 1952.Schwarz, H. A. "Über ein die Flächen kleinsten Flächeninhalts betreffendes Problem der Variationsrechnung." Acta Soc. Scient. Fen. 15, 315-362, 1885. Reprinted in Gesammelte Mathematische Abhandlungen, Vol. 1. New York: Chelsea, pp. 224-269, 1972.

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Schwarz's Inequality

Cite this as:

Weisstein, Eric W. "Schwarz's Inequality." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SchwarzsInequality.html

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